Global and local regularity of Fourier integral operators on weighted and unweighted spaces
aa r X i v : . [ m a t h . A P ] M a y GLOBAL AND LOCAL REGULARITY OF FOURIERINTEGRAL OPERATORS ON WEIGHTED ANDUNWEIGHTED SPACES
DAVID DOS SANTOS FERREIRA AND WOLFGANG STAUBACH
Abstract.
We investigate the global continuity on L p spaces with p ∈ [1 , ∞ ] of Fourier integral operators with smooth and rough amplitudesand/or phase functions subject to certain non-degeneracy conditions.We initiate the investigation of the continuity of smooth and roughFourier integral operators on weighted L p spaces, L pw with 1 < p < ∞ and w ∈ A p , (i.e. the Muckenhoput weights), and establish weightednorm inequalities for operators with rough and smooth amplitudes andphase functions satisfying a suitable rank condition. These results arethen applied to prove weighted and unweighted estimates for the com-mutators of Fourier integral operators with functions of bounded meanoscillation BMO, then to some estimates on weighted Triebel-Lizorkinspaces, and finally to global unweighted and local weighted estimatesfor the solutions of the Cauchy problem for m -th and second order hy-perbolic partial differential equations on R n . Contents
Introduction and main results 2Chapter 1. Prolegomena 10Chapter 2. Global boundedness of Fourier integral operators 27Chapter 3. Global and local weighted L p boundedness of Fourierintegral operators 51Chapter 4. Applications in harmonic analysis and partial differentialequations 68References 76 Mathematics Subject Classification.
Introduction and main results
A Fourier integral operator is an operator that can be written locally inthe form T a,ϕ u ( x ) = (2 π ) − n Z R n e iϕ ( x,ξ ) a ( x, ξ ) b u ( ξ ) d ξ, (0.0.1)where a ( x, ξ ) is the amplitude and ϕ ( x, ξ ) is the phase function . Histori-cally, a systematic study of smooth Fourier integral operators with ampli-tudes in C ∞ ( R n × R n ) with | ∂ αξ ∂ βξ a ( x, ξ ) | ≤ C αβ (1 + | ξ | ) m − ̺ | α | + δ | β | (i.e. a ( x, ξ ) ∈ S m̺,δ ), and phase functions in C ∞ ( R n × R n \
0) homogenous ofdegree 1 in the frequency variable ξ and with non-vanishing determinant ofthe mixed Hessian matrix (i.e. non-degenerate phase functions ), was initi-ated in the classical paper of L. H¨ormander [20]. Furthermore, I. Eskin [11](in the case a ( x, ξ ) ∈ S , ) and H¨ormander [20] (in the case a ( x, ξ ) ∈ S ̺, − ̺ , < ̺ ≤
1) showed the local L boundedness of Fourier integral operatorswith non-degenerate phase functions. Later on, H¨ormander’s local L resultwas extended by R. Beals [4] and A. Greenleaf and G. Uhlmann [15] to thecase of amplitudes in S , . After the pioneering investigations by M. Beals [3], the optimal resultsconcerning local continuity properties of smooth Fourier integral operators(with non-degenerate and homogeneous phase functions) in L p for 1 ≤ p ≤∞ , were obtained in the seminal paper of A. Seeger, C. D. Sogge and E.M.Stein [33]. This also paved the way for further investigations by G. Mock-enhaupt, Seeger and Sogge in [26] and [27], see also [34] and [35]. In theseinvestigations the boundedness, from L p comp to L p loc and from L p comp to L q loc of smooth Fourier integral operators with non-degenerate phase functionshave been established, and furthermore it was shown that the maximal op-erators associated with certain Fourier integral operators (and in particularconstant and variable coefficient hypersurface averages) are L p bounded.In the context of H¨ormander type amplitudes and non-degenerate ho-mogeneous phase functions which are most frequently used in applicationsin partial differential equations, it has been comparatively small amountof activity concerning global L p boundedness of Fourier integral operators.Among these, we would like to mention the global L boundedness of Fourierintegral operators with homogeneous phases in C ∞ ( R n × R n \
0) and am-plitudes in the H¨ormander class S , , due to D. Fujiwara [12]; the global L boundedness of operators with inhomogeneous phases in C ∞ ( R n × R n )and amplitudes in S , , due to K. Asada and D. Fujiwara [2]; the global L p boundedness of operators with smooth amplitudes in the so called SG classes, due to E. Cordero, F. Nicola and L. Rodino in [8]; and finally, S.Coriasco and M. Ruzhansky’s global L p boundedness of Fourier integral op-erators [9], with smooth amplitudes in a suitable subclass of the H¨ormander EGULARITY OF FOURIER INTEGRAL OPERATORS 3 class S , , where certain decay of the amplitudes in the spatial variables areassumed. We should also mention that before the appearance of the pa-per [9], M. Ruzhansky and M. Sugimoto had already proved in [31] certainweighted L boundedness (with some power weights) as well as the globalunweighted L boundedness of Fourier integrals operators with phases in C ∞ ( R n × R n ) that are not necessarily homogeneous in the frequency vari-ables, and amplitudes that are in the class S , . In all the aforementionedresults, one has assumed ceratin bounds on the derivatives of the phase func-tions and also a stronger non-degeneracy condition than the one required inthe local L p estimates.In this paper, we shall take all these results as our point of departureand make a systematic study of the global L p boundedness of Fourier in-tegral operators with amplitudes in S m̺,δ with ̺ and δ in [0 , , which coverall the possible ranges of ̺ ’s and δ ’s. Furthermore we initiate the study ofweighted norm inequalities for Fourier integral operators with weights in the A p class of Muckenhoupt and use our global unweighted L p results to provea sharp weighted L p boundedness theorem for Fourier integral operators.The weighted results in turn will be used to establish the validity of certainvector-valued inequalities and more importantly to prove the weighted andunweighted boundedness of commutators of Fourier integral operators withfunctions of bounded mean oscillation BMO. Thus, all the results of thispaper are connected and each chapter uses the results of the previous ones.This has been reflected in the structure of the paper and the presentationof the results.Concerning the specific conditions that are put in this paper on the phasefunctions, it has been known at least since the appearance of the papers[12], [2], [31] and [9], that one has to assume stronger conditions, than merenon-degeneracy, on the phase function in order to obtain global L p bounded-ness results. In fact it turns out that the assumption on the phase function,referred to in this paper as the strong non-degeneracy condition , which re-quires a nonzero lower bound on the modulus of the determinant of themixed Hessian of the phase, is actually necessary for the validity of globalregularity of Fourier integral operators, see section 1.2.5. Furthermore, wealso introduce the class Φ k of homogeneous (of degree 1) phase functionswith a specific control over the derivatives of orders greater than or equalto k, and assume our phases to be strongly non-degenerate and belong toΦ k for some k . At first glance, these conditions might seem restrictive, butfortunately they are general enough to accommodate the phase functionsarising in the study of hyperbolic partial differential equations and will stillapply to the most generic phases in practical applications. DAVID DOS SANTOS FERREIRA AND WOLFGANG STAUBACH
Concerning our choice of amplitudes, there are some features that set ourinvestigations apart from those made previously, for example partly moti-vated by the investigation of C.E. Kenig and the second author of the presentpaper [23], of the L p boundedness of the so called pseudo-pseudodifferentialoperators , we consider the global and local L p boundedness of Fourier in-tegral operators when the amplitude a ( x, ξ ) belongs to the class L ∞ S m̺ ,wherein a ( x, ξ ) behaves in the spatial variable x like an L ∞ function, and inthe frequency variable ξ, the behaviour is that of a symbol in the H¨ormanderclass S m̺, . It is worth mentioning that the conditions defining classes Φ k , L ∞ S m̺ andthe assumption of strong non-degeneracy make the global results obtainedhere natural extensions of the local boundedness results of Seeger, Sogge andStein’s in [33]. Apart from the obvious local to global generalizations, thisis because on one hand, our methods can handle the singularity of the phasefunction in the frequency variables at the origin and therefore the usual as-sumption that ξ = 0 in the support of the amplitude becomes obsolete. Onthe other hand, we do not require any regularity (and therefore no decay ofderivatives) in the spatial variables of the amplitudes. Therefore, our am-plitudes are close to, and in fact are spatially non-smooth versions of thosein the Seeger-Sogge-Stein’s paper [33]. Indeed, in [33] the authors althoughdealing with spatially smooth amplitudes, assume neither any decay in thespatial variables nor the vanishing of the amplitude in a neighbourhood ofthe singularity of the phase function.There are several steps involved in the proof of the results of the pa-per and there are discussions about various conditions that we impose onthe operators as well as some motivations for the study of rough operators.Moreover, giving examples and counterexamples when necessary, we havestrived to give motivations for our assumptions in the statements of the the-orems. Here we will not mention all the results that have been proven inthis paper, instead we chose to highlight those that are the main outcomesof our investigations.In Chapter 1, we set up the notations and give the definitions of the classes ofamplitudes, phase functions and weights that will be used througt the paper.We also include the tools that we need in proving our global boundednessresults, in this chapter. We close the chapter with a discussion about theconnections between rough amplitudes and global boundedness of Fourierintegral operators.Chapter 2 is devoted to the investigation of the global boundedness ofFourier integral operators with smooth or rough phases, and smooth orrough amplitudes. To achieve our global boundedness results, we split theoperators in low and high frequency parts and show the boundedness of each EGULARITY OF FOURIER INTEGRAL OPERATORS 5 and one of them separately. In proving the L p boundedness of the low fre-quency portion, see Theorem 1.2.11, we utilise Lemma 1.2.10 which yieldsa favourable kernel estimate for the low frequency operator, thereafter weuse the phase reduction of Lemma 1.2.3 to bring the operator to a canonicalform, and finally we use the L p boundedness of the non-smooth substationin Corollary 1.2.9 to conclude the proof. Thus, we are able to establish theglobal L p boundedness of the frequency-localised portion of the operator,for all p ’s in [1 , ∞ ] simultaneously.The global boundedness of the high frequency portion of the Fourier in-tegral operator needs to be investigated in three steps. First we show the L − L boundedness then we proceed to the L − L boundedness and finallywe prove the L ∞ − L ∞ boundedness.In order to show the L boundedness of Theorem 2.1.1, we use a semi-classical reduction from Subsection 1.2.1 and Lemma 1.2.1, which will beused throughout the paper. Thereafter we use the semiclassical version ofthe Seeger-Sogge-Stein decomposition which was introduced in a microlocalform in [33].For our global L boundedness result, we also consider amplitudes with ̺ < δ which appear in the study of Fourier integral operators on nilpotentLie groups and also in scattering theory. In Theorem 2.2.6, we show aglobal L boundedness result for operators with smooth H¨ormander classamplitudes in S min(0 , n ( ̺ − δ )) ̺,δ , ̺ ∈ [0 , , δ ∈ [0 , . Also, in Theorem 2.2.7we prove the L boundedness of operators with amplitudes belonging to S m̺, , with m < n ( ̺ − h x, ξ i isin class Φ and satisfies the strong non-degeneracy condition and thereforeour L boundedness result completes the L boundedness theory of smoothFourier integral operators with homogeneous non-degenerate phases.Finally, in Theorem 2.3.1 we prove the sharp global L ∞ boundedness ofFourier integral operators, where in the proof we follow almost the same lineof argument as in the proof of the L boundedness case, but to obtain thesharp result which we desire, we make a more detailed analysis of the kernelestimates which bring us beyond the result implied by the mere utilisationof the Seeger-Sogge-Stein decomposition. Furthermore, in this case, no non-degeneracy assumption on the phase is required. Our results above aresummerised in the following global L p boundedness theorem, see Theorem2.4.1: DAVID DOS SANTOS FERREIRA AND WOLFGANG STAUBACH
A. Global L p boundedness of smooth Fourier integral operators. Let T be a Fourier integral operator given by 0.0.1 with amplitude a ∈ S m̺,δ and a strongly non-degenerate phase function ϕ ( x, ξ ) ∈ Φ . Setting λ :=min(0 , n ( ̺ − δ )) , suppose that either of the following conditions hold:(a) 1 ≤ p ≤ , ≤ ̺ ≤ , ≤ δ ≤ , and m < n ( ̺ − (cid:18) p − (cid:19) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) + λ (cid:18) − p (cid:19) ;or(b) 2 ≤ p ≤ ∞ , ≤ ̺ ≤ , ≤ δ ≤ , and m < n ( ̺ − (cid:18) − p (cid:19) + ( n − (cid:18) p − (cid:19) + λp ;or(c) p = 2 , ≤ ̺ ≤ , ≤ δ < , and m = λ . Then there exists a constant
C > k T u k L p ≤ C k u k L p . For Fourierintegral operators with rough amplitudes we show in Theorem 2.4.2 thefollowing:
B. Global L p boundedness of rough Fourier integral operators. Let T be a Fourier integral operator given by (0.0.1) with amplitude a ∈ L ∞ S m̺ , ≤ ̺ ≤ ϕ ( x, ξ ) ∈ Φ . Suppose that either of the following conditions hold:(a) 1 ≤ p ≤ m < np ( ̺ −
1) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) ;or(b) 2 ≤ p ≤ ∞ and m < n ̺ −
1) + ( n − (cid:18) p − (cid:19) . Then there exists a constant
C > k T u k L p ≤ C k u k L p . We alsoextend both of the results above, i.e. the L p − L p regularity of smooth andrough operators, to the L p − L q regularity, in Theorem 2.4.4.After proving the global regularity of Fourier integral operators with smoothphase functions, we turn to the problem of local and global boundedness ofoperators which are merely bounded in the spatial variables in both theirphases and amplitudes. A motivation for this investigation stems from thestudy of maximal estimates involving Fourier integral operators, where astandard stopping time argument (here referred to as linearisation) reducesthe problem to a Fourier integral operator with a non-smooth phase and EGULARITY OF FOURIER INTEGRAL OPERATORS 7 sometimes also a non-smooth amplitude. For instance, estimates for themaximal spherical average operator Au ( x ) = sup t ∈ [0 , (cid:12)(cid:12)(cid:12) Z S n − u ( x + tω ) d σ ( ω ) (cid:12)(cid:12)(cid:12) which is directly related to the rough Fourier integral operator T u ( x ) = (2 π ) − n Z R n a ( x, ξ ) e it ( x ) | ξ | + i h x,ξ i b u ( ξ ) d ξ where t ( x ) is a measurable function in x , with values in [0 ,
1] and a ( x, ξ ) ∈ L ∞ S − n − . Here, the phase function of the Fourier integral operator is ϕ ( x, ξ ) = it ( x ) | ξ | + i h x, ξ i which is merely an L ∞ function in the spatial variables x ,but is smooth outside the origin in the frequency variables ξ. As we shall seelater, according to Definition 1.1.5, this phase belongs to the class L ∞ Φ . In our investigation of local or global L p boundedness of the rough Fourierintegral operators above for p = 2, the results obtained are similar to thoseof the local results for amplitudes in the class S m , obtained in [26], [27] and[35] for (2.2.45). However, we consider more general classes of amplitudes(i.e. the S m̺,δ class) and also require only measurability and boundedness inthe spatial variables (i.e. the L ∞ S m̺ class). The main results in this contextare the L boundedness results which apart from the case of Fourier integraloperators in dimension n = 1 , yield a problem of considerable difficulty incase one desires to prove a L regularity result under the sole assumption ofrough non-degeneracy, see Definition 1.1.6.Using the geometric conditions (imposed on the phase functions) whichare the rough analogues of the non-degeneracy and corank conditions forsmooth phases (the rough corank condition 2.2.2), we are able to prove alocal L boundedness result with a certain loss of derivatives depending onthe rough corank of the phase. More explicitly we prove in Theorem 2.2.9: C. Local L boundedness of Fourier integral operators with roughamplitudes and phases. Let T be a Fourier integral operator given by(0.0.1) with amplitude a ∈ L ∞ S m̺ and phase function ϕ ∈ L ∞ Φ . Sup-pose that the phase satisfies the rough corank condition 2.2.2, then T canbe extended as a bounded operator from L to L provided m < − n + k − + ( n − k )( ̺ − . Despite the lack of sharpness in the above theorem, the proof is rather tech-nical. However, in case n = 1 this theorem can be improved to yield a local L boundedness result with m < − , and if the assumptions on the phasefunction are also strengthen with a Lipschitz condition on the ξ derivativesof order 2 and higher of the phase, then the above theorem holds with a loss m < − k + ( n − k )( ̺ − . In Chapter 3 we turn to the problem of weighted norm inequalities forFourier integral operators. To our knowledge this question has never been
DAVID DOS SANTOS FERREIRA AND WOLFGANG STAUBACH investigated previously in the context of Muckenhoupt’s A p weights whichare the most natural class of weights in harmonic analysis. Here we startthis investigation by establishing sharp boundedness results for a fairly wideclass of Fourier integral operators, somehow modeled on the parametrices ofhyperbolic partial differential equations. One notable feature of our investi-gation is that we also prove the results for Fourier integral operators whosephase functions and amplitudes are only bounded and measurable in thespatial variables and exhibit suitable symbol type behavior in the frequencyvariables.As before, we begin by discussing the weighted estimates for the low fre-quency portion of the Fourier integral operators which can be handled byLemma 1.2.10. As a matter of fact, the weighted L p boundedness of lowfrequency parts of Fourier integral operators is merely an analytic issue in-volving the right decay rates of the phase function and does not involve anyrank condition on the phase. The situation in the high frequency case isentirely different. Here, there is also a significant distinction between theweighted a and unweighted case, in the sense that, if one desires to provesharp weighted estimates, then a rank condition on the phase function isabsolutely crucial. This fact has been discussed in detail in Section 3.2,where one finds various examples, including one related to the wave equa-tion, and counterexamples which will lead us to the correct condition on thephase. Then we will proceed with the local high frequency and global highfrequency boundedness estimates. As a rule, in the investigation of bound-edness of Fourier integral operators, the local estimates require somewhatmilder conditions on the phase functions compared to the global estimatesand our case study of the weighted norm inequalities here is no exception tothis rule. Furthermore, we are able to formulate the local weighted bound-edness results in an invariant way involving the canonical relation of theFourier integral operator in question. Our main results in this context arecontained in Theorem 3.4.4: D. Weighted L p boundedness of Fourier integral operators. Let a ( x, ξ ) ∈ L ∞ S − n +12 ̺ + n ( ̺ − ̺ and ̺ ∈ [0 , . Suppose that either(a) a ( x, ξ ) is compactly supported in the x variable and the phase func-tion ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ ξ and satisfies, det ∂ xξ ϕ ( x, ξ ) = 0 as well as rank ∂ ξξ ϕ ( x, ξ ) = n − ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ , ϕ satisfies the rough non-degeneracy con-dition as well as | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > T a,ϕ is bounded on L pw for p ∈ (1 , ∞ ) and all w ∈ A p .Furthermore, for ̺ = 1 this result is sharp.Here, it is worth mentioning that in the non-endpoint case, i.e. if a ( x, ξ ) ∈ L ∞ S m̺ with m < − n +12 ̺ + n ( ̺ − , we can prove a result that requires no EGULARITY OF FOURIER INTEGRAL OPERATORS 9 non-degeneracy assumption on the phase function. The proof of these state-ments are long and technical and use several steps involving careful kernelestimates, uniform pointwise estimates on certain oscillatory integrals, un-weighted local and global L p boundedness, interpolation, and extrapolation.In Chapter 4 we are motivated by the fact that weighted norm inequali-ties with A p weights can be used as an efficient tool in proving vector valuedinequalities and also boundedness of commutators of operators with func-tions of bounded mean oscillation BMO. Therefore, we start the chapterby showing boundedness of certain Fourier integral operators in weightedTriebel-Lizorkin spaces (see (4.1.4)). This is based on a vector valued in-equality for Fourier integral operators.But more importantly we prove for the first time, in Theorems 4.2.4 and4.2.5, the boundedness and weighted boundedness of BMO commutators ofFourier integral operators, namely E. L p boundedness of BMO commutators of Fourier integral op-erators. Suppose either(a) T ∈ I m̺, comp ( R n × R n ; C ) with ≤ ̺ ≤ m < ( ̺ − n ) | p − | , satisfies all the conditions of Theorem 3.4.5 or;(b) T a,ϕ with a ∈ S m̺,δ , ≤ ̺ ≤
1, 0 ≤ δ ≤ , λ = min(0 , n ( ̺ − δ )) and ϕ ( x, ξ ) is a strongly non-degenerate phase function with ϕ ( x, ξ ) −h x, ξ i ∈ Φ , where in the range 1 < p ≤ ,m < n ( ̺ − (cid:18) p − (cid:19) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) + λ (cid:18) − p (cid:19) ;and in the range 2 ≤ p < ∞ m < n ( ̺ − (cid:18) − p (cid:19) + ( n − (cid:18) p − (cid:19) + λp ;or(c) T a,ϕ with a ∈ L ∞ S m̺ , ≤ ̺ ≤ ϕ is a strongly non-degeneratephase function with ϕ ( x, ξ ) − h x, ξ i ∈ Φ , where in the range 1
1) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) , and for the range 2 ≤ p < ∞ m < n ̺ −
1) + ( n − (cid:18) p − (cid:19) . Then for b ∈ BMO, the commutators [ b, T ] and [ b, T a,ϕ ] are bounded on L p with 1 < p < ∞ . Here we like to mention that once again, the global L p bounded in Theorem A above is used in the proof of the L p boundedness ofthe BMO commutators. Finally, the weighted norm inequalities with weightsin all A p classes have the advantage of implying weighted boundedness ofrepeated commutators, namely one has F. Weighted L p boundedness of k-th BMO commutators of Fourierintegral operators. Let a ( x, ξ ) ∈ L ∞ S − n +12 ̺ + n ( ̺ − ̺ and ̺ ∈ [0 , . Sup-pose that either(a) a ( x, ξ ) is compactly supported in the x variable and the phase func-tion ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ ξ and satisfies, det ∂ xξ ϕ ( x, ξ ) = 0 as well as rank ∂ ξξ ϕ ( x, ξ ) = n − ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ , ϕ satisfies the rough non-degeneracy con-dition as well as | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > b ∈ BMO and k a positive integer, the k -th commutator definedby T a,b,k u ( x ) := T a (cid:0) ( b ( x ) − b ( · )) k u (cid:1) ( x )is bounded on L pw for each w ∈ A p and p ∈ (1 , ∞ ).These BMO estimates have no predecessors in the literature and are usefulin connection to the study of hyperbolic partial differential equations withrough coefficients.In the last section of Chapter 4, we also briefly discuss global unweightedand local weighted estimates for the solutions of the Cauchy problem for m -th and second order hyperbolic partial differential equations. Acknowledgements.
Part of this work was undertaken while one of the au-thors was visiting the department of Mathematics of the Heriot-Watt Uni-versity. The first author wishes to express his gratitude for the hospitalityof Heriot-Watt University.
Chapter Prolegomena
In this chapter, we gather some results which will be useful in the studyof boundedness of Fourier integral operators. We also illustrate some of theconnections between global boundedness results for operators with smoothphases and amplitudes and local boundedness results for operators withrough phases and amplitudes, thus justifying a joint study of those operators.1.1.
Definitions, notations and preliminaries.
Phases and amplitudes.
In our investigation of the regularity proper-ties of Fourier integral operators, we will be concerned with both smoothand non-smooth amplitudes and phase functions. Below, we shall recallsome basic definitions and fix some notations which will be used throughoutthe paper. Also, in the sequel we use the notation h ξ i for (1 + | ξ | ) . Thefollowing definition which is due to H¨ormander [17], yields one of the mostwidely used classes of smooth symbols/amplitudes.
EGULARITY OF FOURIER INTEGRAL OPERATORS 11
Definition 1.1.1.
Let m ∈ R , ≤ δ ≤ , ≤ ̺ ≤ . A function a ( x, ξ ) ∈ C ∞ ( R n × R n ) belongs to the class S m̺,δ , if for all multi-indices α, β it satisfies sup ξ ∈ R n h ξ i − m + ̺ | α |− δ | β | | ∂ αξ ∂ βx a ( x, ξ ) | < + ∞ . We shall also deal with the class L ∞ S m̺ of rough symbols/amplitudesintroduced by Kenig and Staubach in [23]. Definition 1.1.2.
Let m ∈ R and ≤ ̺ ≤ . A function a ( x, ξ ) whichis smooth in the frequency variable ξ and bounded measurable in the spatialvariable x , belongs to the symbol class L ∞ S m̺ , if for all multi-indices α itsatisfies sup ξ ∈ R n h ξ i − m + ̺ | α | k ∂ αξ a ( · , ξ ) k L ∞ ( R n ) < + ∞ . We also need to describe the type of phase functions that we will dealwith. To this end, the class Φ k defined below, will play a significant role inour investigations. Definition 1.1.3.
A real valued function ϕ ( x, ξ ) belongs to the class Φ k ,if ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ , is positively homogeneous of degree in thefrequency variable ξ , and satisfies the following condition: For any pair ofmulti-indices α and β , satisfying | α | + | β | ≥ k , there exists a positive constant C α,β such that sup ( x, ξ ) ∈ R n × R n \ | ξ | − | α | | ∂ αξ ∂ βx ϕ ( x, ξ ) | ≤ C α,β . In connection to the problem of local boundedness of Fourier integraloperators, one considers phase functions ϕ ( x, ξ ) that are positively homoge-neous of degree 1 in the frequency variable ξ for which det[ ∂ x j ξ k ϕ ( x, ξ )] = 0 . The latter is referred to as the non-degeneracy condition . However, for thepurpose of proving global regularity results, we require a stronger conditionthan the aforementioned weak non-degeneracy condition.
Definition 1.1.4. ( The strong non-degeneracy condition ) . A real valuedphase ϕ ∈ C ( R n × R n \ satisfies the strong non-degeneracy condition, ifthere exists a positive constant c such that (cid:12)(cid:12)(cid:12) det ∂ ϕ ( x, ξ ) ∂x j ∂ξ k (cid:12)(cid:12)(cid:12) ≥ c, for all ( x, ξ ) ∈ R n × R n \ . The phases in class Φ satisfying the strong non-degeneracy conditionarise naturally in the study of hyperbolic partial differential equations, in-deed a phase function closely related to that of the wave operator, namely ϕ ( x, ξ ) = | ξ | + h x, ξ i belongs to the class Φ and is strongly non-degenerate.We also introduce the non-smooth version of the class Φ k which will be usedthroughout the paper. Definition 1.1.5.
A real valued function ϕ ( x, ξ ) belongs to the phase class L ∞ Φ k , if it is homogeneous of degree and smooth on R n \ in the frequencyvariable ξ , bounded measurable in the spatial variable x, and if for all multi-indices | α | ≥ k it satisfies sup ξ ∈ R n \ | ξ | − | α | k ∂ αξ ϕ ( · , ξ ) k L ∞ ( R n ) < + ∞ . We observe that if t ( x ) ∈ L ∞ then the phase function ϕ ( x, ξ ) = t ( x ) | ξ | + h x, ξ i belongs to the class L ∞ Φ , hence phase functions originating from thelinearisation of the maximal functions associated with averages on surfaces,can be considered as members of the L ∞ Φ class. We will also need a roughanalogue of the non-degeneracy condition, which we define below. Definition 1.1.6. ( The rough non-degeneracy condition ) . A real valuedphase ϕ satisfies the rough non-degeneracy condition, if it is C on R n \ inthe frequency variable ξ , bounded measurable in the spatial variable x, andthere exists a constant c > depending only on the dimension ) such thatfor all x, y ∈ R n and ξ ∈ R n \ | ∂ ξ ϕ ( x, ξ ) − ∂ ξ ϕ ( y, ξ ) | ≥ c | x − y | . Basic notions of weighted inequalities.
Our main reference for thematerial in this section are [14] and [36]. Given u ∈ L p loc , the L p maximalfunction M p ( u ) is defined by(1.1.2) M p ( u )( x ) = sup B ∋ x (cid:18) | B | Z B | u ( y ) | p d y (cid:19) p where the supremum is taken over balls B in R n containing x . Clearly then,the Hardy-Littlewood maximal function is given by M ( u ) := M ( u ) . An immediate consequence of H¨older’s inequality is that M ( u )( x ) ≤ M p ( u )( x )for p ≥
1. We shall use the notation u B := 1 | B | Z B | u ( y ) | d y for the average of the function u over B . One can then define the class ofMuckenhoupt A p weights as follows. Definition 1.1.7.
Let w ∈ L be a positive function. One says that w ∈ A if there exists a constant C > such that (1.1.3) M w ( x ) ≤ Cw ( x ) , for almost all x ∈ R n . One says that w ∈ A p for p ∈ (1 , ∞ ) if (1.1.4) sup B balls in R n w B (cid:0) w − p − (cid:1) p − B < ∞ . EGULARITY OF FOURIER INTEGRAL OPERATORS 13
The A p constants of a weight w ∈ A p are defined by (1.1.5) [ w ] A := sup B balls in R n w B k w − k L ∞ ( B ) , and (1.1.6) [ w ] A p := sup B balls in R n w B (cid:0) w − p − (cid:1) p − B . Example 1.
The function | x | α is in A if and only if − n < α ≤ A p with 1 < p < ∞ iff − n < α < n ( p − u ( x ) = log | x | when | x | < e and u ( x ) = 1 otherwise, is an A weight.1.1.3. Additional conventions.
As is common practice, we will denote con-stants which can be determined by known parameters in a given situation,but whose value is not crucial to the problem at hand, by C . Such parame-ters in this paper would be, for example, m , ̺ , p , n , [ w ] A p , and the constants C α in Definition 1.1.2. The value of C may differ from line to line, but ineach instance could be estimated if necessary. We sometimes write a . b asshorthand for a ≤ Cb . Our goal is to prove estimates of the form k T u k L p ≤ C k u k L p , u ∈ S ( R n )when a ∈ L ∞ S m̺ , ϕ ∈ L ∞ Φ k and m < − σ ≤ k T u k L p ≤ C k u k H s,p , u ∈ S ( R n )when a ∈ L ∞ S ̺ and s > σ and H s,p := { u ∈ S ′ ; ( I − ∆) s u ∈ L p } . We willuse indifferently one or the other equivalent formulation and we will refer to σ as the loss of derivatives in the L p boundedness of T .1.2. Tools in proving L p boundedness. Semi-classical reduction and decomposition of the operators.
It is con-venient to work with semi-classical estimates: let A be the annulus A = (cid:8) ξ ∈ R n ; ≤ | ξ | ≤ (cid:9) and χ ∈ C ∞ ( A ) be a cutoff function, we will prove estimates on the followingsemi-classical Fourier integral operator T h u = (2 πh ) − n Z R n e ih ϕ ( x,ξ ) χ ( ξ ) a ( x, ξ/h ) b u ( ξ/h ) d ξ with h ∈ (0 , T u = (2 π ) − n Z R n e iϕ ( x,ξ ) χ ( ξ ) a ( x, ξ ) b u ( ξ ) d ξ where χ ∈ C ∞ ( B (0 , L p spaces with weights in the Muckenhoupt’s A p class.This extent of generality will be needed when we deal with the weightedboundedness of Fourier integral operators. Lemma 1.2.1.
Let a ∈ L ∞ S m̺ and ϕ ∈ L ∞ Φ k , suppose that for all h ∈ (0 , and w ∈ A p , there exist constants C , C > (only depending on the A p constants of w ) such that the following estimates hold k T u k L pw ≤ C k u k L pw , k T h u k L pw ≤ C h − m − s k u k L pw , u ∈ S ( R n ) . This implies the bound k T u k L pw ≤ C k u k L pw , u ∈ S ( R n ) provided m < − s .Proof. We start by taking a dyadic partition of unity χ ( ξ ) + + ∞ X j =1 χ j ( ξ ) = 1 , where χ ∈ C ∞ ( B (0 , χ j ( ξ ) = χ (2 − j ξ ) when j ≥ χ ∈ C ∞ ( A ) andwe decompose the operator T as T = T χ ( D ) + + ∞ X j =1 T χ j ( D ) . (1.2.1)The first term in (1.2.1) is bounded from L pw to itself by assumption. Aftera change of variables, we have T χ j ( D ) u = (2 π ) − n jn Z R n e i j ϕ ( x,ξ ) χ ( ξ ) a ( x, j ξ ) b u (2 j ξ ) d ξ therefore using the semi-classical estimate with h = 2 − j we obtain k T χ j ( D ) u k L pw ≤ C ( m + s ) j k u k L pw . This finally gives k T u k L pw ≤ C k u k L pw + C ∞ X j =1 ( m + s ) j k u k L pw since the series is convergent when m < − s . This completes the proof ofour lemma. (cid:3) Seeger-Sogge-Stein decomposition.
To get useful estimates for the sym-bol and the phase function, one imposes a second microlocalization on theformer semi-classical operator in such a way that the annulus A is parti-tioned into truncated cones of thickness roughly √ h . Roughly h − ( n − / such pieces are needed to cover the annulus A . For each h ∈ (0 ,
1] we fix acollection of unit vectors { ξ ν } ≤ ν ≤ J which satisfy:(1) | ξ ν − ξ µ | ≥ h − , if ν = µ ,(2) If ξ ∈ S n − , then there exists a ξ ν so that | ξ − ξ ν | ≤ h . EGULARITY OF FOURIER INTEGRAL OPERATORS 15
Let Γ ν denote the cone in the ξ space with aperture √ h whose centraldirection is ξ ν , i.e.(1.2.2) Γ ν = n ξ ∈ R n ; (cid:12)(cid:12)(cid:12) ξ | ξ | − ξ ν (cid:12)(cid:12)(cid:12) ≤ √ h o . One can construct an associated partition of unity given by functions ψ ν ,each homogeneous of degree 0 in ξ and supported in Γ ν with(1.2.3) J X ν =1 ψ ν ( ξ ) = 1 , for all ξ = 0and sup ξ ∈ R n | ∂ α ψ ν ( ξ ) | ≤ C α h − | α | . (1.2.4)We decompose the operator T h as T h = J X ν =1 T h ψ ν ( D ) = J X ν =1 T νh (1.2.5)where the kernel of the operator T νh is given by T νh ( x, y ) = (2 πh ) − n Z R n e ih ϕ ( x,ξ ) − ih h y,ξ i χ ( ξ ) ψ ν ( ξ ) a ( x, ξ/h ) d ξ (1.2.6) = (2 πh ) − n Z R n e ih h∇ ξ ϕ ( x,ξ ν ) − y,ξ i b ν ( x, ξ, h ) d ξ with amplitude b ν ( x, ξ, h ) = e ih h∇ ξ ϕ ( x,ξ ) −∇ ξ ϕ ( x,ξ ν ) ,ξ i χ ( ξ ) ψ ν ( ξ ) a ( x, ξ/h ). Wechoose our coordinates on R n = R ξ ν ⊕ ξ ν ⊥ in the following way ξ = ξ ξ ν + ξ ′ , ξ ′ ⊥ ξ ν . Also it is worth noticing that the symbol χ ( ξ ) a ( x, ξ/h ) satisfies the followingbound(1.2.7) sup ξ k ∂ αξ (cid:0) χ ( ξ ) a ( · , ξ/h ) (cid:1) k L ∞ ≤ C α h − m −| α | (1 − ̺ ) . Lemma 1.2.2.
Let a ∈ L ∞ S m̺ and ϕ ( x, ξ ) ∈ L ∞ Φ . Then the symbol b ν ( x, ξ, h ) = e ih h∇ ξ ϕ ( x,ξ ) −∇ ξ ϕ ( x,ξ ν ) ,ξ i ψ ν ( ξ ) χ ( ξ ) a ( x, ξ/h ) satisfies the estimates sup ξ (cid:13)(cid:13) ∂ αξ b ν ( · , ξ, h ) (cid:13)(cid:13) L ∞ ≤ C α h − m −| α | (1 − ̺ ) − | α ′| . Proof.
We first observe that the bounds (1.2.4) may be improved tosup ξ ∈ A (cid:12)(cid:12) ∂ αξ ψ ν ( ξ ) (cid:12)(cid:12) ≤ C α h − | α ′| . (1.2.8)This can be seen by induction on | α | ; by Euler’s identity, we have ∂ ξ ∂ αξ ψ ν = −h| ξ | − ξ − ξ ν , ∇ ∂ αξ ψ ν i + | α | ∂ αξ ψ ν from which we deduce | ∂ ξ ∂ αξ ψ ν | ≤ (cid:12)(cid:12)(cid:12) ξ | ξ | − ξ ν (cid:12)(cid:12)(cid:12) |∇ ∂ αξ ψ ν | + | α || ∂ αξ ψ ν | . h h − | α ′| + h − | α ′| . This ends the induction. Similarly we havesup ξ ∈ A ∩ Γ ν (cid:13)(cid:13) ∂ αξ (cid:0) e ih h∇ ξ ϕ ( · ,ξ ) −∇ ξ ϕ ( · ,ξ ν ) ,ξ i (cid:1)(cid:13)(cid:13) L ∞ . h − | α ′| . (1.2.9)To prove this bound, we proceed by induction on | α | , we have ∇ ξ ∂ αξ (cid:16) e ih h∇ ξ ϕ ( x,ξ ) −∇ ξ ϕ ( x,ξ ν ) ,ξ i (cid:17) = ih ∂ αξ (cid:16)(cid:0) ∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( x, ξ ν ) (cid:1) e ih h∇ ξ ϕ ( x,ξ ) −∇ ξ ϕ ( x,ξ ν ) ,ξ i (cid:17) and by the Leibniz rule, it suffices to verify that for | β | ≤ ξ ∈ A ∩ Γ ν (cid:13)(cid:13) ∂ βξ (cid:0) ∂ ξ ′ ϕ ( · , ξ ) − ∂ ξ ′ ϕ ( · , ξ ν ) (cid:1)(cid:13)(cid:13) L ∞ . h −| β ′| sup ξ ∈ A ∩ Γ ν (cid:13)(cid:13) ∂ βξ (cid:0) ∂ ξ ϕ ( · , ξ ) − ∂ ξ ϕ ( · , ξ ν ) (cid:1)(cid:13)(cid:13) L ∞ . h − | β ′| , where for the case β = 0 one simply uses the mean value theorem on ∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( x, ξ ν ), which due the condition ϕ ∈ L ∞ Φ yields the de-sired estimates. We note that a homogeneous function which vanishes at ξ = ξ ν may be written in the form (cid:16) ξ | ξ | − ξ ν (cid:17) r ( x, ξ ) = O ( √ h ) on A ∩ Γ ν and this gives the first bound for β = 1. We also have ∂ ξ ∂ ξ ϕ ( x, ξ ν ) = 0 byEuler’s identity, therefore the former remark yields ∂ ξ ∂ ξ ϕ ( x, ξ ) = O ( √ h )which is the first bound for β = 1 (as well as the second bound for β ′ = 0).It remains to prove the second bound for β ′ = 0: by the mean value theoremand the bounds we have already obtained | ∂ ξ ϕ ( x, ξ ) − ∂ ξ ϕ ( x, ξ ν ) | . √ h (cid:12)(cid:12)(cid:12) ξ | ξ | − ξ ν (cid:12)(cid:12)(cid:12) . h. The estimates on b ν are consequences of (1.2.7), (1.2.8) and (1.2.9) and ofLeibniz’s rule. (cid:3) Phase reduction.
In our definition of class L ∞ Φ k we have only re-quired control of those frequency derivatives of the phase function which aregreater or equal to k . This restriction is motivated by the simple model casephase function ϕ ( x, ξ ) = t ( x ) | ξ | + h x, ξ i , t ( x ) ∈ L ∞ , for which the first order ξ -derivatives of the phase are not bounded but all the derivatives of orderequal or higher than 2 are indeed bounded and so ϕ ( x, ξ ) ∈ L ∞ Φ . Howeverin order to deal with low frequency portions of Fourier integral operators EGULARITY OF FOURIER INTEGRAL OPERATORS 17 one also needs to control the first order ξ derivatives of the phase. The fol-lowing phase reduction lemma will reduce the phase of the Fourier integraloperators to a linear term plus a phase for which the first order frequencyderivatives are bounded. Lemma 1.2.3.
Any Fourier integral operator T of the type (0.0.1) withamplitude σ ( x, ξ ) ∈ L ∞ S m̺ and phase function ϕ ( x, ξ ) ∈ L ∞ Φ , can bewritten as a finite sum of operators of the form (1.2.10) 1(2 π ) n Z a ( x, ξ ) e iθ ( x,ξ )+ i h∇ ξ ϕ ( x,ζ ) ,ξ i b u ( ξ ) d ξ where ζ is a point on the unit sphere S n − , θ ( x, ξ ) ∈ L ∞ Φ , and a ( x, ξ ) ∈ L ∞ S m̺ is localized in the ξ variable around the point ζ .Proof. We start by localizing the amplitude in the ξ variable by introducingan open convex covering { U l } Ml =1 , with maximum of diameters d , of the unitsphere S n − . Let Ξ l be a smooth partition of unity subordinate to thecovering U l and set a l ( x, ξ ) = σ ( x, ξ ) Ξ l ( ξ | ξ | ) . We set(1.2.11) T l u ( x ) := 1(2 π ) n Z a l ( x, ξ ) e iϕ ( x,ξ ) b u ( ξ ) d ξ, and fix a point ζ ∈ U l . Then for any ξ ∈ U l , Taylor’s formula and Euler’shomogeneity formula yield ϕ ( x, ξ ) = ϕ ( x, ζ ) + h∇ ξ ϕ ( x, ζ ) , ξ − ζ i + θ ( x, ξ )= θ ( x, ξ ) + h∇ ξ ϕ ( x, ζ ) , ξ i Furthermore, for ξ ∈ U l , ∂ ξ k θ ( x, ξ ) = ∂ ξ k ϕ ( x, ξ | ξ | ) − ∂ ξ k ϕ ( x, ζ ), so the meanvalue theorem and the definition of class L ∞ Φ yield | ∂ ξ k θ ( x, ξ ) | ≤ Cd andfor | α | ≥ | ∂ αξ θ ( x, ξ ) | ≤ C | ξ | −| α | . Here we remark in passing that indealing with function θ ( x, ξ ) , we only needed to control the second andhigher order ξ − derivatives of the phase function ϕ ( x, ξ ) and this gives afurther motivation for the definition of the class L ∞ Φ . We shall now extendthe function θ ( x, ξ ) to the whole of R n × R n \
0, preserving its propertiesand we denote this extension by θ ( x, ξ ) again. Hence the Fourier integraloperators T l defined by(1.2.12) T l u ( x ) := 1(2 π ) n Z a l ( x, ξ ) e iθ ( x,ξ )+ i h∇ ξ ϕ ( x,ζ ) ,ξ i b u ( ξ ) d ξ, are the localized pieces of the original Fourier integral operator T and there-fore T = P Ml =1 T l as claimed. (cid:3) Necessary and sufficient conditions for the non-degeneracy of smoothphase functions.
The smoothness of phases of Fourier integral operatorsmakes the study of boundedness considerably easier in the sense that theconditions of a phase being strongly non-degenerate and belonging to theclass Φ are enough to secure L p boundedness for a wide range of rough amplitudes. The following proposition which is useful in proving global L boundedness of Fourier integral operators, establishes a relationship betweenthe strongly non-degenerate phases and the lower bound estimates for thegradient of the phases in question. Proposition 1.2.4.
Let ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ be a real valued phasethen following statements hold true:(1) Assume that (cid:12)(cid:12)(cid:12) det ∂ ϕ ( x, ξ ) ∂x j ∂ξ k (cid:12)(cid:12)(cid:12) ≥ C , for all ( x, ξ ) ∈ R n × R n \ , and that (cid:13)(cid:13)(cid:13) ∂ ϕ ( x, ξ ) ∂x∂ξ (cid:13)(cid:13)(cid:13) ≤ C , for all ( x, ξ ) ∈ R n × R n \ and some constant C > , where k · k denotes matrix norm. Then (1.2.13) |∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( y, ξ ) | ≥ C | x − y | , for x, y ∈ R n and ξ ∈ R n \ and some C > . (2) Assume that |∇ ξ ϕ ( x, ξ ) − ∇ ξ ( y, ξ ) | ≥ C | x − y | for x, y ∈ R n and ξ ∈ R n \ and some C > . Then there exists a constant C suchthat (cid:12)(cid:12)(cid:12) det ∂ ϕ ( x, ξ ) ∂x j ∂ξ k (cid:12)(cid:12)(cid:12) ≥ C , for all ( x, ξ ) ∈ R n × R n \ .Proof. (1) We consider the map R n ∋ x → ∇ ξ ϕ ( x, ξ ) ∈ R n and using ourassumptions on ϕ , Schwartz’s global inverse function theorem [32] yieldsthat this map is a global C -diffeomorphism whose inverse λ ξ satisfies(1.2.14) | λ ξ ( z ) − λ ξ ( w ) | ≤ sup [ z,w ] k λ ′ ξ k × | z − w | . Furthermore, λ ′ ξ ( z ) = [( λ − ξ ) ′ ] − ◦ λ ξ ( z ) = [ ∂ x,ξ ϕ ( λ ξ ( z ) , ξ )] − . Thereforeusing the wellknown matrix inequality k A − k ≤ c n | det A | − k A k n − which isvalid for all A ∈ GL ( n, R ), we obtain using the assumption k ∂ ϕ ( x,ξ ) ∂x∂ξ k ≤ C that k λ ′ ξ ( z ) k ≤ c n | det[ ∂ x,ξ ϕ ( λ ξ ( z ) , ξ )] | − k ∂ x,ξ ϕ ( λ ξ ( z ) , ξ ) k n − ≤ c n C C ≤ C .
This yields that | λ ξ ( z ) − λ ξ ( w ) | ≤ C | z − w | and setting z = ∇ ξ ϕ ( x, ξ ) and w = ∇ ξ ϕ ( y, ξ ), we obtain (1.2.13).(2) Given the lower bound on the difference of the gradients as in the state-ment of the second part of the proposition, setting y = x + hv with v ∈ R n yields, |∇ ξ ϕ ( x + hv, ξ ) − ∇ ξ ϕ ( x, ξ ) | h ≥ C | v | EGULARITY OF FOURIER INTEGRAL OPERATORS 19 and letting h tend to zero we have for any v ∈ R n (1.2.15) | ∂ x,ξ ϕ ( x, ξ ) · v | ≥ C | v | . This means that ∂ x,ξ ϕ ( x, ξ ) is invertible and | [ ∂ x,ξ ϕ ( x, ξ )] − · w | ≤ | w | C . There-fore, taking the supremum we obtain k [ ∂ x,ξ ϕ ( x,ξ )] − k n ≥ C n . Now using thewellknown matrix inequality(1.2.16) 1 γ n k A − k n ≤ | det A | ≤ γ n k A k n , which is a consequence of the Hadamard inequality, yields for A = ∂ ϕ ( x,ξ ) ∂x j ∂ξ k (1.2.17) (cid:12)(cid:12)(cid:12)(cid:12) det ∂ ϕ ( x, ξ ) ∂x j ∂ξ k (cid:12)(cid:12)(cid:12)(cid:12) ≥ γ n C n . This completes the proof. (cid:3)
Remark 1.2.5.
Proposition 1.2.4 gives a motivation for our rough non-degeneracy condition in Definition 1.1.5, when there is no differentiabilityin the spatial variables.1.2.5.
Necessity of strong non-degeneracy for global regularity.
We shall nowdiscuss a simple example which illustrates the necessity of the strong non-degeneracy condition for the validity o global L p boundedness of Fourierintegral operators. To this end, we take a smooth diffeomorphism κ : R n → R n with everywhere nonzero Jacobian determinant, i.e. det κ ′ ( x ) = 0 forall x ∈ R n . Now, if we let ϕ ( x, ξ ) = h κ ( x ) , ξ i and take a ( x, ξ ) = 1 ∈ S , , then the Fourier integral operator T a,ϕ u ( x ) is nothing but the compositionoperator u ◦ κ ( x ) . Therefore(1.2.18) k T a,ϕ u k L p = k u ◦ κ k L p = { Z R n | u ( y ) | p | det κ ′− ( κ − ( y )) | dy } p , from which we see that T a,ϕ is L p bounded for any p , if and only if there ex-ists a constant C > | det κ ′− ( x ) | ≤ C for all x ∈ R n . The latteris equivalent to | det κ ′ ( x ) | ≥ C > . Now since | det ∂ ϕ ( x,ξ ) ∂x∂ξ | = | det κ ′ ( x ) | itfollows at once that a necessary condition for the L p boundedness of the op-erator T a,ϕ is the strong non-degeneracy of the phase function ϕ . We observethat if we instead had chosen a ( x, ξ ) to be equal to a smooth compactly sup-ported function in x, then the L p boundedness of T a,ϕ would have followedfrom the mere non-degeneracy condition | det ∂ ϕ ( x,ξ ) ∂x∂ξ | = | det κ ′ ( x ) | 6 = 0 . Non smooth changes of variables.
In dealing with rough Fourier in-tegral operators we would need at some point to make changes of variableswhen the substitution is not differentiable. This issue is problematic in gen-eral but in our setting, thanks to the rough non-degeneracy assumption onthe phase, we can show that the substitution is indeed valid and furthermorehas the desired boundedness properties. The discussion below is an abstract approach to the problem of non smooth substitution and we refer the readerinterested in related substitution results to De Guzman [16].
Lemma 1.2.6.
Let U be a measurable set and let t : U → R n be a boundedmeasurable map satisfying | t ( x ) − t ( y ) | ≥ c | x − y | (1.2.19) for almost every x, y ∈ U . Then there exists a function J t ∈ L ∞ ( R n ) supported in t ( U ) such that the substitution formula Z U u ◦ t ( x ) d x = Z u ( z ) J t ( z ) d z (1.2.20) holds for all u ∈ L ( R n ) and the Jacobian J t satisfies the estimate k J t k L ∞ ≤ √ nc . Remark 1.2.7.
If one works with a representative t in the equivalence classof functions equal almost everywhere, then possibly after replacing U with U \ N (where N is a null-set where (1.2.19) does not hold), one may assumethat t is an injective map with (1.2.19) holding everywhere on U .For the convenience of the reader, we provide a proof of this simple lemma. Proof.
As observed in Remark 1.2.7, we may assume that t is an injectivemap from U to R n for which (1.2.19) holds on U . The formula µ t ( f ) = Z U f ◦ t ( x ) d x, f ∈ C ( R n )defines a non-negative Radon measure, which by the Riesz representationtheorem is associated to a Borel measure. In this case, the latter measure isexplicitly given by µ t ( A ) = | t − ( A ) ∩ U | on all Lebesgue measurable sets A ⊂ R n , where we use the notation | · | forthe Lebesgue measure of a set. By the Lebesgue decomposition theorem,this measure can be split into an absolutely continuous and a singular part,i.e. µ t = µ ac t + µ sing t . Now assumption (1.2.19) yields t − (cid:0) B ∞ ( w, r ) (cid:1) ⊂ B ∞ ( x, √ nr/c ) , if t ( x ) ∈ B ∞ ( w, r )where B ∞ ( w, r ) is a ball of center w and radius r for the supremum norm.This implies that whenever A ∩ t ( U ) ⊂ ∞ [ k =0 B ∞ ( w k , r k )it follows that t − ( A ) ∩ U ⊂ ∞ [ k =0 B ∞ ( x k , √ nr k /c ) EGULARITY OF FOURIER INTEGRAL OPERATORS 21 where the centers x k have been chosen in t − ( B ∞ ( w k , r k )) when this set isnonempty. Furthermore, it is wellknown that the Lebesgue measure of a setcan be computed using | Ω | = inf (cid:26) ∞ X k =0 | Q k | , Ω ⊂ ∞ [ k =0 Q k (cid:27) where the infimum is taken over all possible sequences ( Q k ) k ∈ N of cubeswith faces parallel to the axes. Therefore µ t ( A ) ≤ √ nc | A ∩ t ( U ) | ≤ √ nc | A | (1.2.21)for all Lebesgue measurable sets A in R n . In particular, Lebesgue null-setsare also null-sets with respect to µ t , which in turn implies that the mea-sure µ t is absolutely continuous with respect to the Lebesgue measure. Bythe Radon-Nikodym theorem, there exists a positive Lebesgue measurablefunction J t ∈ L such that µ t has density J t µ t ( A ) = Z A J t ( x ) d x. By Lebesgue’s differentiation theorem, we may compute the Jacobian func-tion J t from the measure µ t by a limiting process on balls B , namely J t ( x ) = lim B →{ x } | B | Z B J t ( y ) d y = lim B →{ x } µ t ( B ) | B | . (1.2.22)Equality (1.2.22) together with the estimate (1.2.21) yields that J t is boundedand k J t k L ∞ ≤ √ nc . Moreover, from the definition of µ t it is clear that it is supported in t ( U ).Finally, (1.2.20) follows from Z U u ◦ t ( x ) d x = µ t ( u ) = Z u d µ t = Z u ( z ) J t ( z ) d z for all u ∈ C ( U ), and this extends to functions u ∈ L ( R n ). (cid:3) Remark 1.2.8.
Note that if there is a representative t in the equivalenceclass such that (1.2.19) holds everywhere on U and such that t ( U ) is an opensubset of R n , then t − : t ( U ) → U is a Lipschitz bijection. Furthermore,any open subset V ⊂ t ( U ) is open in R n and by Brouwer’s theorem on theinvariance of the domain t − ( W ) is open. This means that the map t isactually continuous. Corollary 1.2.9.
Let t : R n → R n be a map satisfying the assumptionsin Lemma U = R n , then u u ◦ t is a bounded map on L p for p ∈ [1 , ∞ ] . Proof.
This easily follows from Lemma 1.2.6: Z | u ◦ t ( x ) | p d x = Z | u ( z ) | p J t ( z ) d z ≤ k J t k L ∞ k u k L p when p ∈ [1 , ∞ ). The L ∞ estimate is similar. (cid:3) L p boundedness of the low frequency portion of rough Fourier integraloperators. Here we will prove the L p boundedness for p ∈ [1 , ∞ ] of Fourierintegral operators whose amplitude contains a smooth compactly supportedfunction factor, the support of which lies in a neighbourhood of the origin.There are a couple of difficulties to overcome here, the first being the sin-gularity of the phase function in the frequency variable at the origin. Thesecond problem is the one caused by the lack of smoothness in the spatialvariables. In order to handle these problems we need the following lemma Lemma 1.2.10.
Let b ( x, ξ ) be a bounded function which is C n +1 ( R nξ \ andcompactly supported in the frequency variable ξ and L ∞ ( R nx ) in the spacevariable x satisfying sup ξ ∈ R n \ | ξ | − | α | k ∂ αξ b ( · , ξ ) k L ∞ < + ∞ , | α | ≤ n + 1 . Then for all ≤ µ < we have sup x,y ∈ R n h y i n + µ (cid:12)(cid:12)(cid:12) Z e − i h y,ξ i b ( x, ξ ) d ξ (cid:12)(cid:12)(cid:12) < + ∞ . (1.2.23) Proof.
Since b ( x, ξ ) is assumed to be bounded, the integral in (1.2.23) whichwe denote by B ( x, y ) , is uniformly bounded and therefore it suffices to con-sider the case | y | ≥ . Integrations by parts yield B ( x, y ) = | y | − n Z e − i h y,ξ i h y, D ξ i n b ( x, ξ ) d ξ and therefore we have the estimate | B ( x, y ) | ≤ C | y | − n Z | ξ | 1, we have | y | n B ( x, y ) = Z e − i h y,ξ i χ ( ξ/ε ) β ( x, y, ξ ) d ξ + Z e − i h y,ξ i (cid:0) − χ ( ξ/ε ) (cid:1) β ( x, y, ξ ) d ξ EGULARITY OF FOURIER INTEGRAL OPERATORS 23 the first term is bounded by a constant times ε , while the second term isequal to i | y | − Z e − i h y,ξ i (cid:0) ε − h y, ∂ ξ i χ ( ξ/ε ) β − (cid:0) − χ ( ξ/ε ) (cid:1) h y, ∂ ξ i β (cid:1) d ξ which may be bounded by | y | − ( C − C log ε ) . We minimize the bound C ε + | y | − ( C − C log ε ) by taking ε = | y | − , andobtain | B ( x, y ) | ≤ C | y | − n − (cid:0) | y | (cid:1) ≤ C ′ | y | − n − µ for all 0 ≤ µ < 1. This is the desired estimate. (cid:3) Having this in our disposal we can show that the low frequency portionof the Fourier integral operators are L p bounded, more precisely we have Theorem 1.2.11. Let a ( x, ξ ) ∈ L ∞ S m̺ with m ∈ R and ̺ ∈ [0 , and let thephase function ϕ ( x, ξ ) ∈ L ∞ Φ satisfy the rough non-degeneracy condition ( according to Definition ) . Then for all χ ( ξ ) ∈ C ∞ supported aroundthe origin, the Fourier integral operator T u ( x ) = 1(2 π ) n Z R n e iϕ ( x,ξ ) a ( x, ξ ) χ ( ξ ) b u ( ξ ) d ξ is bounded on L p for p ∈ [1 , ∞ ] .Proof. In proving the L p boundedness, according to the reduction of thephase procedure in Lemma 1.2.10, there is no loss of generality to assumethat our Fourier integral operator is of the form T u ( x ) = 1(2 π ) n Z a ( x, ξ ) χ ( ξ ) e iθ ( x,ξ )+ i h∇ ξ ϕ ( x,ζ ) ,ξ i b u ( ξ ) d ξ, for some ζ ∈ S n − , a ∈ L ∞ S m̺ and θ ∈ L ∞ Φ . In the proof of the L p boundedness of T we only need to analyze the kernel of the operator Z a ( x, ξ ) χ ( ξ ) e iθ ( x,ξ )+ i h∇ ξ ϕ ( x,ζ ) ,ξ i b u ( ξ ) d ξ, which is given by T ( x, y ) := Z e i h∇ ξ ϕ ( x,ζ ) − y,ξ i e iθ ( x,ξ ) a ( x, ξ ) χ ( ξ ) d ξ. Now the estimates on the ξ derivatives of θ ( x, ξ ) above, yieldsup | ξ |6 =0 | ξ | − | α | | ∂ αξ θ ( x, ξ ) | < ∞ , for | α | ≥ x , and therefore setting b ( x, ξ ) := a ( x, ξ ) χ ( ξ ) e iθ ( x,ξ ) we have that b ( x, ξ ) is bounded and sup | ξ |6 =0 | ξ | − | α | | ∂ αξ b ( x, ξ ) | < ∞ , for | α | ≥ x and using Lemma 1.2.10, we have for all µ ∈ [0 , | T ( x, y ) | ≤ C h∇ ξ ϕ ( x, ζ ) − y i − n − µ . From this it follows that sup x Z | T ( x, y ) | dy < ∞ , and using our rough non-degeneracy assumption and Corollary 1.2.9 in thecase p = 1 , we also have Z | T ( x, y ) | d x . Z h∇ ξ ϕ ( x, ζ ) − y i − n − µ d x . Z h z i − n − µ d z < ∞ , uniformly in y . This estimate and Young’s inequality yield the L p bound-edness of the operator T . (cid:3) Some links between nonsmoothness and global boundedness. In this paragraph, we illustrate some of the relations between boundednessfor rough Fourier integral operators and the global boundedness of operatorswith smooth amplitudes and phases. Our observation is that local estimatesfor non-smooth Fourier integral operators imply global estimates for certainclasses of Fourier integral operators. This can be done either by compact-ification or by using a dyadic decomposition. To see the relation betweencompactification and global boundedness, consider the operator T u ( x ) = (2 π ) − n Z e iϕ ( x,ξ ) a ( x, ξ ) b u ( ξ ) d ξ. (1.3.1)Let χ ∈ C ∞ ( B (0 , B (0 , ω = 1 − χ be supported away from zero. Then T = T + T , T = χT, T = ωT. For the global continuity of T , we are only interested in T since the ampli-tude of T is compactly supported in the space variable and the boundednessof that operator follows from the local theory. Concerning T , we make thechange of variables z = x | x | θ , x = z | z | θ , θ ∈ (0 , Z | T u ( x ) | p d x = θ Z (cid:12)(cid:12)(cid:12) T u (cid:16) z | z | θ (cid:17)(cid:12)(cid:12)(cid:12) p | z | − n (1+ θ ) d z. (1.3.3)Therefore it suffices to study the L p boundedness of the Fourier integraloperator ˜ T u ( z ) = (2 π ) − n Z e iϕ ( z/ | z | θ ,ζ ) | z | − np (1+ θ ) ωa (cid:16) z | z | θ , ζ (cid:17)| {z } =˜ a ( z,ζ ) b u ( ζ ) d ζ. (1.3.4) EGULARITY OF FOURIER INTEGRAL OPERATORS 25 The amplitude ˜ a ( z, ζ ) is compactly supported (in the unit ball), and for asuitable choice of θ belongs to L ∞ S m provided h x i s a ( x, ξ ) ∈ L ∞ S m , s > np . (1.3.5)Now suppose that ϕ satisfies the following (global) non-degeneracy assump-tion: |∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( y, ξ ) | ≥ c | x − y | (1.3.6)for all x, y ∈ R n . Then since (cid:12)(cid:12)(cid:12) z | z | θ − w | w | θ (cid:12)(cid:12)(cid:12) = | z | − θ + | w | − θ − h w, z i| z | θ | w | θ (1.3.7) ≥ | w | , | z | ) θ | z − w | , the phase ˜ ϕ ( z, ζ ) = ϕ ( z/ | z | θ , ζ ) satisfies a similar non-degeneracy condi-tion, namely |∇ ζ ˜ ϕ ( z, ζ ) − ∇ ζ ˜ ϕ ( w, ζ ) | = (cid:12)(cid:12)(cid:12) ∇ ζ ϕ (cid:16) z | z | θ , ζ (cid:17) − ∇ ζ ϕ (cid:16) w | w | θ , ζ (cid:17)(cid:12)(cid:12)(cid:12) (1.3.8) ≥ c max( | w | , | z | ) θ | z − w | ≥ c | z − w | , when | w | , | z | ≤ 1. In order to improve the decay assumption on the ampli-tude (1.3.5), one can consider more general changes of variables which donot affect the angular coordinate in the polar decomposition, i.e. coordinatechanges of the form z = f ( | x | ) x | x | where f : (0 , ∞ ) → (0 , 1) is a diffeomorphism.Then x = g ( | z | ) z/ | z | where g is the inverse function of f, and the Jacobianof such a change of variables is given by | g ′ ( | z | ) | g n − ( | z | ) | z | − n . We would like to choose g in such a way that the singularities of its Jacobianbecome weaker than those in the case of g ( s ) = s − θ . One possible choice isto take g ( s ) = log(1 − s ) This decay assumption is due to the singularity at 0 of | z | − n (1+ θ ) /p of the Jacobian.Note that any improvement on the regularity of ˜ a, ˜ ϕ should translate into decay propertiesat infinity of the original amplitude and phase a, ϕ . In the case of the Kelvin transform θ = 1, it is easy to get a better lower bound (infact an equality): (cid:12)(cid:12)(cid:12) z | z | − w | w | (cid:12)(cid:12)(cid:12) = | z | − + | w | − − h w, z i| z | | w | = | z − w | | z | | w | . for which we have | g ′ ( s ) | g n − ( s ) s − n = log n − (1 − s ) s n − (1 − s ) = O (cid:0) (1 − s ) − − θ (cid:1) if s ∈ (0 , h x i s a ( x, ξ ) ∈ L ∞ S m , s > p . (1.3.9)Furthermore, if one assumes that g/s is decreasing (or increasing) then thephase ˜ ϕ satisfies our non-degeneracy assumption, because (cid:12)(cid:12)(cid:12) z | z | g ( | z | ) − w | w | g ( | w | ) (cid:12)(cid:12)(cid:12) (1.3.10) = g ( | z | ) + g ( | w | ) − h w, z i| z || w | g ( | z | ) g ( | w | ) ≥ (cid:16) gs (cid:17)(cid:0) min( | w | , | z | ) (cid:1) | z − w | ≥ g ′ (0) | z − w | . Alternatively, in order to investigate global boundedness using a dyadicdecomposition, one takes a Littlewood-Paley partition of unity 1 = χ ( x ) + P ∞ j =1 ψ (2 − j x ), which yields T = χT + ∞ X j =1 T j , T j := ψ (2 − j · ) T. (1.3.11)Once again we are only interested in T j and following a change of variables,we want to prove Z (cid:12)(cid:12) T j (cid:0) u (2 − j · (cid:1) (2 j z ) (cid:12)(cid:12) p d z ≤ C p Z | u ( z ) | p d z. (1.3.12)This leads us to the study of the operator˜ T j u ( z ) = T j (cid:0) u (2 − j · (cid:1) (2 j z )(1.3.13) = (2 π ) − n Z e i − j ϕ (2 j z,ζ ) ψ ( z ) a (cid:0) j z, − j ζ (cid:1)| {z } =˜ a j ( z,ζ ) b u ( ζ ) d ζ. The estimate | ∂ αζ ˜ a j ( z, ζ ) | ≤ − j | α | (1 + 2 j | z | ) m | {z } ≃ j ( m −| α | ) (1 + 2 − j | ζ | ) m −| α | (1.3.14) ≤ C α (1 + | ζ | ) m −| α | , yields that the amplitude ˜ a j ( z, ζ ) belongs (uniformly with respect to j ) tothe class L ∞ S m provided h x i − m a ( x, ξ ) ∈ L ∞ S m . (1.3.15)The phase ˜ ϕ j ( z, ζ ) = 2 − j ϕ (2 j z, ζ ) satisfies the non-degeneracy assumption |∇ ζ ˜ ϕ j ( z, ζ ) − ∇ ζ ˜ ϕ j ( w, ζ ) | ≥ c | z − w | . (1.3.16) EGULARITY OF FOURIER INTEGRAL OPERATORS 27 Therefore, once again the problem of establishing the global L p boundednessis reduced to a local problem concerning operators with rough amplitudes. Chapter Global boundedness of Fourier integral operators In this chapter, partly motivated by the investigation in [23] of the L p boundedness of the so called pseudo-pseudodifferential operators where thesymbols of the aforementioned operators are only bounded and measurablein the spatial variables x , we consider the global and local boundedness inLebesgue spaces of Fourier integral operators of the form T u ( x ) = (2 π ) − n Z R n e iϕ ( x,ξ ) a ( x, ξ ) b u ( ξ ) d ξ, (2.0.17)in case when the phase function ϕ ( x, ξ ) is smooth and homogeneous of de-gree 1 in the frequency variable ξ, and the amplitude a ( x, ξ ) is either insome H¨ormander class S m̺,δ , or is a L ∞ function in the spatial variable x andbelongs to some L ∞ S m̺ class. We shall also investigate the L p boundednessproblem for Fourier integral operators with rough phases that are L ∞ func-tions in the spatial variable. In the case of the rough phase, the standardnotion of non-degeneracy of the phase function has no meaning due to lackof differentiability in the x variables. However, there is a non-smooth ana-logue of the non-degeneracy condition which has already been introduced inDefinition 1.1.6 which will be exploited further here.We start by investigating the question of L boundedness of Fourier inte-gral operators with rough amplitudes but smooth phase functions satisfyingthe strong non-degeneracy condition. Thereafter we turn to the problem of L boundedness of the Fourier integral operators with smooth phases, butrough or smooth amplitudes. In the case of smooth amplitudes, we show theanalogue of the Calder´on-Vaillancourt’s L boundedness of pseudodifferen-tial operators in the realm of Fourier integral operators. Next, we considerFourier integral operators with rough amplitudes and rough phase functionsand show a global and a local L result in that context. We also give afairly general discussion of the symplectic aspects of the L boundedness ofFourier integral operators.After concluding our investigation of the L boundedness, we proceed byproving a sharp L ∞ boundedness theorem for Fourier integral operatorswith rough amplitudes and rough phases in class L ∞ Φ , without any non-degeneracy assumption on the phase. Finally, we close this chapter by prov-ing L p − L p and L p − L q estimates for operators with smooth phase function,and smooth or rough amplitudes.2.1. Global L boundedness of rough Fourier integral operators. Aswill be shown below, the global L boundedness of Fourier integral operatorsis a consequence of Theorem 1.2.11, the Seeger-Sogge-Stein decomposition,and elementary kernel estimates. Theorem 2.1.1. Let T be a Fourier integral operator given by (0.0.1) withamplitude a ∈ L ∞ S m̺ and phase function ϕ ∈ L ∞ Φ satisfying the roughnon-degeneracy condition. Then there exists a constant C > such that k T u k L ≤ C k u k L , u ∈ S ( R n ) provided m < − n − + n ( ̺ − and ≤ ̺ ≤ .Proof. Using semiclassical reduction of Subsection 1.2.1, we decompose T into low and high frequency portions T and T h . Then we use the Seeger-Sogge-Stein decomposition of Subsection 1.2.2 to decompose T h into the sum P Jν =1 T νh . The boundedness of T follows at once from Theorem 1.2.11, soit remains to establish suitable semiclassical estimates for T νh . To this endwe consider the following differential operator L = 1 − ∂ ξ − h∂ ξ ′ for which we have according to Lemma 1.2.2(2.1.1) sup ξ k L N b ν ( · , ξ, h ) k L ∞ . h − m − N (1 − ̺ ) . Integrations by parts yield | T νh ( x, y ) | ≤ (2 πh ) − n (cid:0) g ( y − ∇ ξ ϕ ( x, ξ ν ) (cid:1) − N Z | L N b ν ( x, ξ, h ) | d ξ for all integers N , with(2.1.2) g ( z ) = h − z + h − | z ′ | . This further gives | T νh ( x, y ) | ≤ C N h − m − n +12 − N (1 − ̺ ) (cid:0) g ( y − ∇ ξ ϕ ( x, ξ ν ) (cid:1) − N since the volume of the portion of cone | A ∩ Γ ν | is of the order of h ( n − / .By interpolation, it is easy to obtain the former bound when the integer N is replaced by M/ M is any given positive number ; indeed write M/ N + θ where N = [ M ] and θ ∈ [0 , 1) and | T νh ( x, y ) | = | T νh ( x, y ) | θ | T νh ( x, y ) | − θ ≤ C − θN C θN +1 h − m − n +12 − (1 − ̺ ) M (cid:0) g ( y − ∇ ξ ϕ ( x, ξ ν ) (cid:1) − M . (2.1.3)This implies that for any real number M > n sup x Z | T νh ( x, y ) | d y ≤ C M h − m − M (1 − ̺ ) . Furthermore the rough non-degeneracy assumption on the phase function ϕ ( x, ξ ) and Corollary 1.2.9 with p = 1 yieldsup y Z | T νh ( x, y ) | d x ≤ C − θN C θN +1 Z (cid:0) g ( ∇ ξ ϕ ( x, ξ ν ) (cid:1) − M d x ≤ C M h − m − M (1 − ̺ )EGULARITY OF FOURIER INTEGRAL OPERATORS 29 thus using Young’s inequality and summing in ν k T h u k L ≤ J X ν =1 k T νh u k L ≤ C M h − m − n − − M (1 − ̺ ) k u k L since J is bounded (from above and below) by a constant times h − n − . ByLemma 1.2.1 one has k T u k L . k u k L provided m < − n − − M (1 − ̺ ) and M > n , i.e. if m < − n − + n ( ̺ − (cid:3) Local and global L boundedness of Fourier integral operators. In this section we study the local and global L boundedness properties ofFourier integral operators. Here we complete the global L theory of Fourierintegral operators with smooth strongly non-degenerate phase functions inclass Φ and smooth amplitudes in the H¨ormander class S m̺,δ for all rangesof ρ ’s and δ ’s. As a first step we establish global L boundedness of Fourierintegral operators with smooth phases and rough amplitudes in L ∞ S m̺ , thenwe proceed by investigating the L boundedness of Fourier integral operatorswith smooth phases and amplitudes and finally we consider the L regularityof the operators with rough amplitudes in L ∞ S m̺ and rough non-degeneratephase functions in L ∞ Φ . L boundedness of Fourier integral operators with phases in Φ . Theglobal L boundedness of Fourier integral operators which we aim to provebelow, yields on one hand a global version of Eskin’s and H¨ormander’s local L boundedness theorem for amplitudes in S , , and on the other handgeneralises the global L result of Fujiwara’s for amplitudes in S , to thecase of rough amplitudes. Furthermore, as we shall see later, our result issharp. Theorem 2.2.1. Let a ( x, ξ ) ∈ L ∞ S m̺ and the phase ϕ ( x, ξ ) ∈ Φ be stronglynon degenerate. Then the Fourier integral operator T a,ϕ u ( x ) = 1(2 π ) n Z a ( x, ξ ) e iϕ ( x,ξ ) b u ( ξ ) d ξ is a bounded operator from L to itself provided m < n ( ̺ − . The boundon m is sharp.Proof. In light of Theorem 1.2.11, we can confine ourselves to deal with thehigh frequency component T h of T a,ϕ , hence we can assume that ξ = 0 onthe support of the amplitude a ( x, ξ ) . Here we shall use a T h T ∗ h argument,and therefore, the kernel of the operator S h = T h T ∗ h reads S h ( x, y ) = 1(2 πh ) n Z e ih ( ϕ ( x,ξ ) − ϕ ( y,ξ )) χ ( ξ ) a ( x, ξ/h ) a ( y, ξ/h ) d ξ Now the strong non degeneracy assumption on the phase and Proposition1.2.4 yield that there is a constant C > |∇ ξ ϕ ( x, ξ ) −∇ ξ ϕ ( y, ξ ) | ≥ C | x − y | , for x, y ∈ R n and ξ ∈ R n \ . This enables us to use the non-stationary phase estimate in [18] Theorem 7.7.1, and the smoothness of thephase function ϕ ( x, ξ ) in the spatial variable, yield that for all integers N | S h ( x, y ) | ≤ C N h − m − n − (1 − ̺ ) N h h − ( x − y ) i − N , for some constant C N > . Let M be a positive real number, we have M = N + θ where N is the integer part of M and θ ∈ [0 , 1) and therefore | S h ( x, y ) | = | S h ( x, y ) | θ | S h ( x, y ) | − θ ≤ C − θN C θN +1 h − m − n − (1 − ̺ ) M h h − ( x − y ) i − M . (2.2.1)This implies sup x Z | S h ( x, y ) | d y ≤ C M h − m − (1 − ̺ ) M (2.2.2)for all M > n . By Cauchy-Schwarz and Young inequalities, we obtain k T ∗ h u k L ≤ k S h u k L k u k L ≤ Ch − m − (1 − ̺ ) M k u k L . (2.2.3)Therefore, by Lemma 1.2.1 we have the L bound k T u k L . k u k L provided m < − (1 − ̺ ) M/ M > n . This completes the proof of The-orem 2.2.1. For the sharpness of this result we consider the phase function ϕ ( x, ξ ) = h x, ξ i ∈ Φ which is strongly non-degenerate. It was shown in[29] that for m = n ( ̺ − 1) there are symbols a ( x, ξ ) ∈ S m̺, such that thepseudodifferential operator a ( x, D ) u ( x ) = 1(2 π ) n Z a ( x, ξ ) e i h x,ξ i b u ( ξ ) d ξ is not L bounded. Since S m̺, ⊂ L ∞ S m̺ , it turns out that there are ampli-tudes in L ∞ S m̺ which yield an L unbounded operator for a non-degeneratephase function in class Φ . Hence the order m in the theorem is sharp. (cid:3) As a consequence, we obtain an alternative proof for the L boundednessof pseudo-pseudodifferential operators introduced in [23]. More precisely wehave Corollary 2.2.2. Let a ( x, D ) be a pseudo-pseudodifferential operator, i.e.an operator defined on the Schwartz class, given by (2.2.4) a ( x, D ) u = 1(2 π ) n Z R n e i h x,ξ i a ( x, ξ ) b u ( ξ ) d ξ, with symbol a ∈ L ∞ S m̺ , ≤ ̺ ≤ . If m < n ( ̺ − / , then a ( x, D ) extendsas an L bounded operator. EGULARITY OF FOURIER INTEGRAL OPERATORS 31 Theorem 2.2.1 can be used to show a simple local L boundedness resultfor Fourier integral operators with smooth symbols in the H¨ormander class S m̺,δ in those cases when the symbolic calculus of the Fourier integral oper-ators, as defined in [20] breaks down (e.g. in case δ ≥ ̺ ), more precisely wehave Corollary 2.2.3. Let a ( x, ξ ) ∈ S m̺,δ with compact support in x variable andlet ϕ ( x, ξ ) ∈ Φ be strongly non-degenerate. Then for m < n ( ̺ − δ − and ≤ ̺ ≤ , ≤ δ ≤ . Then the corresponding Fourier integral operator isbounded on L . Proof. By Sobolev embedding theorem, for a function f ( x, y ) one has Z | f ( x, x ) | d x ≤ C n X | α |≤ N Z Z | ∂ αy f ( x, y ) | d x d y with N > n/ 2. Now let f ( x, y ) := R a ( y, ξ ) e iϕ ( x,ξ ) ˆ u ( ξ ) d ξ. Since a ( x, ξ ) ∈ S m̺,δ , we have that ∂ αy a ( y, ξ ) ∈ L ∞ S m + δ | α | ̺ . Therefore, Theorem 2.2.1 yields Z | ∂ αy f ( x, y ) | d x . k u k L , provided that m + δ | α | < n ( ̺ − | α | ≤ N and N > n/ , one seesthat it suffices to take m < n ( ̺ − δ − . We also note that in the argumentabove, the integration in the y variable will not cause any problem due tothe compact support assumption of the amplitude. (cid:3) However, as was shown by D. Fujiwara in [12], Fourier integral oper-ators with phases in Φ and amplitudes in S , are bounded in L . Thisresult suggests the possibility of the existence of an analog of the Calder´on-Vaillancourt theorem [6], concerning L boundedness of pseudodifferentialoperators with symbols in S ̺,̺ with ̺ ∈ [0 , , in the realm of smooth Fourierintegral operators. That this is indeed the case will be the content of The-orem 2.2.6 below. However, before proceeding with the statement of thattheorem, we will need two lemmas, the first of which is a continuous versionof the Cotlar-Stein lemma, due to A. Calder´on and R. Vaillancourt, see i.e.[6] for a proof. Lemma 2.2.4. Let H be a Hilbert space, and A ( ξ ) a family of boundedlinear endomorphisms of H depending on ξ ∈ R n . Assume the followingthree conditions hold:(1) the operator norm of A ( ξ ) is less than a number C independent of ξ. (2) for every u ∈ H the function ξ A ( ξ ) u from R n H is contin-uous for the norm topology of H . (3) for all ξ and ξ in R n (2.2.5) k A ∗ ( ξ ) A ( ξ ) k ≤ h ( ξ , ξ ) , and k A ( ξ ) A ∗ ( ξ ) k ≤ h ( ξ , ξ ) , with h ( ξ , ξ ) ≥ is the kernel of a bounded linear operator on L with norm K .Then for every E ⊂ R n , with | E | < ∞ , the operator A E = R E A ( ξ ) d ξ defined by h A E u, v i H = R E h A ( ξ ) u, v i H d ξ, is a bounded linear operator on H with norm less than or equal to K. We shall also use the following useful lemma. Lemma 2.2.5. Let (2.2.6) Lu ( x ) := D − (1 − is ( x ) h∇ x F, ∇ x i ) u ( x ) , with D := (1 + s ( x ) |∇ x F | ) / . Then(1) L ( e iF ( x ) ) = e iF ( x ) (2) if t L denotes the formal transpose of L, then for any positive integer N, ( t L ) N u ( x ) is a finite linear combination of terms of the form (2.2.7) CD − k { p Y µ =1 ∂ α µ x s ( x ) }{ q Y ν =1 ∂ β ν x F ( x ) } ∂ γx u ( x ) , with (2.2.8) 2 N ≤ k ≤ N ; k − N ≤ p ≤ k − N ; | α µ | ≥ p X µ =1 | α µ | ≤ Nk − N ≤ q ≤ k − N ; | β ν | ≥ q X µ =1 | β ν | ≤ q + N ; | γ | ≤ N. Proof. First one notes that ∂ x j D − N = − N D − N − n X k =1 { s ( x ) ∂ x k F ∂ x j x k F + ∂ x j s ( ∂ x k F ) } . This and Leibniz’s rule yield t Lu ( x ) = D − u ( x ) + i n X j =1 ∂ x j ( D − s ( x ) u ( x ) ∂ x j F )= D − u ( x ) − iD − n X k, j =1 u ( x ) s ( x ) ∂ x j F (cid:8) s ( x ) ∂ x k F ∂ x j x k F + ∂ x j s ( ∂ x k F ) + iD − n X j =1 u ( x ) ∂ x j s ( x ) ∂ x j F + iD − n X j =1 s ( x ) u ( x ) ∂ x j F + iD − n X j =1 s ( x ) ∂ x j u ( x ) ∂ x j F. From this it follows that t L is a linear combination of operators of the form(2.2.9) D − × EGULARITY OF FOURIER INTEGRAL OPERATORS 33 (2.2.10) D − s ( x ) ∂ x j F ∂ x k F ∂ x j x k F × (2.2.11) D − s ∂ x j s ∂ x j F ( ∂ x k F ) × (2.2.12) D − ∂ x j s ( x ) ∂ x j F × (2.2.13) D − s ( x ) ∂ x j F × (2.2.14) D − s ( x ) ∂ x j F ∂ x j . If we conventionally say that the term (2.2.7) is of the type (cid:18) k, p, p X µ =1 | α µ | , q, q X ν =1 | β ν | , | γ | (cid:19) , then t L is sum of terms of the types (2 , , , , , , (4 , , , , , , (4 , , , , , , (2 , , , , , , (2 , , , , , 0) and (2 , , , , , . Now operating the opera-tors in (2.2.9), (2.2.10), (2.2.11), (2.2.12), (2.2.13) on a term (2.2.7) of type (cid:18) k, p, p X µ =1 | α µ | , q, q X ν =1 | β ν | , | γ | (cid:19) , increases the types by (2 , , , , , , (4 , , , , , , (4 , , , , , , (2 , , , , , , (2 , , , , , 0) respectively. To see how operating a term of form the 2.2.14on (2.2.7) changes the type we use Leibniz rule to obtain D − s ( x ) ∂ x j F ∂ x j (cid:18) D − k (cid:26) p Y µ =1 ∂ α µ x s ( x ) (cid:27)(cid:26) q Y ν =1 ∂ β ν x F ( x ) (cid:27) ∂ γx u ( x ) (cid:19) = − k (cid:18) D − k − n X l =1 ∂ x j F n s ( x ) ∂ x l F ∂ x j x l F + ∂ x j s ( ∂ x l F ) o(cid:19) × (cid:26) p Y µ =1 ∂ α µ x s ( x ) (cid:27)(cid:26) q Y ν =1 ∂ β ν x F ( x ) (cid:27) ∂ γx u ( x )+ D − k − ∂ x j F p X µ ′ =1 (cid:26) Y µ = µ ′ ∂ α µ x s ( x ) (cid:27)(cid:26) ∂ x j ∂ α µ ′ x s ( x ) (cid:27) q Y ν =1 ∂ β ν x F ( x ) ∂ γx u ( x )+ D − k − ∂ x j F p Y µ =1 ∂ α µ x s ( x ) q X ν ′ =1 (cid:26) Y ν = ν ′ ∂ α ν x F ( x ) (cid:27)(cid:26) ∂ x j ∂ α ν ′ x F ( x ) (cid:27) ∂ γx u ( x )+ D − k − (cid:26) p Y µ =1 ∂ α µ x s ( x ) (cid:27)(cid:26) q Y ν =1 ∂ β ν x F ( x ) (cid:27) ∂ x j ∂ γx u ( x ) . Therefore, upon application of t L to (2.2.7), the types of the resulting termsincrease by (2 , , , , , , (4 , , , , , , (4 , , , , , , (2 , , , , , , (2 , , , , , and (2 , , , , , . Iteration of this process yields( t L ) N u ( x ) = X C D − k (cid:26) p Y µ =1 ∂ α µ x s ( x ) (cid:27)(cid:26) q Y ν =1 ∂ β ν x F ( x ) (cid:27) ∂ γx u ( x ) , where the summation is taken over all non-negative integers N , N , N ,N , N , N with P j =1 N j = N and(2.2.15) (cid:18) k, p, p X µ =1 | α µ | , q, q X ν =1 | β ν | , | γ | (cid:19) = N (2 , , , , , 0) + N (4 , , , , , N (4 , , , , , N (2 , , , , , N (2 , , , , , N (2 , , , , , . Hence,(2.2.16) k = 2 N + 4 N + 4 N + 2 N + 2 N + 2 N (2.2.17) p = 2 N + 2 N + N + N + N (2.2.18) p X µ =1 | α µ | = N + N (2.2.19) q = 3 N + 3 N + N + N + N (2.2.20) q X ν =1 | β ν | = 4 N + 3 N + N + 2 N + N (2.2.21) | γ | = N . From this it also follows that ( k, p, P pµ =1 | α µ | , q, P qν =1 | β ν | , | γ | ) satisfy (2.2.8). (cid:3) Theorem 2.2.6. If m = min(0 , n ( ̺ − δ )) , ≤ ̺ ≤ , ≤ δ < , a ∈ S m̺,δ and ϕ ∈ Φ , satisfies the strong non-degeneracy condition, then the operator T a u ( x ) = R a ( x, ξ ) e iϕ ( x,ξ ) ˆ u ( ξ ) d ξ is bounded on L . Proof. First we observe that since for δ ≤ ̺ , S ̺,δ ⊂ S ̺,̺ , it is enough to showthe theorem for 0 ≤ ̺ ≤ δ < m = n ( ̺ − δ ) . Also, as we have donepreviously, we can assume without loss of generality that a ( x, ξ ) = 0 when ξ is in a a small neighbourhood of the origin. Using the T T ∗ argument, itis enough to show that the operator(2.2.22) T b u ( x ) = Z Z b ( x, y, ξ ) e iϕ ( x,ξ ) − iϕ ( y,ξ ) u ( y ) d y d ξ, where b satisfies the estimate(2.2.23) | ∂ αξ ∂ βx ∂ γy b ( x, y, ξ ) | ≤ C α β γ h ξ i m − ̺ | α | + δ ( | β | + | γ | ) , EGULARITY OF FOURIER INTEGRAL OPERATORS 35 with m = n ( ̺ − δ ) and 0 ≤ ̺ ≤ δ < , is bounded on L . We introduce a differential operator L := D − n − i h ξ i ̺ (cid:0) h∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( y, ξ ) , ∇ ξ i (cid:1)o , with D = (1 + h ξ i ̺ |∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( y, ξ ) | ) . It follows from Lemma 2.2.5that L ( e iϕ ( x,ξ ) − iϕ ( y,ξ ) ) = e iϕ ( x,ξ ) − iϕ ( y,ξ ) and that ( t L ) N ( b ( x, y, ξ )) is a finite sum of terms of the form(2.2.24) D − k (cid:26) p Y µ =1 ∂ α µ ξ h ξ i ̺ (cid:27)(cid:26) q Y ν =1 (cid:0) ∂ β ν ξ ϕ ( x, ξ ) − ∂ β ν ξ ϕ ( y, ξ ) (cid:1)(cid:27) ∂ γξ b ( x, y, ξ ) . Furthermore since ϕ ∈ Φ is assumed to be strongly non-degenerate, we canuse Proposition 1.2.4 to deduce that(2.2.25) |∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( y, ξ ) | ≥ c | x − y | (2.2.26) |∇ z ϕ ( z, ξ ) − ∇ z ϕ ( z, ξ ) | ≥ c | ξ − ξ | . Using (2.2.25), (2.2.8) and (2.2.23), we have(2.2.27) | ∂ σx ( t L ) N ( b ( x, y, ξ )) | ≤ C Λ( h ξ i ̺ ( x − y )) h ξ i m + δ | σ | , where Λ is an integrable function with R Λ( x ) dx . . Integration by partsusing L , N times, in (2.2.22) one has(2.2.28) T b u ( x ) = Z Z c ( x, y, ξ ) e iϕ ( x,ξ ) − iϕ ( y,ξ ) u ( y ) d y d ξ, with c ( x, y, ξ ) = ( t L ) N ( b ( x, y, ξ )) and(2.2.29) | ∂ σx c ( x, y, ξ ) | ≤ C Λ( h ξ i ̺ ( x − y )) h ξ i m + δ | σ | and the same estimate is valid for ∂ σy c ( x, y, ξ ) . From this we get the repre-sentation(2.2.30) T b = Z A ( ξ ) d ξ, where A ( ξ ) u ( x ) := R c ( x, y, ξ ) e iϕ ( x,ξ ) − iϕ ( y,ξ ) u ( y ) d y. Noting that A ( ξ ) = 0for ξ outside some compact set, we observe that condition (1) of Lemma2.2.4 follows from Young’s inequality and (2.2.29) with σ = 0 , and condition(2) of Lemma 2.2.4 follows from the assumption of the compact support ofthe amplitude. To verify condition (3) we confine ourselves to the estimateof k A ∗ ( ξ ) A ( ξ ) k , since the one for k A ( ξ ) A ∗ ( ξ ) k is similar. To this end, acalculation shows that the kernel of A ∗ ( ξ ) A ( ξ ) is given by(2.2.31) K ( x, y, ξ , ξ ) := Z c ( z, x, ξ ) c ( z, y, ξ ) e i [ ϕ ( z,ξ ) − ϕ ( z,ξ )+ ϕ ( x,ξ ) − ϕ ( y,ξ )] d z. The estimate (2.2.29) yields(2.2.32) | K ( x, y, ξ , ξ ) | . h ξ i m h ξ i m Z Λ( h ξ i ̺ ( x − z )) Λ( h ξ i ̺ ( y − z )) d z. Therefore by choosing N large enough, Young’s inequality and using thefact that R Λ( x ) dx . k A ∗ ( ξ ) A ( ξ ) k . h ξ i m − n̺ h ξ i m − n̺ . At this point we introduce another first order differential operator M := G − { − i ( h∇ z ϕ ( z, ξ ) − ∇ z ϕ ( z, ξ ) , ∇ z i ) } , with G = (1 + |∇ z ϕ ( z, ξ ) −∇ z ϕ ( z, ξ ) | ) . Using the fact that M e i ( ϕ ( z,ξ ) − ϕ ( z,ξ )) = e i ( ϕ ( z,ξ ) − ϕ ( z,ξ )) , integration by parts in (2.2.31) yields(2.2.34) Z ( t M ) N ′ { c ( z, x, ξ ) c ( z, y, ξ ) } e i [ ϕ ( z,ξ ) − ϕ ( z,ξ )+ ϕ ( x,ξ ) − ϕ ( y,ξ )] d z. Using the second part of Lemma 2.2.5, we find that ( t M ) N ′ { c ( z, x, ξ ) c ( z, y, ξ ) } is a linear combination of terms of the form(2.2.35) G − k (cid:26) q Y ν =1 ( ∂ β ν z ϕ ( z, ξ ) − ∂ β ν ξ ϕ ( z, ξ )) (cid:27) ∂ γ z c ( z, x, ξ ) ∂ γ z c ( z, y, ξ ) , where k, q, β ν satisfy the inequalities in 2.2.8 and | γ | + | γ | ≤ N ′ . Now,(2.2.26), (2.2.29) and (2.2.35), yield the following estimate for K ( x, y, ξ , ξ ) | K ( x, y, ξ , ξ ) | . h ξ i m h ξ i m (1 + | ξ | + | ξ | ) δN ′ | ξ − ξ | − N ′ (2.2.36) × Z Λ( h ξ i ̺ ( x − z )) Λ( h ξ i ̺ ( y − z )) d z. Once again, choosing N large enough, Young’s inequality yields(2.2.37) k A ∗ ( ξ ) A ( ξ ) k . h ξ i m − n̺ h ξ i m − n̺ (1 + | ξ | + | ξ | ) δN ′ | ξ − ξ | N ′ . Using the fact that for x > , inf(1 , x ) ∼ (1 + x ) − , one optimizes theestimates (2.2.33) and (2.2.37) by k A ∗ ( ξ ) A ( ξ ) k . h ξ i m − n̺ h ξ i m − n̺ (cid:18) | ξ − ξ | N ′ (1 + | ξ | + | ξ | ) δN ′ (cid:19) − (2.2.38) := h ( ξ , ξ ) . Therefore recalling that m = n ( ̺ − δ ) , in applying Lemma 2.2.4, we needto show that(2.2.39) K ( ξ , ξ ) = (1 + | ξ | ) − nδ (1 + | ξ | ) − nδ (cid:18) | ξ − ξ | N ′ (1 + | ξ | + | ξ | ) δN ′ (cid:19) − EGULARITY OF FOURIER INTEGRAL OPERATORS 37 is the kernel of a bounded operator in L . At this point we use Schur’slemma, which yields the desired conclusion provided thatsup ξ Z K ( ξ , ξ ) d ξ , sup ξ Z K ( ξ , ξ ) d ξ are both finite. Due to the symmetry of the kernel, we only need to showthe finiteness of one of these quantities.To this end, we fix ξ and consider the domains A = { ( ξ , ξ ); | ξ | ≥ | ξ |} , B = { ( ξ , ξ ); | ξ | ≤ | ξ | ≤ | ξ |} , and C = { ( ξ , ξ ); | ξ | ≤ | ξ | } . Now weobserve that on the set A , K ( ξ , ξ ) is dominated by(2.2.40) (1 + | ξ | ) − nδ (1 + | ξ | ) − nδ + N ′ ( δ − , on B , K ( ξ , ξ ) is dominated by(2.2.41) (1 + | ξ | ) − nδ (cid:18) | ξ − ξ | N ′ (1 + | ξ | ) δN ′ (cid:19) − , and on C , K ( ξ , ξ ) is dominated by(2.2.42) (1 + | ξ | ) − nδ (1 + | ξ | ) − nδ + N ′ ( δ − . Therefore, if I Ω := R Ω K ( ξ , ξ ) dξ , then choosing N ′ ( δ − < − n, which isonly possible if δ < , we have that I A < ∞ uniformly in ξ . Also,(2.2.43) I C ≤ (1 + | ξ | ) n − nδ + N ′ ( δ − ≤ C, which is again possible by the fact that δ < N ′ . In I B let us make a change of variables to set ξ − ξ = (1 + | ξ | ) δ y , then(2.2.44) I B ≤ Z (1 + | y | N ′ ) − d y < ∞ , by taking N ′ large enough. These estimates yield the desired result and theproof of there theorem is therefore complete. (cid:3) L boundedness of Fourier integral operators with phases in L ∞ Φ . Next we shall turn to the problem of L boundedness of Fourier integral op-erators with non-smooth amplitudes and phases. As was mentioned in theintroduction, a motivation for considering fully rough Fourier integral oper-ators stems from a ”linearisation” procedure which reduces certain maximaloperators to Fourier integral operators with a non-smooth phase and some-times also a non-smooth amplitude. For instance, estimates for the maximalspherical average operator Au ( x ) = sup t ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12) Z S n − u ( x + tω ) d σ ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) are related to those for the maximal wave operator W u ( x ) = sup t ∈ [0 , (cid:12)(cid:12) e it √− ∆ u ( x ) (cid:12)(cid:12) , and can for instance be deduced from those of the linearized operator e it ( x ) √− ∆ u = (2 π ) − n Z R n e it ( x ) | ξ | + i h x,ξ i b u ( ξ ) d ξ, (2.2.45)where t ( x ) is a measurable function in x , with values in [0 , 1] and the phasehere belongs to the class L ∞ Φ . As will be demonstrated later, the validityof the results in the rough case depend on the geometric conditions (imposedon the phase functions) which are the rough analogues of the non-degeneracyand corank conditions for smooth phases. In trying to understand the subtleinterrelations between boundedness, smoothness and geometric conditions,we remark that even if one assumes the phase of the linearized operator(2.2.45) to be smooth, there are cases for which the canonical relation of thisoperator ceases to be the graph of a symplectomorphism. Indeed, contraryto the wave operator e it √− ∆ at fixed time t ∈ [0 , ϕ ( x, ξ ) = h x, ξ i + t ( x ) | ξ | of the linearized operator cannot be a generating function ofa canonical transformation, (see [10]), in certain cases since ∂ ϕ∂x∂ξ ( x, ξ ) = Id + ∇ t ( x ) ⊗ ξ | ξ | , ker ∂ ϕ∂x∂ξ ( x, ξ ) = span ∇ t ( x ) when h ξ, ∇ t ( x ) i + | ξ | = 0 , and this happens when |∇ t ( x ) | ≥ ξ = ̺ ( − ∇ t ( x ) |∇ t ( x ) | + η ) with ̺ ∈ R ∗ + and η is a vector orthogonal to ∇ t ( x ) of norm (1 − |∇ t ( x ) | − ) / . There-fore, one can not expect L boundedness of (2.2.45) even when the function t ( x ) is smooth. Nevertheless, in this case the rank of the Hessian ∂ ϕ/∂x∂ξ drops by one with respect to its maximal possible value, and one could stillestablish L estimates with loss of derivatives (see section 2.2.3 for moredetails). The operators that we intend to study will fall into this category.Before we investigate the local L boundedness of operators based on geo-metric conditions on their phase, we state and prove a purely analytic global L boundedness result which will be used later. Theorem 2.2.7. Let T be a Fourier integral operator given by (0.0.1) withamplitude a ∈ L ∞ S m̺ , ≤ ̺ ≤ and a phase function ϕ ( x, ξ ) ∈ L ∞ Φ satisfying the rough non-degeneracy condition. Then there exists a constant C > such that k T u k L . k u k L provided m < n ( ̺ − / − ( n − / .Proof. Using semiclassical reduction of Subsection 1.2.1, we decompose T into low and high frequency portions T and T h . The boundedness of T follows at once from Theorem 1.2.11, so it remains to establish suitablesemiclassical estimates for T h . Once again we use the T T ∗ argument. The EGULARITY OF FOURIER INTEGRAL OPERATORS 39 kernel of the operator S h = T h T ∗ h reads S h ( x, y ) = (2 πh ) − n Z e ih ( ϕ ( x,ξ ) − ϕ ( y,ξ )) χ ( ξ ) a ( x, ξ/h ) a ( y, ξ/h ) d ξ. We now use the Seeger-Sogge-Stein decomposition (section 1.2.2) and splitthis operator as the sum P Nj =1 S νh where the kernel of S νh takes the form S νh ( x, y ) = (2 πh ) − n Z e ih h∇ ξ ϕ ( x,ξ ν ) −∇ ξ ϕ ( y,ξ ν ) ,ξ i b ν ( x, ξ, h ) b ν ( y, ξ, h ) d ξ. We consider the following differential operator L = 1 − ∂ ξ − h∂ ξ ′ for which we have according to Lemma 1.2.2(2.2.46) sup ξ k L N b ν ( · , ξ, h ) k L ∞ . h − m − N (1 − ̺ ) . Integration by parts yields | S νh ( x, y ) | ≤ (2 πh ) − n (cid:0) g (cid:0) ∇ ξ ϕ ( y, ξ ν ) − ∇ ξ ϕ ( x, ξ ν ) (cid:1)(cid:1) − N × Z (cid:12)(cid:12) L N (cid:0) b ν ( x, ξ, h ) b ν ( y, ξ, h ) (cid:1)(cid:12)(cid:12) d ξ for all integers N , with(2.2.47) g ( z ) = h − z + h − | z ′ | . The standard interpolation trick gives the same inequality for for all positivenumbers M > | S νh ( x, y ) | ≤ Ch − m − n +12 − M (1 − ̺ ) (cid:0) g (cid:0) ∇ ξ ϕ ( y, ξ ν ) − ∇ ξ ϕ ( x, ξ ν ) (cid:1)(cid:1) − M since the volume of the portion of cone | A ∩ Γ ν | is of the order of h ( n − / .By the non-degeneracy assumption and Lemma 1.2.6, we get Z | S νh ( x, y ) | d y ≤ Ch − m − n +12 − M (1 − ̺ ) Z (cid:0) g ( z ) (cid:1) − M d z | {z } = ch n +12 . By Young’s inequality (remembering that the kernel S νh ( x, y ) is symmetric),we obtain k S νh u k L ≤ Ch − m − M (1 − ̺ ) k u k L and summing the inequalities k T ∗ h u k L ≤ J X ν =1 k S νh u k L k u k L ≤ Ch − m + n − − − ̺ ) M k u k L , since there are roughly h − ( n − / terms in the sum. By Lemma 1.2.1, wehave the L bound k T u k L . k u k L provided m < − ( n − / ̺ − M and M > n/ 2, which yields the desiredresult. (cid:3) Remark 2.2.8. The reason why we were led to perform the Seeger-Sogge-Stein decomposition is that under the rough non-degeneracy assumption(Definition 1.1.6), the non-stationary phase (Theorem 7.7.1 [18]) providesthe bound | S h ( x, y ) | ≤ C N h − m − n + N | x − y | N (2.2.48) ≤ C N h − m − n (cid:0) h − | x − y | ) − N leading, when say ̺ = 1, to a loss of n/ n − / | ∂ αξ ϕ ( x, ξ ) − ∂ αξ ϕ ( y, ξ ) | ≤ C α | x − y | , | α | ≥ . This is indeed the case in dimension n = 1, or if the phase can be decomposedas ϕ ( x, ξ ) = ϕ ♯ ( x, ξ ) + ϕ ♭ ( x, ξ ) where ϕ ♯ is linear in ξ and ϕ ♭ ∈ Φ .Let π denote the projection onto the spatial variables, i.e. π : T ∗ R n → R n ( x, ξ ) x. A geometric condition sufficient for the local L boundedness of roughFourier integral operators with phase functions ϕ ( x, ξ ) and amplitudes a ( x, ξ )is as follows: Rough corank condition. For each x ∈ π (supp a ) and all ξ ∈ S n − thereexists a linear subspace V x,ξ belonging to the Grassmannian Gr( n, n − k ) varying continuously with ( x, ξ ) , and constants c , c > such that if π V x,ξ denotes the projection onto V x,ξ , then | ∂ ξ ϕ ( x, ξ ) − ∂ ξ ϕ ( y, ξ ) | + c | x − y | ≥ c | π V x,ξ ( x − y ) | for all x, y ∈ π (supp a ) . Theorem 2.2.9. Let T be a Fourier integral operator given by (0.0.1) withamplitude a ∈ L ∞ S m̺ and phase function ϕ ∈ L ∞ Φ . Suppose that thephase satisfies the rough corank condition 2.2.2, then T can be extended asa bounded operator from L to L provided m < − n + k − + ( n − k )( ̺ − . Proof. Since we aim to prove a local L boundedness result, we may assumethat the amplitude a is compactly supported in the spatial variable x . Thensince S = T T ∗ has a bounded compactly supported kernel, it extends toa bounded operator on L . It remains to deal with the high frequency partof the operator. Given ( x µ , ξ µ ) ∈ R n × R n , µ = 1 , . . . , J, we consider a EGULARITY OF FOURIER INTEGRAL OPERATORS 41 partition of unity J X µ =1 ψ µ ( x, ξ ) = 1 , ξ = 0given by functions ψ µ homogeneous of degree 0 in the frequency variable ξ supported in conesΓ µ = n ( x, ξ ) ∈ T ∗ R n ; | x − x µ | + (cid:12)(cid:12)(cid:12) ξ | ξ | − ξ µ (cid:12)(cid:12)(cid:12) ≤ ε o where ε is yet to be chosen. We decompose the operator as T h = N X µ =1 T µh (2.2.49)where the kernel of T µh is given by T µh ( x, y ) = (2 πh ) − n Z R n e ih ϕ ( x,ξ ) − ih h y,ξ i ψ µ ( x, ξ ) χ ( ξ ) a ( x, ξ/h ) d ξ. We have the direct sum R n = V x µ ,ξ µ ⊕ V ⊥ x µ ,ξ µ , dim V x µ ,ξ µ = n − k and we decompose vectors x = x ′ + x ′′ (i.e. x = ( x ′ , x ′′ )) according to thissum. Assumption 2.2.2 implies | ∂ ξ ϕ ( x ′ , x ′′ , ξ ) − ∂ ξ ϕ ( y ′ , x ′′ , ξ ) |≥ c | π V x,ξ ( x ′ − y ′ ) | − c | x ′ − y ′ | ≥ c | x ′ − y ′ | (cid:16) − k π V x,ξ − π V xµ,ξµ k − c c | x ′ − y ′ | (cid:17) . Now since ( x, ξ ) π V x,ξ is continuous, we can choose ε in the definition ofthe cone Γ µ small enough so that k π V x,ξ − π V xµ,ξµ k ≤ , | x ′ − y ′ | ≤ | x ′ − x µ ′ | + | y ′ − x µ ′ | ≤ c c and therefore we have | ∂ ξ ϕ ( x ′ , x ′′ , ξ ) − ∂ ξ ϕ ( y ′ , x ′′ , ξ ) | ≥ c | x ′ − y ′ | (2.2.50)when ( x, ξ ) and ( y, ξ ) belong to Γ µ . We fix the x ′′ variable and use a T T ∗ argument on the operator acting in the x ′ variables. We consider S µh ( x ′ , x ′′ , y ′ ) = (2 πh ) − n Z e ih ( ϕ ( x,ξ ) − ϕ ( y ′ ,x ′′ ,ξ )) a µh ( x, ξ ) a µh ( y ′ , x ′′ , ξ ) d ξ. Because of (2.2.50), performing a Seeger-Sogge-Stein decomposition and rea-soning as in the proof of Theorem 2.2.7 we get (cid:18) Z V xµ,ξµ (cid:12)(cid:12)(cid:12)(cid:12) Z V xµ,ξµ S µh ( x ′ , x ′′ , y ′ ) u ( y ′ ) d y ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ (cid:19) ≤ Ch − m − n − k − − k − M (1 − ̺ ) (cid:18) Z | u ( y ′ ) | d y ′ (cid:19) , with a constant C that is independent of x ′′ , provided M > n − k and there-fore Z (cid:12)(cid:12)(cid:12) Z V xµ,ξµ T µh ( x ′ , x ′′ , y ) u ( x ) d x ′ (cid:12)(cid:12)(cid:12) d y ≤ Ch − m − n − − k − M (1 − ̺ ) k u k L . Hence by Minkowski’s integral inequality k T ∗ h u k L ≤ Z V ⊥ xµ,ξµ (cid:18) Z (cid:12)(cid:12)(cid:12) Z V xµ,ξµ T µh ( x ′ , x ′′ , y ) u ( x ) d x ′ (cid:12)(cid:12)(cid:12) d y (cid:19) d x ′′ ≤ Ch − m − n − − k − M (1 − ̺ ) k u k L provided M > n − k and the amplitude is compactly supported in x ′′ . Thisyields the L bound for m < − ( n − k ) / − (1 − ̺ ) M provided M > n − k ,and completes the proof of Theorem 2.2.9 (cid:3) Remark 2.2.10. The phase of the linearized maximal wave operator whichis ϕ ( x, ξ ) = t ( x ) | ξ | + h x, ξ i , satisfies the assumptions of Theorem 2.2.9 sinceit belongs to L ∞ Φ and it also satisfies the rough corank condition 2.2.2. In-deed if ξ ∈ S n − we can take V x,ξ = ξ ⊥ and if π ξ , π ξ ⊥ denote the projectionsonto span ξ and V x,ξ respectively then it is clear that | ∂ ξ ϕ ( x, ξ ) − ∂ ξ ϕ ( y, ξ ) | = | t ( x ) − t ( y ) | + | x − y | + 2( t ( x ) − t ( y )) h ξ, x − y i | {z } = ±| π ξ ( x − y ) | ≥ (cid:12)(cid:12) π ξ ⊥ ( x − y ) (cid:12)(cid:12) + (cid:12)(cid:12) | t ( x ) − t ( y ) | − | π ξ ( x − y ) | (cid:12)(cid:12) ≥ | π ξ ⊥ ( x − y ) | . Therefore, as mentioned earlier, the Fourier integral operators under con-sideration include the linearized maximal wave operator.A consequence of this is a local L boundedness result for Fourier integraloperators with smooth phase functions and rough symbols. Corollary 2.2.11. Suppose that ϕ ( x, ξ ) is a smooth phase function satisfy-ing the non-degeneracy condition (2.2.51) rank ∂ ϕ∂x j ∂ξ k ≥ n − k, on supp a EGULARITY OF FOURIER INTEGRAL OPERATORS 43 and the entries of the Hessian matrix have bounded derivatives with respect toboth x and ξ separately. Assume also that the symbol a belongs to L ∞ S m̺ , ≤ ̺ ≤ . Then the associated Fourier integral operator is bounded from L to L provided m < − k + ( n − k )( ̺ − . This is sharp, for example in the case k = 0 (i.e. pseudodifferentialoperators), since there exists m with m > n ( ̺ − δ ) / S m ̺,δ is not bounded from L to L , see [21]. Now since the phase of a pseudodifferential operatorsatisfies the condition of the above corollary with k = 0 and since obvi-ously m ≥ n ( ̺ − / S m ̺,δ ⊂ L ∞ S m ̺ , it follows that the above L boundedness is sharp.2.2.3. Symplectic aspects of the L boundedness. Here we shall discuss thesymplectic aspects of the L boundedness of Fourier integral operators whichaims to highlight the essentially geometric nature of the problem of L reg-ularity of Fourier integral operators. We begin by recalling some of the wellknown L continuity results in the case of smooth phases and amplitudes.The kernel of the Fourier integral operator T u ( x ) = (2 π ) − n Z R n e iϕ ( x,ξ ) a ( x, ξ ) b u ( ξ ) d ξ (2.2.52)is an oscillatory integral whose wave front set is contained in the closedsubset of ˙ T ∗ R n = T ∗ R n \ T ) ⊂ (cid:8) ( x, ∂ x ϕ ( x, ξ ) , ∂ ξ ϕ ( x, ξ ) , − ξ ) : ( x, ξ ) ∈ supp a, ξ = 0 (cid:9) . (2.2.53)The cotangent space T ∗ R n is endowed with the symplectic form σ = n X j =1 d ξ j ∧ d x j . A canonical relation is a Lagrangian submanifold of the product T ∗ R n × T ∗ R n endowed with the symplectic form σ ⊕ ( − σ ), this means that theaforementioned symplectic form vanishes on the canonical relation. In par-ticular, by rearranging the terms in the closed cone (2.2.53), one obtains acanonical relation C ϕ = (cid:8) ( x, ∂ x ϕ ( x, ξ ) , ∂ ξ ϕ ( x, ξ ) , ξ ) : ( x, ξ ) ∈ supp a (cid:9) in T ∗ R n × T ∗ R n . If C is a canonical relation, we consider the two maps π : ( x, ξ ) ( x, ∂ x ϕ ) and π : ( x, ξ ) ( ∂ ξ ϕ, ξ ) , C ⊂ T ∗ R n × T ∗ R nπ { { π GGGGGGGG T ∗ R n T ∗ R n . The canonical relation C is (locally) the graph of a smooth function χ if andonly if π is a (local) diffeomorphism, and in this case χ = π ◦ π − . This function χ is a diffeomorphism if and only if π is a diffeomorphism. Notethat if this is the case, χ is a symplectomorphism because the submanifold C is Lagrangian for the symplectic form, i.e. σ ⊕ ( − σ )d ξ ∧ d x − d η ∧ d y = 0 when ( y, η ) = χ ( x, ξ ) . The canonical relation C ϕ is locally the graph of a symplectomorphism inthe neighbourhood of ( x , ∂ x ϕ ( x , ξ ) , ∂ ξ ϕ ( x , ξ ) , ξ ) if and only ifdet ∂ ϕ∂x∂ξ ( x , ξ ) = 0 . (2.2.54)It is well-known that the Fourier integral operators of order 0 whose canon-ical relation C ϕ is locally the graph of a symplectic transformation χ , arelocally L bounded. More precisely Theorem 2.2.12. Let a ∈ S , and ϕ be a real valued function in C ∞ ( R n × R n \ which is homogeneous of degree in ξ . Assume that the homogeneouscanonical relation C ϕ is locally the graph of a symplectomorphism betweentwo open neighbourhoods in ˙ T ∗ R n = T ∗ R n \ . Then the Fourier integraloperator (2.2.52) defines a bounded operator from L to L .Proof. This is Theorem 25.3.1 in [19]. (cid:3) But in fact, there are boundedness results even when C is not the graphof a symplectomorphism, i.e. when either the projection π or π is not adiffeomorphism. There is an important instance for which this is the caseand one could still prove local L boundedness with loss of derivatives. Asuggestive example for this situation is the restriction operator to a linearsubspace H = (cid:8) x = ( x ′ , x ′′ ) ∈ R n = R n ′ × R n ′′ : x ′′ = 0 (cid:9) R H u = h D i m u ( x ′ , 0) = (2 π ) − n Z e i h x ′ ,ξ ′ i h ξ i m b u ( ξ ) d ξ where m ≤ 0. We know that this operator is bounded from L to L ;indeed for all a ∈ C ∞ ( R n ) there exists a constant C m,n such that k aR H u k L ≤ C m,n k u k L provided m ≤ − codim H/ 2. The canonical relation of the Fourier integraloperator R H is given by C H = (cid:8) ( x, ξ ′ , x ′ , , ξ ) , ( x, ξ ) ∈ T ∗ R n (cid:9) π | | yyyyyyyyyy π " " EEEEEEEEEE (cid:8) ξ ′′ = 0 (cid:9) ⊂ T ∗ R n (cid:8) x ′′ = 0 (cid:9) ⊂ T ∗ R n . Or equivalently that (2.2.54) holds on supp a . EGULARITY OF FOURIER INTEGRAL OPERATORS 45 By σ C H we denote the pullback of the symplectic form σ, by π , to C H (ofcourse we could equally well consider the pullback π ∗ σ without changinganything) σ C H = π ∗ σ = d ξ ′ ∧ d x ′ . Then we have corank σ C H = 2 n ′′ = 2 codim H and the condition of L boundedness is therefore m ≤ − corank σ C H / 4. Infact, this example models the general situation, and this is Theorem 25.3.8in [19]. Theorem 2.2.13. Let a ∈ S m , and ϕ be a real valued function in C ∞ ( R n × R n \ which is homogeneous of degree in ξ such that dϕ = 0 on supp a .Then the Fourier integral operator (2.2.52) defines a bounded operator from L to L provided m ≤ − corank σ C ϕ / . Here σ C ϕ is the two form on C ϕ obtained by lifting to C ϕ the symplectic form σ on ˙ T ∗ R n by one of theprojections π or π . The fact that the canonical relation is parametrised by F : ( x, ξ ) ( x, ∂ x ϕ ( x, ξ ) , ∂ ξ ϕ ( x, ξ ) , ξ )allows us to compute F ∗ ( π ∗ σ ) = d( π ◦ F ) ∗ ( ξ d x ) = d (cid:0) ∂ x ϕ ( x, ξ ) d x (cid:1) = n X j,k =1 ∂ x j x k ϕ ( x, ξ ) d x j ∧ d x k | {z } =0 + n X j,k =1 ∂ ξ j x k ϕ ( x, ξ ) d ξ j ∧ d x k . Therefore we have F ∗ σ C ϕ = n X j,k =1 ∂ ξ j x k ϕ ( x, ξ ) d ξ j ∧ d x k which yields corank σ C ϕ = 2 corank ∂ ϕ∂x∂ξ . The geometric assumption in Theorem 2.2.13 (which is valid for generalFourier integral operators, not necessarily of the form (2.2.52)) is thereforeequivalent to m ≤ − 12 corank ∂ ϕ∂x∂ξ . (2.2.55) This ensures that C ϕ is a homogeneous canonical relation to which the radial vectorsof ˙ T ∗ R n × × ˙ T ∗ R n are never tangential. Remark 2.2.14. If the function t ( x ) in the linearized maximal wave oper-ator (2.2.45) were smooth, then that operator would fall into the categoryof Fourier integral operators satisfying the assumptions of Theorem 2.2.13.Indeed as already noted in the introduction, the corank of ∂ ϕ/∂x∂ξ when ϕ ( x, ξ ) = t ( x ) | ξ | + h x, ξ i is at most 1. Therefore e it ( x ) √− ∆ defines a boundedoperator from H / to L when t ( x ) is a smooth function on R n .Theorem 2.2.7 for ̺ = 1 is the non-smooth analogue of Theorem 2.2.12where the non-degeneracy condition (2.2.54) which requires smoothness in x has been replaced by Definition 1.1.6. Note nevertheless that Theorem 2.2.7is a global L result. Similarly Theorem 2.2.9 for ̺ = 1 is the non-smoothanalogue of Theorem 2.2.13 with (2.2.55) replaced by assumption 2.2.2.2.3. Global L ∞ boundedness of rough Fourier integral operators. In this section, we establish the L ∞ boundedness of Fourier integral oper-ators. To prove the L ∞ boundedness of the high frequency portion of theoperator, we need to use the semiclassical estimates of Subsection 1.2.2.However, using only the Seeger-Sogge-Stein decomposition yields a loss ofderivatives no better than m < − n − + n ( ̺ − L ∞ boundedness result claimed in Theorem 2.3.1, further analysis is needed. Theorem 2.3.1. Let T be a Fourier integral operator given by (0.0.1) withamplitude a ∈ L ∞ S m̺ and phase function ϕ ∈ L ∞ Φ . Then there exists aconstant C > such that k T u k L ∞ . k u k L ∞ , u ∈ S ( R n ) provided m < − n − + n ( ̺ − and ≤ ̺ ≤ . Furthermore, this result issharp.Proof. As a first step, we use the semiclassical reduction of Subsection 1.2.1to decompose T into T and T h . Thereafter we split the semiclassical piece T h further into P Jν =1 T νh using the Seeger-Sogge-Stein decomposition of Sub-section 1.2.2 applied to the amplitude a ( x, ξ ) and the phase ϕ ( x, ξ ) . Onceagain, the boundedness of T follows from Theorem 1.2.11, but here we don’tneed the rough non-degeneracy of the phase function due to the fact thatwe are dealing with the L ∞ boundedness of T which only requires that theintegral with respect to the y variable of the Schwartz kernel T ( x, y ) beingfinite. See Theorem 1.2.11 for further details.From equation 1.2.6 one deduces that the kernel of the semiclassical highfrequency operator T νh is given by T νh ( x, y ) = (2 πh ) − n Z R n e ih h∇ ξ ϕ ( x,ξ ν ) − y,ξ i b ν ( x, ξ, h ) d ξ, with b ν ( x, ξ, h ) = e ih h∇ ξ ϕ ( x,ξ ) −∇ ξ ϕ ( x,ξ ν ) ,ξ i ψ ν ( ξ ) χ ( ξ ) a ( x, ξ/h ) . Now since k T νh u k L ∞ ≤ k u k L ∞ Z | T νh ( x, y ) | d y, EGULARITY OF FOURIER INTEGRAL OPERATORS 47 it remains to show a suitable estimate for R | T νh ( x, y ) | dy . As in the proof of L boundedness, we use the differential operator L = 1 − ∂ ξ − h∂ ξ ′ for which we have according to Lemma 1.2.2(2.3.1) sup ξ k L N b ν ( · , ξ, h ) k L ∞ . h − m − N (1 − ̺ ) . Setting(2.3.2) g ( z ) = h − z + h − | z ′ | , we have L N e ih h∇ ξ ϕ ( x,ξ ν ) − y,ξ i = (cid:0) g ( y − ∇ ξ ϕ ( x, ξ ν ) (cid:1) N e ih h∇ ξ ϕ ( x,ξ ν ) − y,ξ i for all integers N . Now we observe that(2 πh ) n Z | T νh ( x, y ) | d y = (2 πh ) n Z | T νh ( x, y + ∇ ξ ϕ ( x, ξ ν )) | d y = Z | b b ν ( x, y, h ) | d y = Z √ g ( y ) ≤ h ̺ + Z √ g ( y ) >h ̺ | b b ν ( x, y, h ) | d y := I + I , where b b ν ( x, y, h ) = (2 πh ) − n Z e − ih h y,ξ i b ν ( x, ξ, h ) d ξ is the semiclassical Fourier transform of b ν . To estimate I we use theCauchy-Schwarz inequality, the semiclassical Plancherel theorem, the def-inition of g in (2.3.2) and (2.3.1). Hence remembering the fact that themeasure of the ξ -support of b ν ( x, ξ, h ) is O ( h ( n − ) we have I ≤ (cid:26) Z √ g ( y ) ≤ h ̺ d y (cid:27) (cid:26) Z | b b ν ( x, y, h ) | d y (cid:27) . h n +14 (cid:26) Z | y |≤ h ̺ d y (cid:27) (cid:26) Z | b ν ( x, ξ, h ) | d ξ (cid:27) . h n +14 h n̺ h − m + n − . h n h − m + n̺ . Before we proceed with the estimate of I , we observe that if l is a non-negative integer then the semiclassical Plancherel theorem and (2.3.1) yield (cid:18) Z | b b ν ( x, y, h ) | (1 + g ( y )) l d y (cid:19) ≤ (cid:18) Z | L l b ν ( x, ξ, h ) | d ξ (cid:19) (2.3.3) ≤ h − m − l (1 − ̺ )+ n − . Moreover, any positive real number l which is not an integer can be writtenas [ l ] + { l } where [ l ] denotes the integer part of l and { l } its fractional part, which is 0 < { l } < 1. Therefore, H¨older’s inequality with conjugateexponents { l } and −{ l } yields Z | b b ν ( x, y, h ) | (1 + g ( y )) l d y = Z | b b ν | { l } | b b ν | −{ l } ) (1 + g ( y )) { l } ([ l ]+1) (1 + g ( y )) l ](1 −{ l } ) d y ≤ (cid:18) Z | b b ν | (1 + g ( y )) l ]+1) d y (cid:19) { l } (cid:18) Z | b b ν | (1 + g ( y )) l ] d y (cid:19) −{ l } . Therefore, using (2.3.3) we obtain (cid:18) Z | b b ν ( x, y, h ) | (1 + g ( y )) l d y (cid:19) ≤ (cid:18) Z | L [ l ]+1 b ν ( x, ξ, h ) | d ξ (cid:19) { l } (cid:18) Z | L [ l ] b ν ( x, ξ, h ) | d ξ (cid:19) −{ l } ≤ h { l } ( − m − l ]+1)(1 − ̺ )+ n − ) h (1 −{ l } )( − m − l ](1 − ̺ )+ n − ) ≤ h − m − l (1 − ̺ )+ n − , and hence (2.3.3) is actually valid for all non-negative real numbers l . Turn-ing now to the estimates for I , we use the same tools as in the case of I and (2.3.3) for l ∈ [0 , ∞ ). This yields for any l > n I ≤ (cid:26) Z √ g ( y ) >h ̺ (1 + g ( y )) − l d y (cid:27) × (cid:26) Z | b b ν ( x, y, h ) | (1 + g ( y )) l d y (cid:27) . h n +14 (cid:26) Z | y | >h ̺ | y | − l d y (cid:27) h − m − l (1 − ̺ )+ n − . h n +14 h ̺ ( n − l ) h − m − l (1 − ̺ )+ n − . h n h − m + n̺ − l . Therefore sup x Z | T νh ( x, y ) | d y ≤ C l h − m + n̺ − l (2.3.4)and summing in ν yields k T h u k L ∞ ≤ J X ν =1 k T νh u k L ∞ ≤ C l h − m + n̺ − l − n − k u k L ∞ , since J is bounded (from above and below) by a constant times h − n − . ByLemma 1.2.1 one has k T u k L ∞ . k u k L ∞ provided m < − n − + n̺ − l and l > n , i.e. if m < − n − + n ( ̺ − (cid:3) EGULARITY OF FOURIER INTEGRAL OPERATORS 49 Global L p - L p and L p - L q boundedness of Fourier integral opera-tors. In this section we shall state and prove our main boundedness resultsfor Fourier integral operators. Here, we prove results both for smooth andrough operators with phases satisfying various non-degeneracy conditions.As a first step, interpolation yields the following global L p results: Theorem 2.4.1. Let T be a Fourier integral operator given by (0.0.1) withamplitude a ∈ S m̺,δ , ≤ ̺ ≤ , ≤ δ ≤ , and a phase function ϕ ( x, ξ ) ∈ Φ satisfying the strong non-degeneracy condition. Setting λ := min(0 , n ( ̺ − δ )) , suppose that either of the following conditions hold: ( a ) 1 ≤ p ≤ and m < n ( ̺ − (cid:18) p − (cid:19) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) + λ (cid:18) − p (cid:19) ; or ( b ) 2 ≤ p ≤ ∞ and m < n ( ̺ − (cid:18) − p (cid:19) + ( n − (cid:18) p − (cid:19) + λp ; or ( c ) p = 2 , ≤ ̺ ≤ , ≤ δ < , and m = λ . Then there exists a constant C > such that k T u k L p ≤ C k u k L p . Proof. The proof is a direct consequence of interpolation of the the L boundedness result of Theorem 2.1.1 with the L boundedness of Theo-rem 2.2.6 on one hand, and the interpolation of the latter with the L ∞ boundedness result of Theorem 2.3.1. The details are left to the reader. (cid:3) Theorem 2.4.2. Let T be a Fourier integral operator given by (0.0.1) withamplitude a ∈ L ∞ S m̺ , ≤ ̺ ≤ and a strongly non-degenerate phase func-tion ϕ ( x, ξ ) ∈ Φ . Suppose that either of the following conditions hold: ( a ) 1 ≤ p ≤ and m < np ( ̺ − 1) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) ; or ( b ) 2 ≤ p ≤ ∞ and m < n ̺ − 1) + ( n − (cid:18) p − (cid:19) . Then there exists a constant C > such that k T u k L p ≤ C k u k L p . Proof. The proof follows once again from interpolation of the L bound-edness result of Theorem 2.1.1 with the L boundedness of Theorem 2.2.1on one hand, and the interpolation of the latter with the L ∞ boundednessresult of Theorem 2.3.1. (cid:3) As an immediate consequence of the theorem above one has Corollary 2.4.3. For a Fourier integral operator T with amplitude a ∈ L ∞ S m and a strongly non-degenerate phase function ϕ ( x, ξ ) ∈ Φ , one has L p boundedness for p ∈ [1 , ∞ ] provided m < − ( n − | p − | . Using Sobolev embedding theorem one can also show the following L p − L q estimates for rough Fourier integral operators. Theorem 2.4.4. Suppose that (1) T is a Fourier integral operator with an amplitude a ∈ S m̺,δ , ≤ ̺ ≤ , ≤ δ ≤ and a strongly non-degenerate phase function ϕ ( x, ξ ) ∈ Φ , with either of the following conditions: ( a ) 1 ≤ p ≤ q ≤ and m < n ( ̺ − (cid:18) q − (cid:19) − ( n − (cid:18) p − (cid:19) + λ (cid:18) − q (cid:19) + 1 q − p ; or ( b ) 2 ≤ p ≤ q ≤ ∞ and m < n ( ̺ − (cid:18) − q (cid:19) + ( n − (cid:18) q − p − (cid:19) + λq + 1 q − p . (2) T is a Fourier integral operator with an amplitude a ∈ L ∞ S m̺ , ≤ ̺ ≤ and a strongly non-degenerate phase function ϕ ( x, ξ ) ∈ Φ , with either of the following conditions: ( a ) 1 ≤ p ≤ q ≤ and m < nq ( ̺ − − ( n − (cid:18) p − (cid:19) + 1 q − p ; or ( b ) 2 ≤ p ≤ q ≤ ∞ and m < n ̺ − 1) + ( n − (cid:18) q − p − (cid:19) + 1 q − p . Then there exists a constant C > such that k T u k L q ≤ C k u k L p . Proof. We give the details of the proof only for (1) (a). The rest of the proofis similar to that of (1)(a), through the use of Theorem 2.4.1 part (b) orTheorem 2.4.2. Condition m < n ( ̺ − q − − ( n − p − )+ λ (1 − q )+ q − p yields the existence of of a real number s with(2.4.1) n (cid:18) p − q (cid:19) ≤ s < n ( ̺ − (cid:18) q − (cid:19) + (cid:0) n − (cid:1)(cid:18) − q (cid:19) + λ (cid:18) − q (cid:19) − m. Therefore writing T = T (1 − ∆) s (1 − ∆) − s , Leibniz rule reveals that theamplitude of T (1 − ∆) s belongs to L ∞ S m + s̺ and since m + s < n ( ̺ − (cid:18) q − (cid:19) + (cid:0) n − (cid:1)(cid:18) − q (cid:19) + λ (cid:18) − q (cid:19) , EGULARITY OF FOURIER INTEGRAL OPERATORS 51 Theorem 2.4.1 part (a) yields k T u k L q = k T (1 − ∆) s (1 − ∆) − s u k L q . k (1 − ∆) − s u k L q . k u k L p , where the very last estimate is a direct consequence of (2.4.1) and theSobolev embedding theorem. Hence k T u k L q . k u k L p for the above rangesof p , q and m , and the proof is complete. (cid:3) Chapter Global and local weighted L p boundedness ofFourier integral operators The purpose of this chapter is to establish boundedness results for a fairlywide class of Fourier integral operators on weighted L p spaces with weightsbelonging to Muckenhoupt’s A p class. We also prove these results for Fourierintegral operators whose phase functions and amplitudes are only boundedand measurable in the spatial variables and exhibit suitable symbol typebehavior in the frequency variable. We will start by recalling some factsfrom the theory of A p weights which will be needed in this section. There-after we prove a couple of uniform stationary phase estimates for oscillatoryintegrals and then proceed with the weighted boundedness for the low fre-quency portions of Fourier integral operators. Before proceeding with ourclaims about the weighted boundedness of the high frequency part of Fourierintegral operators, a discussion of a counterexample leads us to a rank con-dition on the phase function ϕ ( x, ξ ) which is crucial for the validity of theweighted boundedness (with A p weights) of Fourier integral operators. Us-ing interpolation and extrapolation, we can prove an endpoint weighted L p boundedness theorem for operators within a specific class of amplitudes andall A p weights, which is shown to be sharp in a case of particular interestand can also be invariantly formulated in the local case. Finally we showthe L p boundedness of a much wider class of operators for some subclassesof the A p weights.3.1. Tools in proving weighted boundedness. The following results arewell-known and can be found in their order of appearance in [13], [22] and[36]. Theorem 3.1.1. Suppose p > and w ∈ A p . There exists an exponent q < p , which depends only on p and [ w ] A p , such that w ∈ A q . There exists ε > , which depends only on p and [ w ] A p , such that w ε ∈ A p . Theorem 3.1.2. For < q < ∞ , the Hardy-Littlewood maximal operatoris bounded on L qw if and only if w ∈ A q . Consequently, for ≤ p < ∞ , M p is bounded on L pw if and only if w ∈ A q/p Theorem 3.1.3. Suppose that ϕ : R n → R is integrable non-increasing andradial. Then, for u ∈ L , we have Z ϕ ( y ) u ( x − y ) d y ≤ k ϕ k L M u ( x ) for all x ∈ R n . The following result of J.Rubio de Francia is also basic in the context ofweighted norm inequalities. Theorem 3.1.4 (Extrapolation Theorem) . If k T u k L p w ≤ C k u k L p w for somefixed p ∈ (1 , ∞ ) and all w ∈ A p , then one has in fact k T u k L pw ≤ C k u k L pw for all p ∈ (1 , ∞ ) and w ∈ A p . A pointwise uniform bound on oscillatory integrals. Before we proceedwith the main estimates we would need a stationary-phase estimate whichwill enable us to control certain integrals depending on various parametersuniformly with respect to those parameters. Here and in the sequel wedenote the Hessian in ξ of the phase function ϕ ( x, ξ ) by ∂ ξξ ϕ ( x, ξ ). Lemma 3.1.5. For λ ≥ , let a λ ( x, ξ ) ∈ L ∞ S with seminorms that areuniform in λ and let supp ξ a λ ( x, ξ ) ⊂ B (0 , λ µ ) for some µ ≥ . Assumethat ϕ ( x, ξ ) ∈ L ∞ S and | det ∂ ξξ ϕ ( x, ξ ) | ≥ c > for all ( x, ξ ) ∈ supp a λ .Then one has (3.1.1) sup x ∈ R n (cid:12)(cid:12)(cid:12) Z e iλϕ ( x,ξ ) a λ ( x, ξ ) d ξ (cid:12)(cid:12)(cid:12) . λ nµ − n Proof. We start with the case µ = 0. The matrix inequality k A − k ≤ C n | det A | − k A k n − with A = ∂ ξξ ϕ ( x, ξ ) and the assumptions on ϕ, yieldthe uniform bound (in x and ξ )(3.1.2) (cid:13)(cid:13) [ ∂ ξξ ϕ ( x, ξ )] − (cid:13)(cid:13) ≤ C n | det ∂ ξξ ϕ ( x, ξ ) | − (cid:13)(cid:13) ∂ ξξ ϕ ( x, ξ ) (cid:13)(cid:13) n − . . Looking at the map κ x : ξ 7→ ∇ ξ ϕ ( x, ξ ), we observe that Dκ x ( ξ ) = ∂ ξξ ϕ ( x, ξ ),where Dκ x ( ξ ) denotes the Jacobian matrix of the map κ x , and that κ x is adiffeomorphism due to the condition on ϕ in the lemma. Therefore Dκ − x ( ˜ ξ ) = (cid:2) ∂ ξξ ϕ ( x, κ − x ( ˜ ξ )) (cid:3) − , which using (3.1.2) yields uniform bounds for k Dκ − x ( ˜ ξ ) k , hence | κ − x ( ˜ ξ ) − κ − x (˜ η ) | ≤ (cid:13)(cid:13) Dκ − x (cid:13)(cid:13) × | ˜ ξ − ˜ η | . | ˜ ξ − ˜ η | . This applied to ˜ ξ = κ x ( ξ ), ˜ η = κ x ( η ) implies that | ξ − η | . | κ x ( ξ ) − κ x ( η ) | = |∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( x, η ) | . (3.1.3)We set I ( λ, x ) := Z e iλϕ ( x,ξ ) a λ ( x, ξ ) d ξ and compute | I ( λ, x ) | = Z Z e iλ ( ϕ ( x,ξ ) − ϕ ( x,ξ + η )) a λ ( x, ξ ) a λ ( x, ξ + η ) d ξ d η. EGULARITY OF FOURIER INTEGRAL OPERATORS 53 We decompose the integral in η into two integrals, one on | η | ≤ δ and theother on | η | > δ , and this yields the estimate | I ( λ, x ) | . δ n + Z ∞ δ Z S n − (cid:12)(cid:12)(cid:12)(cid:12) Z e iλr ϕ ( x,ξ ) − ϕ ( x,ξ + rθ ) r a λ ( x, ξ ) a λ ( x, ξ + rθ ) d ξ (cid:12)(cid:12)(cid:12)(cid:12) d θ r n − d r. Using the uniform lower bound on the gradient of the phase in (3.1.3), weget the uniform lower bound (cid:12)(cid:12)(cid:12)(cid:12) ∇ ξ ϕ ( x, ξ ) − ∇ ξ ϕ ( x, ξ + rθ ) r (cid:12)(cid:12)(cid:12)(cid:12) & . Therefore, applying the non-stationary phase estimate of [18, Theorem 7.7.1]to the right-hand side integral yields | I ( λ, x ) | . δ n + λ − n − Z ∞ δ r − n − r n − d r . δ n + δ − λ − n − . We now optimize this estimate by choosing δ = λ − and obtain the bound | I ( λ, x ) | ≤ Cλ − n , with a constant uniform in x .In the case µ > 0, we cover the ball B (0 , λ µ ) with balls of radius 1 and indoing that, one would need O ( λ nµ ) balls of unit radius. This will obviouslyprovide a covering of the ξ support of a λ with balls of radius 1 and we takea finite smooth partition of unity θ j ( ξ ) , j = 1 , . . . , O ( λ nµ ) , subordinate tothis covering with | ∂ αξ θ j | ≤ C α . Now by the first part of this proof we have(3.1.4) (cid:12)(cid:12)(cid:12) Z e iλϕ ( x,ξ ) a λ ( x, ξ ) θ j ( ξ ) d ξ (cid:12)(cid:12)(cid:12) ≤ Cλ − n with a constant that is uniform in x and j . Finally summing in j andremembering that there are roughly O ( λ nµ ) terms involved yields the desiredestimate. (cid:3) Weighted local and global low frequency estimates. For the low fre-quency portion of the Fourier integral operators studied in this section weare once again able to handle the L p boundedness for all p ∈ [1 , ∞ ], usingLemma 1.2.10 and imposing suitable conditions on the phases. Proposition 3.1.6. Let ̺ ∈ [0 , and suppose either:(a) a ( x, ξ ) ∈ L ∞ S m̺ is compactly supported in the x variable, m ∈ R and ϕ ( x, ξ ) ∈ L ∞ Φ ; or(b) a ( x, ξ ) ∈ L ∞ S m̺ , m ∈ R and ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ Then for all χ ( ξ ) ∈ C ∞ supported near the origin, the Fourier integraloperator T u ( x ) = 1(2 π ) n Z a ( x, ξ ) χ ( ξ ) e iϕ ( x,ξ ) b u ( ξ ) d ξ is bounded on L pw for < p < ∞ and all w ∈ A p .Proof. (a) The operator T can be written as T u ( x ) = R K ( x, y ) u ( x − y ) d y with K ( x, y ) = 1(2 π ) n Z e iψ ( x,ξ ) − i h y,ξ i χ ( ξ ) a ( x, ξ ) d ξ, where ψ ( x, ξ ) := ϕ ( x, ξ ) − h x, ξ i satisfies the estimatesup | ξ |6 =0 | ξ | − | α | | ∂ αξ ψ ( x, ξ ) | ≤ C α , for | α | ≥ 1, on support of the amplitude a . Therefore setting b ( x, ξ ) := a ( x, ξ ) χ ( ξ ) e iψ ( x,ξ ) we have that b is bounded andsup | ξ |6 =0 | ξ | − | α | | ∂ αξ b ( x, ξ ) | < ∞ , for | α | ≥ x and using Lemma 1.2.10, we have for all µ ∈ [0 , | K l ( x, y ) | ≤ C l h y i − n − µ , for all x . From this and Theorem 3.1.3, it follows that | T u ( x ) | . M u ( x )and Theorem 3.1.2 yields the L pw boundedness of T .(b) The only difference from the local case, is that instead of the assumptionof compact support in x , the assumption ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ yields that b ( x, ξ ) in the previous proof satisfies the very same estimate, whereupon thesame argument will conclude the proof. (cid:3) Counterexamples in the context of weighted boundedness. The following counterexample going back to [24], shows that for smoothFourier integral operators (smooth phases and well as amplitudes), the non-degeneracy of the phase function i.e. the non-vanishing of the determinant ofthe mixed hessian of the phase, is not enough to yield weighted L p bound-edness, unless one is prepared to loose a rather unreasonable amount ofderivatives. Counterexample 1. Let ϕ ( x, ξ ) = h x, ξ i + ξ which is non-degenerate butrank ∂ ξξ ϕ = 0 , and let a ( x, ξ ) = h ξ i m with − n < m < 0. Then it has been shown in [40]that for 1 < p < ∞ there exists w ∈ A p and f ∈ L pw such that the Fourierintegral operator T u ( x ) = (2 π ) − n R e i h x,ξ i + iξ h ξ i m b u ( ξ ) d ξ, does not belongto L pw .However, as the following proposition shows, even with a phase of thetype above, one can prove weighted L p boundedness provided certain (com-paratively large) loss of derivatives. Proposition 3.2.1. Let a ( x, ξ ) ∈ L ∞ S m , m ≤ − n and ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ . Then T a,ϕ u ( x ) := R e iϕ ( x,ξ ) σ ( x, ξ ) b u ( ξ ) d ξ is bounded on L pw for w ∈ A p and < p < ∞ . This result is sharp. EGULARITY OF FOURIER INTEGRAL OPERATORS 55 Proof. For the low frequency part of the Fourier integral operator we couldfor example use Proposition 3.1.6. For the high frequency part we mayassume that a ( x, ξ ) = 0 when ξ is in a neighborhood of the origin. The proofin the case m < − n can be done by a simple integration by parts argumentin the integral defining the Schwartz kernel of the operator. Hence the mainpoint of the proof is to deal with the case m = − n. Now the Fourier integraloperator T a,ϕ can be written as(3.2.1) T a,ϕ u ( x ) = Z e iϕ ( x,ξ ) a ( x, ξ ) b u ( ξ ) d ξ = Z σ ( x, ξ ) e i h x,ξ i b u ( ξ ) d ξ with σ ( x, ξ ) = a ( x, ξ ) e i ( ϕ ( x,ξ ) −h x,ξ i ) and we can assume that σ ( x, ξ ) = 0 in aneighborhood of the origin. Therefore, since we have assumed that ϕ ( x, ξ ) −h x, ξ i ∈ L ∞ Φ , the operator T a,ϕ = σ ( x, D ) is a pseudo-pseudodifferentialoperator in the sense of [23], belonging to the class OP L ∞ S − n . We use theLittlewood-Paley partition of unity and decompose the symbol as σ ( x, ξ ) = ∞ X k =1 σ k ( x, ξ )with σ k ( x, ξ ) = σ ( x, ξ ) ϕ k ( ξ ), k ≥ 1. We have σ k ( x, D )( u )( x ) = 1(2 π ) n Z σ k ( x, ξ ) b u ( ξ ) e i h x,ξ i d ξ for k ≥ 1. We note that σ k ( x, D ) u ( x ) can be written as σ k ( x, D ) u ( x ) = Z K k ( x, y ) u ( x − y ) d y with K k ( x, y ) = 1(2 π ) n Z σ k ( x, ξ ) e i h y,ξ i d ξ = ˇ σ k ( x, y ) , where ˇ σ k here denotes the inverse Fourier transform of σ k ( x, ξ ) with respectto ξ . One observes that | σ k ( x, D ) u ( x ) | p = (cid:12)(cid:12)(cid:12) Z K k ( x, y ) u ( x − y ) d y (cid:12)(cid:12)(cid:12) p = (cid:12)(cid:12)(cid:12) Z K k ( x, y ) ω ( y ) 1 ω ( y ) u ( x − y ) d y (cid:12)(cid:12)(cid:12) p , with weight functions ω ( y ) which will be chosen momentarily. Therefore,H¨older’s inequality yields(3.2.2) | σ k ( x, D ) u ( x ) | p ≤ (cid:26) Z | K k ( x, y ) | p ′ | ω ( y ) | p ′ d y (cid:27) pp ′ (cid:26) Z | u ( x − y ) | p | ω ( y ) | p d y (cid:27) , where p + p ′ = 1 . Now for an l > np , we define ω by ω ( y ) = ( , | y | ≤ | y | l , | y | > . By Hausdorff-Young’s theorem and the symbol estimate, first for α = 0 andthen for | α | = l , we have Z | K k ( x, y ) | p ′ d y ≤ (cid:26) Z | σ k ( x, ξ ) | p d ξ (cid:27) p ′ p . (cid:26) Z | ξ |∼ k − npk d ξ (cid:27) p ′ p (3.2.3) . kp ′ ( np − n ) , and Z | K k ( x, y ) | p ′ | y | p ′ l d y . (cid:26) Z |∇ lξ σ k ( x, ξ ) | p d ξ (cid:27) p ′ p (3.2.4) . (cid:26) Z | ξ |∼ k − kpn d ξ (cid:27) p ′ p . kp ′ ( np − n ) . Hence, splitting the integral into | y | ≤ , | y | > (cid:26) Z | K k ( x, y ) | p ′ | ω ( y ) | p ′ d y (cid:27) pp ′ . (cid:8) kp ′ ( np − n ) (cid:9) pp ′ = 2 kp ( np − n ) . Furthermore, using Theorem 3.1.3 we have Z | u ( x − y ) | p | ω ( y ) | p d y . (cid:0) M p u ( x ) (cid:1) p with a constant that only depends on the dimension n . Thus (3.2.2) yields(3.2.5) | σ k ( x, D ) u ( x ) | p . k ( np − n ) (cid:0) M p u ( x ) (cid:1) p Summing in k using (3.2.5) we obtain | T a,ϕ u ( x ) | = | σ ( x, D ) u ( x ) | . (cid:26) ∞ X k =1 | σ k ( x, D ) u ( x ) | p (cid:27) p . (cid:0) M p u ( x ) (cid:1)(cid:18) ∞ X k =1 k ( np − n ) (cid:19) p We observe that the series above converges for any p > (cid:3) The above discussion suggests that without further conditions on thephase, which as it will turn out amounts to a rank condition, the weightednorm inequalities of the type considered in this paper will be false. Thefollowing counterexample shows that, even if the phase function satisfiesan appropriate rank condition, there is a critical threshold on the loss ofderivatives, below which the weighted norm inequalities will fail. Counterexample 2. We consider the following operator T m = e i | D | h D i m EGULARITY OF FOURIER INTEGRAL OPERATORS 57 and the following functions w b ( x ) = | x | − b , f µ ( x ) = Z e − i | ξ | + ix · ξ h ξ i − µ d ξ. As was mentioned in Example 1 in Chapter 1, w b ∈ A if 0 ≤ b < n, fromwhich it also follows that w := w b χ | x | < ∈ A for 0 ≤ b < n , were χ A denotesthe characteristic function of the set A . Now if µ < m + n then we claimthat | T m f µ ( x ) | ≥ C mµ | x | µ − m − n on | x | ≤ 2. Indeed, we have T m f µ ( x ) = Z e ix · ξ h ξ i m − µ d ξ which is a radial function equal to | S n − | | x | µ − m − n Z ∞ c d ω ( r ) (cid:0) | x | + r (cid:1) m − µ r n − d r. If we denote by g µm the function given by the integral, and take a dyadicpartition of unity 1 = ψ + P ∞ j =1 ψ (2 − j · ) then g µm ( s ) = Z c d ω ( r )( s + r ) m − µ r n − ψ ( r ) d r + ∞ X j =1 jn Z ∞ c d ω (2 j r )( s + 2 j r ) m − µ r n − ψ ( r ) d r. The first term is continuous if m − µ + n > c d ω (2 j r ) and integrating by parts yields that the series in thesecond term of g µm is a smooth function of s . Moreover g µm (0) = Z e i h ξ,e i | ξ | m − µ d ξ = C n | e | − n − m + µ = 0 , since the inverse Fourier transform of a radial homogeneous distribution ofdegree α is a radial homogeneous distribution of order − n − α. This provesthe claim. From this claim it follows that Z | T m f µ | p w d x ≥ C µm Z | x |≤ | x | ( µ − m − n ) p − b d x, and therefore T m f µ / ∈ L pw if ( µ − m − n ) p − b ≤ − n. Now, we also have | f µ ( x ) | ≤ A µ (cid:12)(cid:12) − | x | (cid:12)(cid:12) µ − n +12 + B µ on | x | ≤ 2. This isbecause the function f µ is radial f µ ( x ) = | S n − | Z ∞ c d ω ( r | x | ) e − ir (1 + r ) µ r n − d r and using the information on the Fourier transform of the measure of thesphere, f µ ( x ) = | S n − | X ± Z ∞ e − ir (1 ±| x | ) a ± ( r | x | )(1 + r ) µ r n − d r (3.2.6) where | ∂ α a ± ( r ) | ≤ C α h r i − n − − α . We now use a dyadic partition of unity 1 = ψ + P ∞ k =1 ψ (2 − k · ) on the integraland obtain a sum of terms of the form2 kn Z ∞ e − k ir (1 ±| x | ) a ± (2 k r | x | ) ψ ( r )(1 + 2 k r ) µ r n − | {z } = b ± k ( r, | x | ) d r with | ∂ αr b ± k ( r, | x | ) | ≤ C α − ( n − − µ + α ) k . Integration by parts yields | f µ | ≤ C + C X k | −| x ||≤ − ( n +12 − µ ) k + C X k | −| x || > − ( n +12 − µ + N ) k (cid:12)(cid:12) − | x | (cid:12)(cid:12) − N ≤ C + C ′ (cid:12)(cid:12) − | x | (cid:12)(cid:12) µ − n +12 . Hence one has Z | f µ | p w d x ≤ A µ Z | x |≤ (cid:12)(cid:12) − | x | (cid:12)(cid:12) µp − n +12 p | x | − b d x + B µ Z | x |≤ | x | − b d x, which in turn yields f µ ∈ L pw if µ > n +12 − p and 0 ≤ b < n . From theestimates above it follows that if 1 < p < ∞ and T m is bounded on L pw then m ≤ − n − − p . (3.2.7)Indeed if T m is bounded on L pw then we have − m > b − np + n − µ for all 0 ≤ b < n and all µ > n +12 − p . Letting µ tend to n +12 − p we obtain m ≤ − b − np − n − − p for all 0 ≤ b < n , and letting b tend to n yields (3.2.7).Now by Theorem 3.1.4 if T m is bounded on L qw for a fixed q > w ∈ A q , then by extrapolation it is bounded on all L pw for all w ∈ A p and all1 < p < ∞ , therefore since w ∈ A ⊂ A p , we conclude that T m is boundedon L pw for all 1 < p < ∞ , which implies that m has to satisfy the inequality m ≤ − n − − p , for all 1 < p < ∞ . Letting p tend to 1, we obtain m ≤ − n + 12 , which is the desired critical threshold for the validity of the weighted L p boundedness of the class of Fourier integral operators under considerationin this paper. EGULARITY OF FOURIER INTEGRAL OPERATORS 59 We end up with an example showing that in the most general situationone cannot expect global L p weighted estimates unless the phase satisfiessome slightly stronger property than a rank condition. Counterexample 3. Let B be the unit ball in R n , we consider the operator T u ( x ) = (1 B ∗ u )(2 x )and suppose that this operator is bounded on L pw with bound C p = C (cid:0) [ w ] A p (cid:1) only depending on [ w ] A p k T u k L pw ≤ C p k u k L pw , u ∈ S ( R n ) . Note that the A p -constant [ w ] A p is scale invariant. If we apply the estimateto the function u ( ε · ) and the weight w ( ε · ) and scale it, we obtain k T ε u k L pw ≤ C p k u k L pw , u ∈ S ( R n )with T ε u = ε − n (1 εB ∗ u )(2 x ). Since T ε u tends to u (2 x ) in L pw , by letting ε tend to 0 we deduce from the former k u (2 · ) k L pw ≤ C p k u k L pw , u ∈ S ( R n ) . After a change of variable, this would imply2 − n Z | u ( x ) | p w ( x/ 2) d x ≤ C pp Z | u ( x ) | p w ( x ) d x for all u ∈ S ( R n ) , hence w ( x/ ≤ C pp n w ( x ) . This means that one can expect weighted L p estimates for T only if w satisfies a doubling property. Note that T can be written as a sum of Fourierintegral operators with amplitudes in S − n +12 , with phases of the form ϕ ± = 2 h x, ξ i ± | ξ | which satisfy a rank condition. Nevertheless, one has ϕ ± − h x, ξ i / ∈ L ∞ Φ . In particular, one cannot generally expect global weighted estimate forFourier integral operators with phases such that ϕ − h x, ξ i / ∈ L ∞ Φ unlessthe weight satisfies some doubling property.3.3. Invariant formulation in the local boundedness. The aim of thissection is to give an invariant formulation of the rank condition on thephase function, which will be used to prove our weighted norm inequalitiesfor Fourier integral operators. In Counterexample 1 we saw that a rankcondition is necessary for the validity of weighted L p estimates. The fol-lowing discussion will enable us to give an invariant formulation of the localweighted norm inequalities for operators with amplitudes in S − ( n +1)2 ̺ + n ( ̺ − ̺, − ̺ ,̺ ∈ [ , . We refer the reader to [20] for the properties of Fourier integral operators with amplitudes in S m̺, − ̺ , ̺ ∈ ( , 1] and the paper by A. Greenleafand G. Uhlmann for the case when ̺ = . The central object in the theory of Fourier integral operators is the canon-ical relation C ϕ = (cid:8) ( x, ∂ x ϕ ( x, ξ ) , ∂ ξ ϕ ( x, ξ ) , ξ ) : ( x, ξ ) ∈ supp a (cid:9) in T ∗ R n × T ∗ R n . We consider the following projection on the space variables C ϕ ⊂ T ∗ R nx × T ∗ R ny ≃ T ∗ ( R nx,y ) π (cid:15) (cid:15) π ( C ) ⊂ R nxy with π ( x, ξ, y, η ) = ( x, y ) . The differential of this projection is given byd π ( t x , t ξ , t y , t η ) = ( t x , t y ) , t ξ = ∂ xx ϕ t x + ∂ xξ ϕ t η t y = ∂ ξx ϕ t x + ∂ ξξ ϕ t η so that its kernel is given by (cid:8) (0 , ∂ xξ ϕ t η , , t η ) : t η ∈ ker ∂ ξξ ϕ (cid:9) This impliesrank d π = codim ker d π = codim ker ∂ ξξ ϕ = n + rank ∂ ξξ ϕ. Our assumption on the phase rank ∂ ξξ ϕ = n − π = 2 n − . Using these facts, we will later on be able to show that if T is a Fourierintegral operator with amplitude in S − n +12 ̺ + n ( ̺ − ̺, − ̺ with ̺ ∈ [ , 1] whosecanonical relation C is locally the graph of a symplectomorphism, and ifrank d π = 2 n − C , with π : C → R n defined by π ( x, ξ, y, η ) = ( x, η ), thenthere exists a constant C > k T u k L pw, loc ≤ k u k L pw, comp for all w ∈ A p and all 1 < p < ∞ . However, we will actually provelocal weighted L p boundedness of operators with amplitudes in the class L ∞ S − n +12 ̺ + n ( ̺ − ̺ with ̺ ∈ [0 , 1] for which the invariant formulation abovelacks meaning, and therefore to keep the presentation of the statements assimple as possible, we will not always (with the exception of Theorem 3.4.5)state the local boundedness theorems in an invariant form. EGULARITY OF FOURIER INTEGRAL OPERATORS 61 Weighted local and global L p boundedness of Fourier integraloperators. We start this by showing the local weighted L p boundednessof Fourier integral operators. In Counterexample 1 which was related toFourier integral operators with linear phases, the Hessian in the frequencyvariable ξ of the phase function vanishes identically. This suggests thatsome kind of condition on the Hessian of the phase might be required. Itturns out that the condition we need can be formulated in terms of the rankof the Hessian of the phase function in the frequency variable. Further-more Counterexample 2 which was related to the wave operator, suggests acondition on the order of the amplitudes involved. It turns out that theseconditions, appropriately formulated, will indeed yield weighted bounded-ness of a wide range of Fourier integral operators even having rough phasesand amplitudes. Theorem 3.4.1. Let a ( x, ξ ) ∈ L ∞ S m̺ with m < − n +12 ̺ + n ( ̺ − and ̺ ∈ [0 , be compactly supported in the x variable. Let the phase function ϕ ( x, ξ ) ∈ C ∞ ( R × R \ homogeneous of degree in ξ satisfy rank ∂ ξξ ϕ ( x, ξ ) = n − . Then the corresponding Fourier integral operator is L pw bounded for < p < ∞ and all w ∈ A p .Proof. The low frequency part of the Fourier integral operator is handledby Proposition 3.1.6 part (a). For the high frequency portion, once againwe use a Littlewood-Paley decomposition in the frequency variables as inSubsection 1.2.1 to reduce the operator to its semiclassical piece T h u ( x ) = (2 π ) − n Z Z e i ( ϕ ( x,ξ ) −h y,ξ i ) χ ( hξ ) a ( x, ξ ) u ( y ) d y d ξ with χ ( ξ ) smooth and supported in the annulus ≤ | ξ | ≤ . Now if we let θ ( x, ξ ) := ϕ ( x, ξ ) − h x, ξ i , then we have(3.4.1) |∇ ξ θ | . a . Furthermore, if(3.4.2) T h ( x, y ) = (2 π ) − n Z e i ( θ ( x,ξ ) −h y,ξ i ) χ ( hξ ) a (cid:0) x, ξ (cid:1) d ξ, then in order to get useful pointwise estimates for the operator T h we wouldneed to estimate the kernel T h ( x, y ). Localising in frequency and rotatingthe coordinates we may assume that a ( x, ξ ) is supported in a small conicneighbourhood of a ξ = e n . At this point we split the modulus of T h into | T h u ( x ) | ≤ Z | y | > k∇ ξ θ k L ∞ + Z | y |≤ k∇ ξ θ k L ∞ | T h ( x, y ) | | u ( x − y ) | d y := I + II . where there are obviously no critical points on the domain of integrationfor I . Estimate of I. Making the change of variables ξ → h − ξ we obtain T h u ( x ) = (2 π ) − n h − n Z Z e ih ( θ ( x,ξ ) −h y,ξ i ) χ ( ξ ) a (cid:0) x, ξ/h (cid:1) u ( y ) d y d ξ. Here, since 2 k∇ ξ θ k L ∞ < k∇ ξ θ k L ∞ < | y | , we have(3.4.3) |∇ ξ θ ( x, ξ ) − y | ≥ | y | − k∇ ξ θ k L ∞ > | y | . Also, | ∂ αξ ( θ ( x, ξ ) − h y, ξ i ) | ≤ C α for all | α | ≥ x and y . There-fore using the non-stationary phase estimate in [18] Theorem 7.7.1 to (3.4.2),the derivative estimates on a ( x, ξ/h ) and (3.4.3) yield for N > | T h ( x, y ) | . h − n h N X α ≤ N sup n(cid:12)(cid:12) ∂ α ( χa ( x, ξ/h ) (cid:12)(cid:12) |∇ ξ θ ( x, ξ ) − y | | α |− N o . h − m − n + N̺ | y | − N . h − m − n + N̺ h y i − N , where we have used the fact that | y | > I . Hence taking N > n , Theorem3.1.3 yields(3.4.4) I . h − m − n + N̺ Z h y i − N | u ( x − y ) | d y . h − m − n + N̺ M u ( x ) . Estimate of II. Making a change of variables ξ → h − ̺ ξ we obtain T h ( x, y ) = (2 π ) − n h − n̺ Z e ih − ̺ ( θ ( x,ξ ) −h y,ξ i ) χ ( h − ̺ +1 ξ ) a (cid:0) x, h − ̺ ξ (cid:1) d ξ := (2 π ) − n h − n̺ Z e ih − ̺ ( ϕ ( x,ξ ) −h y,ξ i ) a h (cid:0) x, ξ (cid:1) d ξ where a h ( x, ξ ) is compactly supported in x , supported in ξ in the annu-lus h ̺ − ≤ | ξ | ≤ h ̺ − and is uniformly bounded together with all itsderivatives in ξ , by h − m . Here the assumption, rank ∂ ξξ ϕ ( x, ξ ) = n − ξ , yields that ker ∂ ξξ ϕ ( x, ξ ) = span { ξ } = span { e n } . Therefore by thedefinition of θ ( x, ξ )(3.4.5) det ∂ ξ ′ ξ ′ θ ( x, e n ) = 0 . The assumption that a has its ξ -support in a small conic neighborhood of e n implies that if that support is sufficiently small, then(3.4.6) | det ∂ ξ ′ ξ ′ θ ( x, ξ ) | ≥ , ( x, ξ ) ∈ supp a h . Finally, due to the restriction 1 + 2 k∇ ξ θ k L ∞ ≥ | y | and (3.4.1), one has(3.4.7) | ∂ αξ ( θ ( x, ξ ) − h y, ξ i ) | ≤ C α for all | α | ≥ x and y .Hence θ ( x, ξ ) − h y ′ , ξ ′ i and h m a h satisfy all the assumptions of the sta-tionary phase estimate in Lemma 3.1.5 with λ = h − ̺ and λ µ = h ̺ − , weobtain (cid:12)(cid:12)(cid:12)(cid:12) Z e ih − ̺ ( ϕ ( x,ξ ) −h y,ξ i ) a h (cid:0) x, ξ (cid:1) d ξ ′ (cid:12)(cid:12)(cid:12)(cid:12) . h − m h n − ̺ h ( n − ̺ − and using the fact that the integral in ξ n lies on a segment of size h ̺ − , weget(3.4.8) | T h ( x, y ) | . h − n̺ h − m h n − ̺ − (1 − ̺ ) n . h − m − n +12 ̺ − (1 − ̺ ) n . This yields that II . h − m − n +12 ̺ − (1 − ̺ ) n Z | y |≤ k∇ ξ θ k L ∞ | u ( x − y ) | d y . h − m − n +12 ̺ − (1 − ̺ ) n M u ( x )Now adding I and II , taking N > n large enough, using Lemma 1.2.1, theassumption m < − n +12 ̺ + n ( ̺ − 1) and Theorem 3.1.2, we will obtain thedesired result. (cid:3) Here we remark that the condition on the rank of the Hessian on themetric is quite natural and is satisfied by phases like h x, ξ i + | ξ | and h x, ξ i + p | ξ ′ | − | ξ ′′ | where ξ = ( ξ ′ , ξ ′′ ) (with an amplitude supported in | ξ ′ | > | ξ ′′ | ),but if we put a slightly stronger condition than the rank condition on thephase, then it turns out that we would not only be able to extend the localresult to a global one but also lower the regularity assumption on the phasefunction to the sole assumption of measurability and boundedness in thespatial variable x . Therefore the estimates we provide below will aim toachieve this level of generality. Having the uniform stationary phase abovein our disposal we will proceed with the main high frequency estimates, butbefore that let us fix a notation. Notation. Given an n × n matrix M , we denote by det n − M the determinantof the matrix P M P where P is the projection to the orthogonal complementof ker M . Theorem 3.4.2. Let a ( x, ξ ) ∈ L ∞ S m̺ with m < − n +12 ̺ + n ( ̺ − and ̺ ∈ [0 , . Let the phase function ϕ ( x, ξ ) satisfy | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > . Furthermore, suppose that ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ Then the associatedFourier integral operator is bounded on L pw , for < p < ∞ and w ∈ A p .Proof. As before, the low frequency part of the Fourier integral operator istreated using Proposition 3.1.6 part (b). For the high frequency part we fol-low the same line of argument as in the proof of Theorem 3.4.1. More specifi-cally at the level of showing the estimate (3.4.1), the lack of compact supportin x variable lead us to use our assumption ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ instead,which yields that k∇ ξ θ ( x, ξ ) k L ∞ . 1. Splitting the kernel of the Fourier inte-gral operator into the same pieces I and II as in the proof of Theorem 3.4.1.We estimate the I piece exactly in the same way as before but for piece II weproceed as follows. First of all, the assumption | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > x, ξ ), yields in particular that | det n − ( ∂ ξξ ( θ ( x, ξ ) − h y, ξ i ) | ≥ c > k∇ ξ θ k L ∞ ≥ | y | and (3.4.1), one also has | ∂ αξ ( θ ( x, ξ ) − h y, ξ i ) | ≤ C α for all | α | ≥ x and y , which yields that θ ( x, ξ ) − h y, ξ i ∈ L ∞ S . This means that all the assumptions in Lemma 3.1.5 are satisfiedand therefore we get II . h − m − n +12 ̺ − (1 − ̺ ) n × Z | y |≤ k∇ ξ θ k L ∞ | u ( x − y ) | d y . h − m − n +12 ̺ − (1 − ̺ ) n M u ( x )Now adding I and II , using Lemma 1.2.1 and the assumptions N > n , m < − n +12 ̺ + n ( ̺ − 1) and Theorem 3.1.2 all together yield the desiredresult. (cid:3) Endpoint weighted boundedness of Fourier integral operators. The Fol-lowing interpolation lemma is the main tool in proving the endpoint weightedboundedness of Fourier integral operators. Lemma 3.4.3. Let ≤ ̺ ≤ , < p < ∞ and m < m . Suppose that ( a ) the Fourier integral operator T with amplitude a ( x, ξ ) ∈ L ∞ S m ̺ andthe phase ϕ ( x, ξ ) are bounded on L pw for a fixed w ∈ A p , and ( b ) the Fourier integral operator T with amplitude a ( x, ξ ) ∈ L ∞ S m ̺ andthe same type of phases as in ( a ) are bounded on L p ,where the bounds depend only on a finite number of seminorms in Definition1.1.2 . Then, for each m ∈ ( m , m ) , operators with amplitudes in L ∞ S m̺ arebounded on L pw ν , and (3.4.9) ν = m − mm − m . Proof. For a ∈ L ∞ S m̺ we introduce a family of amplitudes a z ( x, ξ ) := h ξ i z a ( x, ξ ), where z ∈ Ω := { z ∈ C ; m − m ≤ Re z ≤ m − m } . It iseasy to see that, for | α | ≤ C with C large enough and z ∈ Ω, | ∂ αξ a z ( x, ξ ) | . (1 + | Im z | ) C h ξ i Re z + m − ̺ | α | , for some C . We introduce the operator T z u := w m − m − zp ( m − m T a z ,ϕ (cid:0) w − m − m − zp ( m − m u (cid:1) . First we consider the case of p ∈ [1 , A p ⊂ A which in turnimplies that both w p and w − p belong to L p loc and therefore for z ∈ Ω, T z isan analytic family of operators in the sense of Stein-Weiss [38]. Now we claimthat for z ∈ C with Re z = m − m , the operator (1 + | Im z | ) − C T a z ,ϕ is bounded on L pw with bounds uniform in z . Indeed the amplitude of thisoperator is (1 + | Im z | ) − C a z ( x, ξ ) which belongs to L ∞ S m ̺ with constantsuniform in z . Thus, the claim follows from assumption (a). Consequently, EGULARITY OF FOURIER INTEGRAL OPERATORS 65 we have k T z u k pL p = (1 + | Im z | ) pC (cid:13)(cid:13)(cid:13) (1 + | Im z | ) − C w m − m − z p ( m − m T a z ,ϕ (cid:0) w − m − m − z p ( m − m u (cid:1)(cid:13)(cid:13)(cid:13) pL p . (1 + | Im z | ) pC (cid:13)(cid:13)(cid:13) w − m − m − z p ( m − m u (cid:13)(cid:13)(cid:13) pL pw = (1 + | Im z | ) pC k u k pL p , where we have used the fact that (cid:12)(cid:12) w m − m − z m − m (cid:12)(cid:12) = w .Similarly if z ∈ C with Re z = m − m , then (cid:12)(cid:12) w m − m − z m − m (cid:12)(cid:12) = 1 andthe amplitude of the operator (1 + | Im z | ) − C T a z belongs to L ∞ S m ̺ withconstants uniform in z . Assumption (b) therefore implies that k T z u k pL p . (1 + | Im z | ) pC k u k pL p . Therefore the complex interpolation of operators in [5] implies that for z = 0we have k T u k pL p = (cid:13)(cid:13)(cid:13) w m − mp ( m − m T a,ϕ (cid:0) w − m − mp ( m − m u (cid:1)(cid:13)(cid:13)(cid:13) pL p ≤ C k u k pL p . Hence, setting v = w − m − mp ( m − m u this reads(3.4.10) k T a,ϕ v k pL pwν ≤ C k u k pL pwν , where ν = ( m − m ) / ( m − m ). This ends the proof in the case 1 ≤ p ≤ T is bounded on L pw ,then its adjoint T ∗ is bounded on L p ′ w − p ′ . Therefore, in the case p > 2, weapply the above proof to T ∗ a,ϕ , with p ′ ∈ [1 , 2) and v = w − p ′ , which yieldsthat T ∗ a,ϕ is bounded on L p ′ v ν and since w ∈ A p , we have v ∈ A p ′ and so T a is bounded on L pv (1 − p ) ν = L pw (1 − p ′ )(1 − p ) ν = L pw ν , which concludes the proof ofthe theorem. (cid:3) Now we are ready to prove our main result concerning weighted bound-edness of Fourier integral operators. This is done by combining our previousresults with a method based on the properties of the A p weights and complexinterpolation. Theorem 3.4.4. Let a ( x, ξ ) ∈ L ∞ S − n +12 ̺ + n ( ̺ − ̺ and ̺ ∈ [0 , . Supposethat either ( a ) a ( x, ξ ) is compactly supported in the x variable and the phase func-tion ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ , is positively homogeneous of degree in ξ and satisfies, det ∂ xξ ϕ ( x, ξ ) = 0 as well as rank ∂ ξξ ϕ ( x, ξ ) = n − ; or ( b ) ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ , ϕ satisfies the rough non-degeneracy condi-tion as well as | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > .Then the operator T a,ϕ is bounded on L pw for p ∈ (1 , ∞ ) and all w ∈ A p .Furthermore, for ̺ = 1 this result is sharp. Proof. The sharpness of this result for ̺ = 1 is already contained in Coun-terexample 2 discussed above. The key issue in the proof is that both as-sumptions in the statement of the theorem guarantee the weighted bound-edness for m < − n +12 ̺ + n ( ̺ − . The rest of the argument is rather abstractand goes as follows. By the extrapolation Theorem 3.1.4, it is enough toshow the boundedness of T a,ϕ in weighted L spaces with weights in theclass A . Let us fix m such that − n +12 ̺ + n ( ̺ − < m < n ( ̺ − . By Theorem 3.1.1, given w ∈ A choose ε such that w ε ∈ A . For this ε take m < − n +12 ̺ + n ( ̺ − 1) in such a way that the straight line L that joins points with coordinates ( m , ε ) and ( m , x = − n +12 ̺ + n ( ̺ − 1) in the ( x, y ) plane in a point with coordinates( − n +12 ̺ + n ( ̺ − , m on the negative x axis as close as we like to thepoint − n +12 ̺ + n ( ̺ − . By Theorem 3.4.1, given ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ ξ, satisfying rank ∂ ξξ ϕ ( x, ξ ) = n − , and a ∈ L ∞ S m ̺ , the Fourier integral operators T a,ϕ are bounded opera-tors on L w ε for w ∈ A and, by Theorem 2.2.1, or rather its proof, theFourier integral operators with amplitudes in L ∞ S m ̺ compactly supportedin the spatial variable, and phases that are positively homogeneous of de-gree 1 in the frequency variable and satisfying the non-degeneracy conditiondet ∂ xξ ϕ ( x, ξ ) = 0 , are bounded on L . Therefore, by Lemma 3.4.3, theFourier integral operators T a,ϕ with phases and amplitudes as in the state-ment of the theorem are bounded operators on L w . The proof of part (b) issimilar and uses instead the interpolation between the weighted bounded-ness of Theorem 3.4.2 and the unweighted L boundedness result of Theorem2.2.7. The details are left to the interested reader. (cid:3) If we don’t insist on proving weighted boundedness results valid for all A p weights then, it is possible to improve on the number of derivatives in theestimates and push the numerology almost all the way to those that guarantyunweighted L p boundedness. Therefore, there is the trade-off between thegenerality of weights and loss of derivatives as will be discussed below. Theorem 3.4.5. Let C ⊂ ( R n × R n \ × ( R n × R n \ , be a conic manifoldwhich is locally a canonical graph, see e.g. [20] for the definitions. Let π : C → R n × R n denote the natural projection. Suppose that for every λ =( x , ξ , y , η ) ∈ C there is a conic neighborhood U λ ⊂ C of λ and a smoothmap π λ : C ∩ U λ → C , homogeneous of degree , with rank dπ λ = 2 n − , such that π = π ◦ π λ . Let T ∈ I m̺, comp ( R n × R n ; C ) ( see [20]) with ≤ ̺ ≤ and m < ( ̺ − n ) | p − | . Then for all w ∈ A p there exists α ∈ (0 , depending on m , ̺ , δ , p and [ w ] A p such that, for all ε ∈ [0 , α ] , the Fourier integral operator T a,ϕ is bounded on L pw ε where < p < ∞ . EGULARITY OF FOURIER INTEGRAL OPERATORS 67 Proof. By the equivalence of phase function theorem, which for < ̺ ≤ ̺ = is due to Greenleaf-Uhlmann [15],we reduce the study of operator T to that of a finite linear combination ofoperators which in appropriate local coordinate systems have the form(3.4.11) T a u ( x ) = (2 π ) − n Z Z e iϕ ( x,ξ ) − i h y,ξ i a ( x, ξ ) u ( y ) dy dξ, where a ∈ S m̺, − ̺ with compact support in x variable, and ϕ homogeneousof degree 1 in ξ, with det ∂ xξ ϕ ( x, ξ ) = 0 and rank ∂ ξξ ϕ ( x, ξ ) = n − . If m ≤− n +12 ̺ + n ( ̺ − , then Theorem 3.4.4 case ( a ) yields the result, so we assumethat m > − n +12 ̺ + n ( ̺ − . Also by assumption of the theorem we can finda m , which we shall fix from now on, such that m < m < ( ̺ − n ) | p − | and m < − n +12 ̺ + n ( ̺ − . Now a result of Seeger-Sogge-Stein, namely Theorem5.1 in [33] yields that operators T a with amplitudes compactly supported inthe x variable in the class S m ̺, − ̺ , and phase functions ϕ ( x, ξ ) satisfyingrank ∂ ξξ ϕ ( x, ξ ) = n − L p . Furthermore by Theorem 3.4.4case ( a ) , The operators T a with a ∈ S m ̺,δ are bounded on L pw , p ∈ (1 , ∞ ) . Therefore, Lemma 3.4.3 yields the desired result. (cid:3) A similar result also holds for operators with amplitudes in S m̺,δ withwithout any rank condition on the phase function. Theorem 3.4.6. Let a ( x, ξ ) ∈ S m̺,δ , ϕ ( x, ξ ) be a strongly non-degeneratephase function with ϕ ( x, ξ ) − h x, ξ i ∈ Φ , and λ := min(0 , n ( ̺ − δ )) , witheither of the following ranges of parameters: (1) 0 ≤ ̺ ≤ , ≤ δ ≤ , ≤ p ≤ and m < n ( ̺ − (cid:18) p − (cid:19) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) + λ (cid:18) − p (cid:19) ; or (2) 0 ≤ ̺ ≤ , ≤ δ ≤ , ≤ p ≤ ∞ and m < n ( ̺ − (cid:18) − p (cid:19) + ( n − (cid:18) p − (cid:19) + λp . Then for all w ∈ A p there exists α ∈ (0 , depending on m , ̺ , δ , p and [ w ] A p such that, for all ε ∈ [0 , α ] , the Fourier integral operator T a,ϕ is bounded on L pw ε where < p < ∞ . Proof. The proof is similar to that of Theorem 3.4.5 and we only considerthe proof in case (1), since the other case is similar. We observe that sinceΦ ⊂ Φ and h x, ξ i ∈ Φ , the assumption ϕ ( x, ξ ) − h x, ξ i ∈ Φ , implies that ϕ ( x, ξ ) ∈ Φ . To proceed with the proof we can assume that m > − n becauseotherwise by Proposition 3.2.1 there is nothing to prove. The assumptionof the theorem, enables us to find m such that m < m < n ( ̺ − (cid:18) p − (cid:19) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) + λ (cid:18) − p (cid:19) and m < − n. Now Theorem 2.4.1 yields that operators T a with amplitudesin the class S m ̺,δ , and strongly non-degenerate phase functions ϕ ( x, ξ ) ∈ Φ are bounded on L p . Furthermore Proposition 3.2.1 yields that operators T a with b ∈ S m ̺,δ are bounded on L pw . Therefore, Lemma 3.4.3 yields once againthe desired result for the range 1 < p ≤ . (cid:3) Finally, for operators with non-smooth amplitudes we can prove the fol-lowing: Theorem 3.4.7. Let a ( x, ξ ) ∈ L ∞ S m̺ , ≤ ̺ ≤ , and let ϕ ( x, ξ ) − h x, ξ i ∈ Φ , with a strongly non-degenerate ϕ and either of the following ranges ofparameters: ( a ) 1 ≤ p ≤ and m < np ( ̺ − 1) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) ; or ( b ) 2 ≤ p ≤ ∞ and m < n ̺ − 1) + ( n − (cid:18) p − (cid:19) . Then for all w ∈ A p there exists α ∈ (0 , depending on m , ̺ , p and [ w ] A p such that, for all ε ∈ [0 , α ] , the Fourier integral operator T a,ϕ is bounded on L pw ε . Proof. The proof is a modification of that of Theorem 3.4.6, where onealso uses Theorem 2.4.2. The straightforward modifications are left to theinterested reader. (cid:3) Here we remark that if in the proofs of Theorems 3.4.6 and 3.4.7 wewould have used Theorem 3.4.4 case (b) instead of Proposition 3.2.1 in theproof above, then we would obtain a similar result, under the condition | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > α as com-pared to those in the statements of Theorems 3.4.6 and 3.4.7. Chapter Applications in harmonic analysis and partialdifferential equations In this chapter, we use our weighted estimates proved in the previouschapter to show the boundedness of constant coefficient Fourier integral op-erators in weighted Triebel-Lizorkin spaces. This is done using vector-valuedinequalities for the aforementioned operators. We proceed by establishingweighted and unweighted L p boundedness of commutators of a wide classof Fourier integral operators with functions of bounded mean oscillation(BMO), where in some cases we also show the weighted boundedness of it-erated commutators. The boundedness of commutators are proven using theweighted estimates of the previous chapter and a rather abstract complexanalytic method. Finally in the last section, we prove global unweighted and EGULARITY OF FOURIER INTEGRAL OPERATORS 69 local weighted estimates for the solutions of the Cauchy problem for m -thand second order hyperbolic partial differential equations on R n . Estimates in weighted Triebel-Lizorkin spaces. In this section,we investigate the problem of the boundedness of certain classes on Fourierintegral operators in weighted Triebel-Lizorkin spaces. The result obtainedhere can be viewed as an example of the application of weighted norm in-equalities for FIO’s. The main reference for this section is [13] and we willrefer the reader to that monograph for the proofs of the statements concern-ing vector-valued inequalities. Definition 4.1.1. An operator T defined in L p ( µ ) ( this denotes L p spaceswith measure dµ ) is called linearizable if there exits a linear operator U defined on L p ( µ ) whose values are Banach space-valued functions such that (4.1.1) | T u ( x ) | = k U u ( x ) k B , u ∈ L p ( µ )We shall use the following theorem due to Garcia-Cuerva and Rubio deFrancia, whose proof can be found in [13]. Theorem 4.1.2. Let T j be a sequence of linearizable operators and supposethat for some fixed r > and all weights w ∈ A r (4.1.2) Z | T j u ( x ) | r w ( x ) d x ≤ C r ( w ) Z | u ( x ) | r w ( x ) d x, with C r ( w ) depending on the weight w . Then for < p, q < ∞ and w ∈ A p one has the following weighted vector-valued inequality (4.1.3) (cid:13)(cid:13)(cid:13)n X j | T j u j | q o q (cid:13)(cid:13)(cid:13) L pw ≤ C p,q ( w ) (cid:13)(cid:13)(cid:13)n X j | u j | q o q (cid:13)(cid:13)(cid:13) L pw . Next we recall the definition of the weighted Triebel-Lizorkin spaces, seee.g. [39] Definition 4.1.3. Start with a partition of unity P ∞ j =0 ψ j ( ξ ) = 1 , where ψ ( ξ ) is supported in | ξ | ≤ , ψ j ( ξ ) for j ≥ is supported in j − ≤ | ξ | ≤ j +1 and | ∂ α ψ j ( ξ ) | ≤ C α − j | α | , for j ≥ . Given s ∈ R , ≤ p ≤ ∞ , ≤ q ≤ ∞ , and w ∈ A p , a tempered distribution u belongs to the weightedTriebel-Lizorkin space F s,pq, w if (4.1.4) k u k F s,pq, w := (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) ∞ X j =0 | js ψ j ( D ) u | q (cid:27) q (cid:13)(cid:13)(cid:13)(cid:13) L pw < ∞ , From this it follows that for a linear operator T the estimate(4.1.5) k T u k F s ′ ,pq, w . k u k F s,pq, w , is implied by(4.1.6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) ∞ X j =0 | js ′ ψ j ( D ) T u | q (cid:27) q (cid:13)(cid:13)(cid:13)(cid:13) L pw . (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) ∞ X j =0 | js ψ j ( D ) u | q (cid:27) q (cid:13)(cid:13)(cid:13)(cid:13) L pw . Now if one is in the situation where [ T, ψ j ] = 0 , then (4.1.6) is equivalent to(4.1.7) (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) ∞ X j =0 | j ( s ′ − s ) T (2 js ψ j ( D ) u ) | q (cid:27) q (cid:13)(cid:13)(cid:13)(cid:13) L pw . (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) ∞ X j =0 | js ψ j ( D ) u | q (cid:27) q (cid:13)(cid:13)(cid:13)(cid:13) L pw . Therefore, setting T j := 2 j ( s ′ − s ) T and u j := 2 js ψ j u and assuming that s ≥ s ′ , (4.1.7) has the same form as the the vector-valued inequality (4.1.3)and follows from (4.1.2). Using these facts yields the following result, Theorem 4.1.4. Let a ( ξ ) ∈ S − n +12 , and ϕ ∈ Φ with | det n − ∂ ξξ ϕ ( ξ ) | ≥ c > . Then for s ≥ s ′ , < p < ∞ , < q < ∞ , and w ∈ A p , the Fourier integraloperator T u ( x ) = 1(2 π ) n Z R n e iϕ ( ξ )+ i h x,ξ i a ( ξ )ˆ u ( ξ ) dξ satisfies the estimate (4.1.8) k T u k F s ′ ,pq, w . k u k F s,pq, w Proof. We only need to check that T j = 2 j ( s ′ − s ) T satisfies (4.1.2). But thisfollows from the assumption s ≥ s ′ and Theorem 3.4.4 part (b) concerningthe global weighted boundedness of Fourier integral operators. (cid:3) Commutators with BMO functions. In this section we show howour weighted norm inequalities can be used to derive the L p boundednessof commutators of functions of bounded mean oscillation with a wide rangeof pseudodifferential operators. We start with the precise definition of afunction of bounded mean oscillation. Definition 4.2.1. A locally integrable function b is of bounded mean oscil-lation if (4.2.1) k b k BMO := sup B | B | Z B | b ( x ) − b B | d x < ∞ , where the supremum is taken over all balls in R n . We denote the set of suchfunctions by BMO . For b ∈ BMO it is well-known that for any γ < n e , there exits a constant C n,γ so that for u ∈ BMO and all balls B ,(4.2.2) 1 | B | Z B e γ | b ( x ) − b B | / k u k BMO d x ≤ C n,γ . For this see [14, p. 528]. The following abstract lemma will enable us to provethe L p boundedness of the BMO commutators of Fourier integral operators. Lemma 4.2.2. For < p < ∞ , let T be a linear operator which is boundedon L pw ε for all w ∈ A p for some fixed ε ∈ (0 , . Then given a function b ∈ BMO , if Ψ( z ) := R e zb ( x ) T ( e − zb ( x ) u )( x ) v ( x ) d x is holomorphic in a disc | z | < λ , then the commutator [ b, T ] is bounded on L p . EGULARITY OF FOURIER INTEGRAL OPERATORS 71 Proof. Without loss of generality we can assume that k b k BMO = 1. Wetake u and v in C ∞ with k u k L p ≤ k v k L p ′ ≤ 1, and an applicationof H¨older’s inequality to the holomorphic function Ψ( z ) together with theassumption on v yield | Ψ( z ) | p ≤ Z e p Re zb ( x ) | T ( e − z b ( x ) u ) | p d x. Our first goal is to show that the function Ψ( z ) defined above is bounded ona disc with centre at the origin and sufficiently small radius. At this pointwe recall a lemma due to Chanillo [7] which states that if k b k BMO = 1, thenfor 2 < s < ∞ , there is an r depending on s such that for all r ∈ [ − r , r ], e rb ( x ) ∈ A s . Taking s = 2 p in Chanillo’s lemma, we see that there is some r depending on p such that for | r | < r , e rb ( x ) ∈ A p . Then we claim that if R := min ( λ, εr p )and | z | < R then | Ψ( z ) | . 1. Indeed since R < εr p we have | Re z | < εr p andtherefore | p Re zε | < r . Therefore Chanillo’s lemma yields that for | z | < R , w := e p Re zε b ( x ) ∈ A p and since e p Re zb ( x ) = w ε , the assumption of weightedboundedness of T and the L p bound on u , imply that for | z | < R | Ψ( z ) | p ≤ Z e p Re zb ( x ) | T ( e − z b ( x ) u ) | p d x = Z w ε | T ( e − z b ( x ) u ) | p d x . Z w ε | e − z b ( x ) u | p d x = Z w ε w − ε | u | p d x . , and therefore | Ψ( z ) | . | z | < R . Finally, using the holomorphicity ofΨ( z ) in the disc | z | < R , Cauchy’s integral formula applied to the circle | z | = R ′ < R , and the estimate | Ψ( z ) | . 1, we conclude that | Ψ ′ (0) | ≤ π Z | z | = R ′ | Ψ( ζ ) || ζ | | d ζ | . . By construction of Ψ( z ), we actually have that Ψ ′ (0) = R v ( x )[ b, T ] u ( x ) d x and the definition of the L p norm of the operator [ b, T ] together with theassumptions on u and v yield at once that [ f, T ] is a bounded operator from L p to itself for p . (cid:3) The following lemma guarantees the holomorphicity ofΨ( z ) := Z e zb ( x ) T a,ϕ ( e − zb ( x ) u )( x ) v ( x ) d x, when T a,ϕ is a L bounded Fourier integral operator. Lemma 4.2.3. Assume that ϕ is a strongly non-degenerate phase functionin the class Φ and suppose that either: ( a ) T a,ϕ is a Fourier integral operator with a ∈ S m̺,δ , ≤ ̺ ≤ , ≤ δ < , m = min(0 , n ( ̺ − δ )) or ( b ) T a,ϕ is a Fourier integral operator with a ∈ L ∞ S m̺ , ≤ ̺ ≤ ,m < n ( ̺ − . Then given b ∈ BMO with k b k BMO = 1 and u and v in C ∞ , there exists λ > such that the function Ψ( z ) := R e zb ( x ) T a,ϕ ( e − zb ( x ) u )( x ) v ( x ) d x isholomorphic in the disc | z | < λ .Proof. (a) From the explicit representation of Ψ( z )(4.2.3) Ψ( z ) = Z Z Z a ( x, ξ ) e iϕ ( x,ξ ) − i h y,ξ i e zb ( x ) − zb ( y ) v ( x ) u ( y ) d y d ξ d x we can without loss of generality assume that a ( x, ξ ) has compact x − support.For f ∈ S and ε ∈ (0 , 1) we take χ ( ξ ) ∈ C ∞ ( R n ) such that χ (0) = 1 andset(4.2.4) T a ε ,ϕ f ( x ) = Z a ( x, ξ ) χ ( εξ ) e iϕ ( x,ξ ) ˆ f ( ξ ) dξ. Using this and the assumption of the compact x − support of the amplitude,one can see that for f ∈ S , lim ε → T a ε ,ϕ f = T a,ϕ f in the Schwartz class S and also lim ε → k T a ε ,ϕ f − T a,ϕ f k L = 0 . Since a ( x, ξ ) χ ( εξ ) ∈ S m̺,δ withseminorms that are independent of ε, it follows from our assumptions onthe amplitude and the phase and Theorem 2.2.6 that k T a ε ,ϕ f k L . k f k L with a L bound that is independent of ε. Therefore, the density of S in L yields(4.2.5) lim ε → k T a ε ,ϕ f − T a,ϕ f k L = 0 , for all f ∈ L . Now if we defineΨ ε ( z ) := Z e zb ( x ) T a ε ,ϕ ( e − zb ( x ) u )( x ) v ( x ) d x (4.2.6) = Z Z Z a ( x, ξ ) χ ( εξ ) e iϕ ( x,ξ ) − i h y,ξ i e zb ( x ) − zb ( y ) v ( x ) u ( y ) d y d ξ d x, then the integrand in the last expression is a holomorphic function of z .Furthermore, from (4.2.2) and the assumption k b k BMO = 1 one can deducethat for all p ∈ [1 , ∞ ) and | z | < γp , and all compact sets K one has(4.2.7) Z K e ± p Re z b ( x ) d x ≤ C γ ( K ) . This fact shows that ue − z b and ve z b both belong to L p for all p ∈ [1 , ∞ )provided | z | < γp . These together with the compact support in ξ of theintegrand defining Ψ ε ( z ) and uniform bounds on the amplitude in x , yield EGULARITY OF FOURIER INTEGRAL OPERATORS 73 the absolute convergence of the integral in (4.2.6) and therefore Ψ ε ( z ) is aholomorphic function in | z | < . Now we claim that for γ as in (4.2.2),lim ε → sup | z | < γ | Ψ ε ( z ) − Ψ( z ) | = 0 . Indeed, since γ < , one has for | z | < γ | Ψ ε ( z ) − Ψ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z v ( x ) e zb ( x ) (cid:2) T a ε ,ϕ − T a,ϕ (cid:3) ( e − zb u )( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k v e zb k L (cid:13)(cid:13) [ T a ε ,ϕ − T a,ϕ ]( u e − zb ) (cid:13)(cid:13) L ≤ k v k L ∞ (cid:26)Z supp v e zb ( x ) d x (cid:27) k [ T a ε ,ϕ − T a,ϕ ]( u e − zb ) k L . Therefore, using (4.2.7) with p = 2 and (4.2.5) yield thatlim ε → sup | z | < γ | Ψ ε ( z ) − Ψ( z ) | = 0and hence Ψ( z ) is a holomorphic function in | z | < λ with λ ∈ (0 , γ ) . (b) Using the semiclassical reduction in the proof of Theorem 2.2.1, wedecompose the operator T a,ϕ into low and high frequency parts, T and T h . From this it follows that Ψ ( z ) := R e zb ( x ) T ( e − zb ( x ) u )( x ) v ( x ) d x can bewritten as(4.2.8) Ψ ( z ) = Z (cid:26) Z Z e iϕ ( x,ξ ) − i h y,ξ i χ ( ξ ) a ( x, ξ ) u ( y ) e − zb ( y ) d y d ξ (cid:27) v ( x ) e zb ( x ) d x and Ψ h ( z ) := R e zb ( x ) T h ( e − zb ( x ) u )( x ) v ( x ) d x is given by(4.2.9) Ψ h ( z ) = h − n Z (cid:26) Z Z e ih ϕ ( x,ξ ) − ih h y,ξ i χ ( ξ ) a ( x, ξ/h ) u ( y ) e − zb ( y ) d y d ξ (cid:27) v ( x ) e zb ( x ) d x, Now we claim that for Ψ ( z ) and Ψ h ( z ) are holomorphic in | z | < 1. To seethis, we reason in a way similar to the proof of part (a). Namely, using thecompact support in ξ of the integrands of (4.2.8) and (4.2.9) and uniformbounds on the amplitude in x , yield the absolute convergence of the integralsin (4.2.8) and (4.2.9) and therefore Ψ ( z ) and Ψ h ( z ) are holomorphic func-tions in | z | < . Next we proceed with a uniform estimate (in z ) for Ψ h ( z ).For this we use once again that ue − z b and ve z b both belong to L provided | z | < γ . Therefore the Cauchy-Schwartz inequality and (2.2.3) yield | Ψ h ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z v ( x ) e zb ( x ) T h ( e − zb u )( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) (4.2.10) ≤ k u e − zb k L k T ∗ h ( v e zb ) k L ≤ k u e − zb k L k T h T ∗ h ( v e zb ) k / L k v e zb k / L ≤ h − m − (1 − ̺ ) M/ k u e − zb k L k v e zb k L . h − m − (1 − ̺ ) M/ Hence, | Ψ h ( z ) | . h − m − (1 − ̺ ) M/ and setting h = 2 − j , using m < n ( ̺ − 1) andsumming in j we would have a uniformly convergent series of holomorphicfunctions which therefore converges to a holomorphic function and by takinga λ in the interval (0 , γ ) we conclude the holomorphicity of Ψ( z ) in | z | < λ. (cid:3) Lemmas 4.2.2 and 4.2.3 yield our main result concerning commutatorswith BMO functions, namely Theorem 4.2.4. Suppose either ( a ) T ∈ I m̺, comp ( R n × R n ; C ) with ≤ ̺ ≤ and m < ( ̺ − n ) | p − | , satisfies all the conditions of Theorem 3.4.5 or; ( b ) T a,ϕ with a ∈ S m̺,δ , ≤ ̺ ≤ , ≤ δ ≤ , λ = min(0 , n ( ̺ − δ )) and ϕ ( x, ξ ) is a strongly non-degenerate phase function with ϕ ( x, ξ ) −h x, ξ i ∈ Φ , where in the range < p ≤ ,m < n ( ̺ − (cid:18) p − (cid:19) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) + λ (cid:18) − p (cid:19) ; and in the range ≤ p < ∞ m < n ( ̺ − (cid:18) − p (cid:19) + ( n − (cid:18) p − (cid:19) + λp ; or ( c ) T a,ϕ with a ∈ L ∞ S m̺ , ≤ ̺ ≤ and ϕ is a strongly non-degeneratephase function with ϕ ( x, ξ ) − h x, ξ i ∈ Φ , where in the range < p ≤ , m < np ( ̺ − 1) + (cid:0) n − (cid:1)(cid:18) − p (cid:19) , and for the range ≤ p < ∞ m < n ̺ − 1) + ( n − (cid:18) p − (cid:19) . Then for b ∈ BMO , the commutators [ b, T ] and [ b, T a,ϕ ] are bounded on L p with < p < ∞ . Proof. (a) One reduces T to a finite sum of operators of the form T a asin the proof of Theorem 3.4.5. That theorem also yields the existence ofan ε ∈ (0 , 1) such that T a with a ∈ S m̺, − ̺ and m < ( ̺ − n ) | p − | is EGULARITY OF FOURIER INTEGRAL OPERATORS 75 L pw ε − bounded. Moreover, since m < ( ̺ − n ) | p − | ≤ , and 1 − ̺ ≤ ̺, Theorem 2.2.6 yields that T a is L bounded. Hence, ifΨ( z ) = Z e zb ( x ) T a ( e − zb ( x ) u )( x ) v ( x ) d x with u and v in C ∞ , then Lemma 4.2.3 yields that Ψ( z ) is holomorphic ina neighbourhood of the origin. Therefore, Lemma 4.2.2 implies that thecommutator [ b, T a ] is bounded on L p and the linearity of the commutator in T allows us to conclude the same result for a finite linear combinations ofoperators of the same type as T a . This ends the proof of part (a).(b) The proof of this part is similar to that of part (a). Here we observethat for any ranges of p in the statement of the theorem, the order of theamplitude m < min(0 , n ( ̺ − δ )) and so T a,ϕ is L bounded. Now, applicationof 3.4.6, Theorem 2.2.6 and Lemma 4.2.3 part (a), concludes the proof.(c) The proof of this part is similar to that of part (b). For any rangesof p, the order of the amplitude m < n ( ̺ − 1) and so T a,ϕ is L bounded.Therefore, Theorem 3.4.7, Theorem 2.2.1 and Lemma 4.2.3 part (b), yieldthe desired result. (cid:3) Finally, the weighted norm inequalities with weights in all A p classes havethe advantage of implying weighted boundedness of repeated commutators.Namely, one has Theorem 4.2.5. Let a ( x, ξ ) ∈ L ∞ S − n +12 ̺ + n ( ̺ − ̺ and ̺ ∈ [0 , . Supposethat either ( a ) a ( x, ξ ) is compactly supported in the x variable and the phase func-tion ϕ ( x, ξ ) ∈ C ∞ ( R n × R n \ , is positively homogeneous of degree in ξ and satisfies, det ∂ xξ ϕ ( x, ξ ) = 0 as well as rank ∂ ξξ ϕ ( x, ξ ) = n − ; or ( b ) ϕ ( x, ξ ) − h x, ξ i ∈ L ∞ Φ , ϕ satisfies either the strong or the roughnon-degeneracy condition ( depending on whether the phase is spa-tially smooth or not ) , as well as | det n − ∂ ξξ ϕ ( x, ξ ) | ≥ c > .Then, for b ∈ BMO and k a positive integer, the k -th commutator definedby T a,b,k u ( x ) := T a (cid:0) ( b ( x ) − b ( · )) k u (cid:1) ( x ) is bounded on L pw for each w ∈ A p and p ∈ (1 , ∞ ) .Proof. The claims in (a) and (b) are direct consequences of Theorem 3.4.4and Theorem 2.13 in [1]. (cid:3) Applications to hyperbolic partial differential equations. It iswellknown, see e.g. [10], that the Cauchy problem for a strictly hyperbolicpartial differential equation(4.3.1) ( D mt + P mj =1 P j ( x, t, D x ) D m − jt , t = 0 ∂ jt u | t =0 = f j ( x ) , ≤ j ≤ m − can be solved locally in time and modulo smoothing operators by(4.3.2) u ( x, t ) = m − X j =0 m X k =1 Z R n e iϕ k ( x,ξ,t ) a jk ( x, ξ, t ) b f j ( ξ ) d ξ where a jk ( x, ξ, t ) are suitably chosen amplitudes depending smoothly on t and belonging to S − j , , and the phases ϕ k ( x, ξ, t ) also depend smoothly on t, are strongly non-degenerate and belong to the class Φ . This yields thefollowing: Theorem 4.3.1. Let u ( x, t ) be the solution of the hyperbolic Cauchy problem (4.3.1) with initial data f j . Let m p = ( n − | p − | , for a given p ∈ [1 , ∞ ] . If f j ∈ H s + m p − j,p ( R n ) and T ∈ (0 , ∞ ) is fixed, then for any ε > the solution u ( · , t ) ∈ H s − ε,p ( R n ) , satisfies the global estimate (4.3.3) k u ( · , t ) k H s − ε,p ≤ C T m − X j =0 k f j k H s + mp − j,p , t ∈ [ − T, T ] , p ∈ [1 , ∞ ] Proof. The result follows at once from the Fourier integral operator repre-sentation (4.3.2) and Corollary 2.4.3. (cid:3) The representation formula (4.3.2) also yields the following local weightedestimate for the solution of the Cauchy problem for the second order hyper-bolic equation above and in particular for variable coefficient wave equation.In this connection we recall that H sw := { u ∈ S ′ ; (1 − ∆) s u ∈ L pw , w ∈ A p } . Theorem 4.3.2. Let u ( x, t ) be the solution of the hyperbolic Cauchy problem (4.3.1) with m = 2 and initial data f j . 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Yabuta, Calder´on-Zygmund operators and pseudodifferential operators , Comm.Partial Differential Equations (1985), no. 9, 1005–1022 Universit´e Paris 13, Cnrs, umr 7539 Laga, 99 avenue Jean-Baptiste Cl´ement,F-93430 Villetaneuse, France E-mail address : [email protected] Department of Mathematics and the Maxwell Institute for MathematicalSciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom E-mail address ::