Sharp L^p decay estimates for degenerate and singular oscillatory integral operators
aa r X i v : . [ m a t h . C A ] J a n SHARP L p DECAY ESTIMATES FOR DEGENERATE AND SINGULAROSCILLATORY INTEGRAL OPERATORS
SHAOZHEN XU
Abstract.
We consider the following model of degenerate and singular oscillatory integraloperators introduced in [4]:
T f p x q “ ż R e iλS p x,y q K p x, y q ψ p x, y q f p y q dy, (0.1)where the phase functions are homogeneous polynomials of degree n and the singular kernel K p x, y q satisfies suitable conditions related to a real parameter µ . We show that the sharpdecay estimates on L spaces, obtained in [4], can be preserved on more general L p spaceswith an additional condition imposed on the singular kernel. In fact, we obtain that } T f } L p ď C E,S,ψ,µ,n,p λ ´ ´ µn } f } L p , n ´ µn ´ ´ µ ď p ď n ´ µ ´ µ . The case without the additional condition is also discussed.
Contents
1. Introduction 22. First step toward Theorem 1.1 33. Damped operators 54. Local Riesz-Thorin interpolation 115. The remaining case: T ∆ Mathematics Subject Classification. Introduction
The central topic of oscillatory integrals is to find the optimal decay estimates. It was revealedthat property of phase function plays a dominant role. For nondegenerate scalar oscillatoryintegrals, stationary phase method gives precise asymptotics which naturally implies the opti-mal decay estimate. Generally speaking, it is much more complicated to get optimal decay indegenerate cases. For the degenerate oscillatory integrals with real-analytic phases, it was con-jectured by Arnold that the optimal decay is determined by the Newton distance of the phasefunction. By using Hironaka’s celebrated theorem of resolution of singularities [2], Varchenko[15] confirmed Arnold’s conjecture for real-analytic phase functions with some nonsingularconditions and thus built deep connections between real algebraic geometry and oscillatoryintegrals. In harmonic analysis, like Fourier transform, more attentions are paid to those oper-ators with oscillatory kernels. Actually, a general oscillatory integral operator can be writtenas T λ f p x q “ ż R n e iλS p x,y q ψ p x, y q f p y q dy. (1.1)If the phase function is nondegenerate in the sense that the Hessian of S p x, y q is nonvanish-ing in the support of ψ p x, y q . By using T T ˚ -method globally, H¨ormander [3] showed that } T λ } L Ñ L has the sharp decay λ ´ n . However, for the oscillatory integral operators with de-generate phases, T T ˚ -method no longer suits. For polynomial-like phase functions, by using T T ˚ -method locally, Phong and Stein successfully established a uniform oscillatory integralestimate, called Operator van der Corput Lemma, details can be found in [6][7]. By meansof the local operator van der Corput lemma and orthogonality argument, Phong and Steinsucceeded to give the sharp L decay of the oscillatory integral operators with real-analyticphase functions. They also clarified the relation between decay rate and the Newton distanceof the phase function. Unlike Varchenko’s results, which require some nonsingular conditionson phase functions, Phong-Stein’s results in operator setting are verified for all real-analyticphase functions. The difference has an intuitive explanation which has been mentioned in [9]that operators of the form (1.1) fix the directions of the axes, and this turns out to eliminatethe excess freedom in the choice of coordinate systems which was essential in original scalaroscillatory integrals. Based on the pre-mentioned works, Rychkov[11] and Greenblatt [1] fi-nally extended similar sharp results to the oscillatory integral operators with smooth phases.Since L case has been understood well, an interesting question is whether or not the sharpdecay estimate can be preserved on general L p spaces. In [17], Yang got an affirmative answerfor oscillatory integral operators with special homogeneous polynomial phases. Actually, byembedding these operators into a family of analytic operators and establishing the correspond-ing boundedness result between critical spaces H and L , he used the damped estimates in[8] and Stein’s interpolation to give sharp L p decay. In [14], Shi and Yan established sharpendpoint L p decay for arbitrary homogeneous polynomial phase functions. Later, Xiao ex-tended this result to arbitrary analytic phases in [16], another proof see also [13]. It shouldbe pointed out that our argument relies heavily on the techniques developed in the previousarticles.Inserting a singular kernel into (1.1) will produce (0.1) which is exactly what this paper isconcerned with. Motivated by studying harmonic analysis on nilpotent groups, Ricci andStein [10] first considered this kind of polynomial-phase oscillatory operators with a standardCalder´on-Zygmund kernel. They established a result that T is bounded from L p to itself withthe bound independent of the coefficients of the phase function, this shows that the decayproduced by the oscillatory term is in some sense cancelled by the singular kernel. In [4],the author proposed that if some modified size and derivative conditions are imposed on thesingular kernel, decay estimates also exist. One direction to extend this result is to study more p DECAY OF OSCILLATORY INTEGRAL OPERATORS 3 complicated phases, see for instance [18]. Another direction is to see if the same sharp decaycan be preserved on more general L p spaces. Our main result tried in the latter direction andstates as follows. Theorem 1.1.
For (0.1) , if S p x, y q is a homogeneous polynomial of degree n and can bewritten as S p x, y q “ n ´ ÿ k “ a k x n ´ k y k , (1.2) where a a n ´ ‰ , K p x, y q is a C function away from the diagonal satisfying | K p x, y q| ď E | x ´ y | ´ µ , ˇˇ B iy K p x, y q ˇˇ ď E | x ´ y | ´ µ ´ i , (1.3) where the ă µ ă and i “ , or . Moreover, we impose an additional condition on K p x, y q , ˇˇ B ix K p x, y q ˇˇ ď E | x ´ y | ´ µ ´ i . (AC) Then the sharp decay estimate } T f } L p ď C E,S,ψ,µ,n,p λ ´ ´ µn } f } L p (1.4) holds for n ´ µn ´ ´ µ ď p ď n ´ µ ´ µ . Remark 1.2.
The purpose of imposing the condition (AC) is to make duality argument inestablishing Theorem 1.1 and Remark 3.4 possible.The usual Riesz-Thorin interpolation may give a L p estimate but it also bears the loss ofsharp decay. Applying Stein’s complex interpolation method to damped oscillatory integraloperators may provide a sharp L p decay estimate but the proof of boundedness result betweencritical spaces is complicated in general. Inspired by [16][12][13], we shall separate the operatorinto three parts and then apply different interpolations to each one, thus we can get an elegantproof. Notation:
For two positive constants C and C , C À C or C Á C means that there existsan constant C independent of λ and the test function f such that C ď CC or C ě CC .In this paper, all parameters, some constant depends on, shall be listed in the subscript.2. First step toward Theorem 1.1
Along the lines of the proof of [4], (0.1) is decomposed into two main parts:
T f p x q “ ż R e iλS p x,y q K p x, y q ψ p x, y q f p y q dy “ ż R e iλS p x,y q K p x, y q φ ´ p x ´ y q λ n ¯ ψ p x, y q f p y q dy ` ż R e iλS p x,y q K p x, y q ” ´ φ ´ p x ´ y q λ n ¯ı ψ p x, y q f p y q dy : “ T f p x q ` T f p x q , where φ P C p R q and φ p x q ” , | x | ě , , | x | ď . SHAOZHEN XU
To prove our Theorem 1.1, it suffices to verify that (1.4) holds for both T and T . Observethat the kernel of T K p x, y q “ e iλS p x,y q K p x, y q φ ´ p x ´ y q λ n ¯ ψ p x, y q is absolutely integrable. Thus we may ignore the oscillatory term and employ the followingSchur test to prove (1.4) for T . Lemma 2.1.
If the operator
V f p x q “ ż K p x, y q f p y q dy, has a kernel K p x, y q satisfying sup x ż | K p x, y q| dy ď A , sup y ż | K p x, y q| dx ď A , then we have } V } L p Ñ L p ď ˆ A p ` A p ˙ , where ď p ď `8 . To make the present paper self-contained, we give the details of the proof.
Proof. } V } L p Ñ L p “ sup } f } Lp ď } V f } L p “ sup } f } Lp ď sup } g } Lp ď |x V f, g y|ď sup } f } Lp ď sup } g } Lp ď ż ż | K p x, y q| | f | | g | dxdy. Convex inequality and Fubini’s theorem imply ż ż | K p x, y q| | f p y q| | g p x q| dxdy ď ż ż | K p x, y q| ˜ | f p y q| p p ` | g p x q| p p ¸ dxdy ď A p ż | f p y q| p dy ` A p ż | g p x q| p dx. It follows } V } L p Ñ L p ď ´ A p ` A p ¯ , then we complete the proof. (cid:3) Note that sup x ż | K p x, y q| dy ď C E,ψ,µ λ ´ ´ µn , sup y ż | K p x, y q| dx ď C E,ψ,µ λ ´ ´ µn . Thus Lemma 2.1 yields the desired result for T . For T , we may suppose supp p ψ q Ăr´ , s ˆ r´ , s (if not we may impose a dilation on all variables and the dilation factor p DECAY OF OSCILLATORY INTEGRAL OPERATORS 5 can be incorporated into λ ). Choose a cut-off function Ψ P C such that supp p Ψ q Ă r , s and ř l P Z Ψ p l x q ”
1. Thus T can be dyadically decomposed as T f p x q “ ÿ σ ,σ “˘ ÿ j,k ż R e iλS p x,y q K p x, y q ” ´ φ ´ p x ´ y q λ n ¯ı Ψ j p σ x q Ψ k p σ y q ψ p x, y q f p y q dy : “ ÿ j,k T σ ,σ j,k f p x q where Ψ j p x q “ Ψ p j x q , Ψ k p x q “ Ψ p k x q . For convenience, we focus only on the case σ “` , σ “ ` , the remaining cases can be dealt with similarly. We shall still use T and T j,k todenote ř j,k T ` , ` j,k and T ` , ` j,k respectively. Suppose that S xy p x, y q “ c s ź l “ p y ´ α l x q m l r ź l “ Q l p x, y q , (2.1)where c and α j are nonzero and each Q l p x, y q is a positive definite quadratic form. Since wehave restricted our attention to the first quadrant, we may suppose that 0 ă α ă α ă ¨ ¨ ¨ ă α s . From this we know that the Hessian of S vanishes on some lines crossing the origin. Italso reveals an obvious fact s ÿ l “ m l ` r “ n ´ . It should be noted that there is an implicit variety y “ x because of the singular kernel K p x, y q .Before proceeding further, let us introduce some notations. Assume that K is a positiveconstant depending on α , ¨ ¨ ¨ , α s thus on S . Let j " k ( j ! k ) represent j ą k ` K ( j ă k ´ K )such that the size of y -variable( x -variable) is dominant in the Hessian S xy , while j „ k naturally means | j ´ k | ď K . To make full use of the nondegeneracy of the Hessian whenlocalized on dyadic areas, we divide T into three groups as follows. T f p x q “ ÿ j " k T j,k f p x q ` ÿ j „ k T j,k f p x q ` ÿ j ! k T j,k f p x q“ T Y f p x q ` T ∆ f p x q ` T X f p x q . If we can establish (1.4) for T X , T ∆ and T Y individually, then the proof of our main result iscomplete. 3. Damped operators
This section is devoted to proving some damped estimates which will be used to deal with T X and T Y . In fact, our main strategy is to insert T X or T Y into the following two families of SHAOZHEN XU operators T zY f p x q “ ÿ j " k ż R e iλS p x,y q K p x, y q ˇˇˇ S xy ˇˇˇ z ” ´ φ ´ p x ´ y q λ n ¯ı ¨ φ j p x q φ k p y q ψ p x, y q f p y q dy, : “ ÿ j " k D Yj,k f p x q , (3.1) T zX f p x q “ ÿ j ! k ż R e iλS p x,y q K p x, y q ˇˇˇ S xy ˇˇˇ z ” ´ φ ´ p x ´ y q λ n ¯ı ¨ φ j p x q φ k p y q ψ p x, y q f p y q dy : “ ÿ j ! k D Xj,k f p x q . (3.2)For each operator we establish the sharp L decay estimate as well as the endpoint estimate,by adequate interpolations we get the desired results. This idea first appeared in [12] andlater was used in [13] to give a new proof of sharp L p decay of real-analytic oscillatory integraloperators. The sharp L decay estimates for T zX and T zY state as follows. Theorem 3.1.
If the Hessian of the phase function S p x, y q is of the form (2.1) , then for Re p z q “ we have } T zY f } L ď C E, K ,ψ,n,z λ µn ´ } f } L , (3.3) } T zX f } L ď C E, K ,ψ,n,z λ µn ´ } f } L . (3.4)To establish these two estimates, we need to give a local oscillatory estimate. Consider thefollowing damped operator D p B q f p x q “ ż R e iλS p x,y q ˇˇˇ S xy ˇˇˇ K p x, y q ” ´ φ ´ p x ´ y q λ n ¯ı ψ p x, y q f p y q dy, (3.5)where K p x, y q is defined in (1.3), ψ P C and supp ψ Ă B . Now we consider two operators D p B q and D p B q with supports B and B respectively. Here both B and B are rectangularboxes with sides parallel to the axes; in addition, the minor box B will be contained in ahorizontal translate of the major box B . As demonstrated in [8], we give the precise definitionsand assumptions. B “ tp x, y q : a ă x ă b , c ă y ă d u , ρ “ d ´ c ; Ă B “ " p x, y q : a ´ p b ´ a q ă x ă b ` p b ´ a q , c ă y ă d * ; B ˚ “ tp x, y q : a ´ p b ´ a q ă x ă b ` p b ´ a q , c ă y ă d u . We also have the minor box B B “ tp x, y q : a ă x ă b , c ă y ă d u , ρ “ d ´ c . For these two boxes, we have the following assumptions:(A1) We define the span span ( B , B ), as the union of all line segments parallel to the x -axis,which joints a point p x, y q P B with a point p z, y q P B . While we also assume that p DECAY OF OSCILLATORY INTEGRAL OPERATORS 7 S xy does not change sign in the span span ( B , B ) and satisfies ν ď min Ă B ˇˇˇ S xy ˇˇˇ ď Aν, (3.6)max span p B , B q ˇˇˇ S xy ˇˇˇ ď Aν. (3.7)(A2) B Ă B ˚ , this implies ρ ď ρ .For the cut-off functions ψ j p x, y q , we also assume that(A3) ř k ρ kj ˇˇ B ky ψ j ˇˇ ď B .Now we formulate an almost-orthogonality principle which we will rely on to establish Theorem3.1. Lemma 3.2.
Under the assumptions (A1)-(A3), we have } D p B q D p B q ˚ } L ď C E,A,B,µ,n λ µn ´ sup B ˇˇˇ S xy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ , (3.8) } D p B q D p B q ˚ } L ď C E,A,B,µ,n λ µn ´ sup B ˇˇˇ S xy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ (3.9) where the constant C A,B,n depends only on A , B and n . Remark 3.3.
In Lemma 3.2, if we set B “ B which we denote by B , we have } D p B q} L ď C E,A,B,µ,n λ µn ´ . This corresponds to Lemma 1 in [4].
Remark 3.4.
Interchanging the roles of x and y in assumptions (A1)-(A3), (3.8) and (3.9)also hold for operators D p B q ˚ D p B q and D p B q ˚ D p B q respectively.Before the formal proof of Lemma Lemma 3.2, we first show how Lemma 3.2 implies Theorem3.1. Proof that Lemma 3.2 implies Theorem 3.1.
Recall that T zX f p x q “ ÿ j ! k D Xj,k f p x q ,T zY f p x q “ ÿ j " k D Yj,k f p x q . We rewrite them as T zX f p x q “ ÿ j ! k D Xj,k f p x q : “ ÿ j D Xj f p x q , (3.10) T zY f p x q “ ÿ j " k D Yj,k f p x q : “ ÿ k D Yk f p x q . (3.11) SHAOZHEN XU
Evidently, the amplitude of D Xj is supported in a rectangle of the form t x „ ´ j u ˆ t ă y ă ´ j u . For j ‰ j , from Lemma 3.2 we know that } D Xj ` D Xj ˘ ˚ } L ď C E, K ,ψ,µ,n λ µn ´ ´ | j ´ j | p n ´ q , } D Xj ` D Xj ˘ ˚ } L ď C E, K ,ψ,µ,n λ µn ´ ´ | j ´ j | p n ´ q . For k ‰ k , from Remark 3.4, we also have } ` D Yk ˘ ˚ D Yk } L ď C E, K ,ψ,µ,n λ µn ´ ´ | k ´ k | p n ´ q , } ` D Yk ˘ ˚ D Yk } L ď C E, K ,ψ,µ,n λ µn ´ ´ | k ´ k | p n ´ q . On the other hand, t D Xj u and t D Yk u are two sequences of operators that are pairwise essentiallydisjoint in x ´ variable and y ´ variable respectively. By Cotlar-Stein Lemma, we can concludeTheorem 3.1 and finish the proof. (cid:3) Now we return to the proof of Lemma 3.2.
Proof.
First, we compute the kernel of D p B q D p B q ˚ as follows. Ker p D p B q D p B q ˚ qp x, z q “ ż e iλ r S p x,y q´ S p z,y qs ” ´ φ ´ p x ´ y q λ n ¯ı ” ´ φ ´ p z ´ y q λ n ¯ı ¨ K p x, y q K p z, y q ˇˇˇ S xy ˇˇˇ ˇˇˇ S zy ˇˇˇ ψ p x, y q ψ p z, y q dy, where supp p ψ q Ă B , supp p ψ q Ă B . SetΦ p x, y, z q “ S y p x, y q ´ S y p z, y q “ ż xz S uy p u, y q du, Λ p x, y, z q “ ” ´ φ ´ p x ´ y q λ n ¯ı ” ´ φ ´ p z ´ y q λ n ¯ı K p x, y q K p z, y q , Λ p x, y, z q “ ˇˇˇ S xy ˇˇˇ ˇˇˇ S zy ˇˇˇ ψ p x, y q ψ p z, y q . For fixed y , if both p x, y q and p z, y q are contained in Ă B , from assumption (A1), we know that | x ´ z | ď | Φ p x, y, z q| ď Aν | x ´ z | , (3.12)In view of the fact that S p x, y q is a polynomial of degree at most n , from Lemma 1.2 in [6],we also have ˇˇˇ B ky Φ p x, y, z q ˇˇˇ ď C A,n ν | x ´ z | ρ ´ k . (3.13)For fixed y , if p z, y q is outside Ă B , we can see that | Φ p x, y, z q| ě ˇˇˇˇż x ˜ x S uy p u, y q du ˇˇˇˇ , where ˜ x is between x and z , in addition, p ˜ x, y q is also on the edge of Ă B . Obviously, | Φ p x, y, z q| ě ν | x ´ ˜ x | . Recall the definition of Ă B , since B Ă B ˚ , then Aν | x ´ z | ě | Φ p x, y, z q| ě ν | x ´ ˜ x | ě ν ¨ b ´ a ě ν ¨ | x ´ z | . (3.14) p DECAY OF OSCILLATORY INTEGRAL OPERATORS 9
Similarly, in this case, ˇˇˇ B ky Φ p x, y, z q ˇˇˇ ď C A,n ν | x ´ z | ρ ´ k . (3.15)On one hand, we have the trivial size estimate | Ker p D p B q D p B q ˚ qp x, z q| ď C E,B,µ λ µn ρ sup B ˇˇˇ S xy ˇˇˇ sup B ˇˇˇ S xy ˇˇˇ ď C E,A,B,µ νρ sup B ˇˇˇ S zy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ . (3.16)On the other hand, we also claim that | Ker p D p B q D p B q ˚ qp x, z q| ď C A,B,n λ µn ρ ´ sup B ˇˇˇ S xy ˇˇˇ sup B ˇˇˇ S xy ˇˇˇ λ ν | x ´ z | (3.17) ď C A,B,n λ µn ρ ´ νλ ν | x ´ z | ¨ sup B ˇˇˇ S zy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ . (3.18)On account of the assumptions, the second inequality is easy. For the first inequality, integra-tion by parts yields Ker p D p B q D p B q ˚ qp x, z q “ iλ ż e iλ r S p x,y q´ S p z,y qs ddy ˆ Λ Λ Φ ˙ dy “ ´ λ ż e iλ r S p x,y q´ S p z,y qs ddy „ ddy ˆ Λ Λ Φ ˙ dy “ ´ λ ż e iλ r S p x,y q´ S p z,y qs ddy „ d Λ dy ¨ Λ Φ ` Λ Φ ¨ ddy ˆ Λ Φ ˙ dy “ ´ λ ż e iλ r S p x,y q´ S p z,y qs d Λ dy ¨ „ ¨ ddy ˆ Λ Φ ˙ ` ddy ˆ Λ Φ ˙ ` ddy „ ¨ ddy ˆ Λ Φ ˙ ¨ Λ ` Λ Φ ¨ d Λ dy dy. From [4], we know that | Λ | ď C E,µ λ µn , ż ˇˇˇˇ d Λ dy ˇˇˇˇ dy ď C E,µ λ µn , ż ˇˇˇˇ d Λ dy ˇˇˇˇ dy ď C E,µ λ µn ρ ´ . The other terms in the integrand can be dealt with using (3.12)-(3.15), details can also befound in [6], here we only list the facts ˇˇˇˇ ¨ ddy ˆ Λ Φ ˙ ` ddy ˆ Λ Φ ˙ˇˇˇˇ ď C A,B,n ρ ´ sup B ˇˇˇ S xy ˇˇˇ sup B ˇˇˇ S xy ˇˇˇ ν | x ´ z | , ˇˇˇˇ ddy „ ¨ ddy ˆ Λ Φ ˙ˇˇˇˇ ď C A,B,n ρ ´ sup B ˇˇˇ S xy ˇˇˇ sup B ˇˇˇ S xy ˇˇˇ ν | x ´ z | , ˇˇˇˇ Λ Φ ˇˇˇˇ ď C A,B sup B ˇˇˇ S xy ˇˇˇ sup B ˇˇˇ S xy ˇˇˇ ν | x ´ z | . Collecting all estimates above, we can conclude (3.17). Also, in view of (3.16), we obtain | Ker p D p B q D p B q ˚ qp x, z q| ď C E,A,B,µ,n λ µn ´ λνρ ` λ ν ρ | x ´ z | ¨ sup B ˇˇˇ S zy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ . It follows that | D p B q D p B q ˚ f p x q| ď ż | Ker p DD ˚ qp x, z q| | f p z q| dz ď C E,A,B,µ,n λ µn ´ ż λνρ ` λ ν ρ | x ´ z | | f p z q| dz ď C E,A,B,µ,n λ µn ´ sup B ˇˇˇ S zy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ M f p x q , here M is the Hardy-Littlewood maximal operator. Due to the L boundedness of the Hardy-Littlewood maximal operator, we can conclude that } D p B q D p B q ˚ f } L ď C E,A,B,µ,n λ µn ´ sup B ˇˇˇ S zy ˇˇˇ sup Ă B ˇˇ S xy ˇˇ } f } L . This gives (3.8). The inequality (3.9) follows by taking adjoints. Thus we complete the proofof Theorem 3.1. (cid:3)
To get the L p estimate we also need the following endpoint estimates. Theorem 3.5.
With the same requirements as Theorem 3.1 for the phase function, then for Re p z q “ ´ ´ µn ´ we have } T zX f } L , ď C E,ψ } f } L , (3.19) } T zY f } L ď C E,ψ } f } L . (3.20) Proof.
Observe that | T zX f p x q| ď C ψ ż | y |ď| x | | K p x, y q| ˇˇˇ S xy ˇˇˇ ´ ´ µn ´ | f p y q| dy ď C E,ψ ż | y |ď| x | | x | ´ µ | x | ´ ` µ | f p y q| dy “ C E,ψ | x | ´ ż | y |ď| x | | f p y q| dy ď C E,ψ | x | ´ } f } L . p DECAY OF OSCILLATORY INTEGRAL OPERATORS 11 If f P L , we can easily conclude (3.19). Now we turn to prove (3.20). Similarly, | T zY f p x q| ď C ψ ż | y |ě| x | | K p x, y q| ˇˇˇ S xy ˇˇˇ ´ ´ µn ´ | f p y q| dy ď C E,ψ ż | y |ě| x | | y | ´ µ | y | ´ ` µ | f p y q| dy “ C E,ψ ż | y |ě| x | | y | ´ | f p y q| dy. By Fubini’s theorem, we have } T zY f } L ď C E,ψ ij | y |ě| x | | y | ´ | f p y q| dydx “ C E,ψ ż | y | ´ | f p y q| ż | y |ě| x | dxdy ď C E,ψ ż | y | ´ | y | | f p y q| dy “ C E,ψ } f } L . This implies (3.20). (cid:3)
For the sake of interpolation, we also need the following lemma with change of power weights.An earlier version of this lemma appeared in [5], see also [14],[12] for details of proof.
Lemma 3.6.
Let dx be the Lebesgue measure on R . Assume V is a linear operator definedon all simple functions with respect to dx . If there exist two constant A , A ą such that(1) } V f } L p dx q ď A } f } L p dx q for all simple functions f ,(2) } | x | a V f } L p p dx q ď A } f } L p p dx q for some ă p , a P R satisfying ap ‰ ´ ,then for any θ P p , q , there exists a constant C “ C p a, p , θ q such that } | x | b V f } L p p dx q ď CA θ A ´ θ } f } L p p dx q (3.21) for all simple function f , where b and p satisfy b “ ´ θ ` p ´ θ q a and p “ θ ` ´ θp . Local Riesz-Thorin interpolation
Now, we will establish the sharp L p decay estimates for T X and T Y whenever n ´ µn ´ ´ µ ă p ă n ´ µ ´ µ . It reads } T X f } L p ď C E,ψ λ ´ ´ µn } f } L p , n ´ µn ´ ´ µ ă p ă `8 (4.1) } T Y f } L p ď C E,ψ λ ´ ´ µn } f } L p , ă p ă n ´ µ ´ µ . (4.2)In fact, the main strategy, the local interpolation with trivial endpoint estimates, was essen-tially introduced in [16]. In this paper, we modify it to adapt to our model. Lemma 4.1.
Consider the singular oscillatory integral operator r Df p x q “ ż R e iλS p x,y q K p x, y q ” ´ φ ´ p x ´ y q λ n ¯ı ψ p x, y q f p y q dy, with the same assumptions of Lemma 1 in [4] , furthermore, we also denote the upper boundsof x ´ cross section and y ´ cross section of the support of ψ p x, y q by δ and δ respectively. Ifwe denote the kernel of r D by K λ p x, y q and | K λ p x, y q| ď C K , then we have } r D } L p ď C E,ψ,n,p min " λ p µn ´ q ¨ p ν ´ p C p ´ K δ p ´ , C K δ p δ p * , ă p ă
2; (4.3) } r D } L p ď C E,ψ,n,p min " λ p µn ´ q ¨ p ν ´ p C ´ p K δ ´ p , C K δ p δ p * , ă p ă `8 . (4.4)This can be deduced by interpolating the L estimates, shown in [4], } r D } L ď C E,ψ,n λ µn ´ ν ´ , } r D } L ď C K p δ δ q , with the endpoint estimates } r D } L ď C K δ , } r D } L ď C K δ . Now we turn to prove (4.1) and (4.2).
Proof.
For j " k , we can see ν “ C K ´ k p n ´ q , δ « ´ j ; C K “ C E, K ,ψ µk , δ « ´ k . (4.5)If p ą
2, by invoking (4.4), we can deduce that } T j,k } L p ď C E, K ,ψ,n,p min " λ p µn ´ q ¨ p ´ ´ k p n ´ q ¯ ´ p ´ µk ¯ ´ p ´ ´ k ¯ ´ p , µk ´ jp ´ kp * . Since j " k , we may set j “ k ` M , then from above we have } T j,k } L p ď C E, K ,ψ,n,p min λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ ´ p ¯ , p µ ´ q k ´ Mp + . Therefore, } T Y } L p “ } ÿ j " k T j,k } L p ď C E, K ,ψ,n,p `8 ÿ M “ `8 ÿ k “ min λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ ´ p ¯ , p µ ´ q k ´ Mp + . We derive from λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ ´ p ¯ « p µ ´ q k ´ Mp that 2 k « λ n M µ ´ n . p DECAY OF OSCILLATORY INTEGRAL OPERATORS 13
It implies `8 ÿ k “ min λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ ´ p ¯ , p µ ´ q k ´ Mp + « ´ Mp ´ λ n M µ ´ n ¯ µ ´ “ λ µ ´ n ´ µ ´ µ ´ n ´ p ¯ ¨ M . To make the sum above converge, we require that µ ´ µ ´ n ´ p ă , which equals p ă n ´ µ ´ µ . It should be noted that p ą n ą
2. Actually, the case n “ L p estimate of T Y whenever 1 ă p ă
2. Onaccount of (4.3) and (4.5), we can see } T j,k } L p ď C E, K ,ψ,n,p min " λ p µn ´ q ¨ p ´ ´ k p n ´ q ¯ ´ p ´ µk ¯ p ´ ` ´ j ˘ p ´ , µk ´ jp ´ kp * “ C E, K ,ψ,n,p min λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ p ´ ¯ ´ M ´ p ´ ¯ , p µ ´ q k ´ Mp + . Similarly, if λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ p ´ ¯ ´ M ´ p ´ ¯ « p µ ´ q k ´ Mp , then 2 k « λ n M µ ´ n . Therefore } T Y } L p “ } ÿ j " k T j,k } L p ď C E, K ,ψ,n,p `8 ÿ M “ `8 ÿ k “ min λ p µn ´ q ¨ p ´ k ¯ np ´ ` µ ´ p ´ ¯ ´ M ´ p ´ ¯ , p µ ´ q k ´ Mp + ď C E, K ,ψ,n,p `8 ÿ M “ λ µ ´ n ´ µ ´ µ ´ n ´ p ¯ ¨ M . Taking 1 ă p ă µ ´ µ ´ n ´ p ă . The sum converges, thus we complete the proof of (4.2). Repeating the argument above for T X , we can get (4.1) similarly. The proof is finished. (cid:3) The remaining case: T ∆ This section is devoted to establishing the sharp L p decay estimates for T ∆ . Preceding theformal proof, we state a useful almost-orthogonality principle which was introduced in [9]. Lemma 5.1.
For the bilinear operator V p f, g q “ ş Ω ˆ Ω K p x, y q f p x q g p y q dxdy , assume that tp x, y q : K p x, y q ‰ u Ă ď k “ p I k ˆ J k q , where t I k u and t J k u are two groups of mutually disjoint measurable subsets of Ω and Ω . Let V k be the bilinear operator with kernel χ I k p x q χ J k p y q K p x, y q , let } V k } and } V } be the norm ofthe T k and T as bilinear operators on L p p Ω q ˆ L p p Ω q . Then } V } ď sup k } V k } . (5.1) Remark 5.2.
In fact, if both t I k u and t J k u are groups of sets having finite overlaps, we alsohave } V } ď C sup k } V k } , where the implicit constant depends only on the overlapping numbers.Now we turn to deal with T ∆ . We claim that } T ∆ f } L p ď C E, K ,ψ,n,p λ ´ ´ µn } f } L p , n ´ µn ´ ´ µ ď p ď n ´ µ ´ µ . (5.2)Note that T ∆ f p x q “ ř j „ k T j,k f p x q , thus from the remark above, we know that } T ∆ } L p ď C K sup j „ k } T j,k } L p . Thus we are reduced to estimating each T j,k . Recalling the Hessian of the phase function, wecan list the varieties as follows y ´ α x “ , y ´ α x “ , ¨ ¨ ¨ , y ´ α s x “ , while 0 ă α ă α ă ¨ ¨ ¨ ă α s . Case I : There exists a l such that α l “ α “
1. Along the variety y ´ x “
0, wefurther decompose the support of T j,k dyadically. T j,k f p x q “ ÿ l ż R e iλS p x,y q K p x, y q ” ´ φ ´ p x ´ y q λ n ¯ı φ l p y ´ x q¨ φ k p x q φ k p y q ψ p x, y q f p y q dy : “ ÿ l T l j,k f p x q . Given that j „ k , in the support of T l k,k , we know l ě j „ k . So we can divide the sum intotwo parts as follows T j,k f p x q “ ÿ l „ j „ k T l j,k f p x q ` ÿ l " j „ k T l j,k f p x q . (5.3)To get the final result, it suffices to establish L p estimates for the above two parts respectively.For the latter one, since l " j „ k , we may set l “ k ` M , then ν “ C K ´ k p n ´ q ´ M , δ « ´ k ´ M ; C K “ C E, K ,ψ µ p k ` M q , δ « ´ k ´ M . p DECAY OF OSCILLATORY INTEGRAL OPERATORS 15
From (4.4), we know that for p ą } T l j,k } L p ď C E, K ,ψ,n,p λ p µn ´ q ¨ p ´ ´ k p n ´ q´ M ¯ ´ p ´ p µ ´ qp k ` M q ¯ ´ p , } T l j,k } L p ď C E, K ,ψ,p p µ ´ qp k ` M q . By convex combination, for 0 ď θ ď
1, we have } T l j,k } L p ď C E, K ,ψ,n,p „ λ p µn ´ q ¨ p ´ ´ k p n ´ q´ M ¯ ´ p ´ p µ ´ qp k ` M q ¯ ´ p θ ¨ ” p µ ´ qp k ` M q ı ´ θ . We choose suitable θ to eliminate k . This leads to θ „ n ´ p ` p µ ´ q ˆ ´ p ˙ ` p µ ´ qp ´ θ q “ . Solve this equation about θ and get the solution θ “ p p ´ µ q n ´ µ . The restriction 0 ď θ ď p ď n ´ µ ´ µ . Plugging this into the above convex combination,we obtain } ÿ l " k T l j,k } L p ď C E, K ,ψ,n,p ÿ M “ λ µ ´ nnp ¨ θ M ” p `p µ ´ q ´ ´ p ¯ı ¨ θ p µ ´ qp ´ θ q M “ C E, K ,ψ,n,p ÿ M “ λ µ ´ n ´ M p n ´ qp ´ µ q n ´ µ . If n ą
3, the sum above converges. If n “
3, we shall use (4.4) instead of the convexcombination to give the L p estimate. } T l j,k } L p ď C E, K ,ψ,n,p min " λ p µ ´ q ¨ p ´ ´ l ¯ ´ p ´ p µ ´ q l ¯ ´ p , p µ ´ q l * “ C E, K ,ψ,n,p min " λ p µ ´ q ¨ p ´ l ¯ p `p µ ´ qp ´ p q , p µ ´ q l * . Hence } ÿ l " k T l j,k } L p ď C E, K ,ψ,n,p ÿ l “ min " λ p µ ´ q ¨ p ´ l ¯ p `p µ ´ qp ´ p q , p µ ´ q l * ď C E, K ,ψ,n,p λ µ ´ . For p ă
2, from (4.3), we know that } T l j,k } L p ď C E, K ,ψ,n,p λ p µn ´ q ¨ p ´ ´ k p n ´ q´ M ¯ ´ p ´ p µ ´ qp k ` M q ¯ p ´ , } T l j,k } L p ď C E, K ,ψ,n,p p µ ´ qp k ` M q . Similarly, by convex combination, we know that for 0 ď θ ď } T l j,k } L p ď C E, K ,ψ,n,p „ λ p µn ´ q ¨ p ´ ´ k p n ´ q´ M ¯ ´ p ´ p µ ´ qp k ` M q ¯ p ´ θ ¨ ” p µ ´ qp k ` M q ı ´ θ . Again, choose suitable θ to eliminate k , this requires θ „ n ´ p ` p µ ´ q ˆ p ´ ˙ ` p µ ´ qp ´ θ q “ . It equals θ “ p p ´ µ q n ´ µ . Given that 0 ď θ ď
1, we can obtain p ď n ´ µ ´ µ , i.e., p ě n ´ µn ´ ´ µ . Therefore } ÿ l " k T l j,k } L p ď C E, K ,ψ,n,p ÿ M “ λ µ ´ nnp ¨ θ M ” p `p µ ´ q ´ p ´ ¯ı ¨ θ p µ ´ qp ´ θ q M “ C E, K ,ψ,n,p ÿ M “ λ µ ´ n ´ M p n ´ qp ´ µ q n ´ µ . This sum converges if n ą
3, next we will treat the case n “ } T l j,k } L p ď C E, K ,ψ,n,p min " λ p µn ´ q ¨ p ´ ´ l ¯ ´ p ´ p µ ´ q l ¯ p ´ , p µ ´ q l * “ C E, K ,ψ,n,p min " λ p µ ´ q ¨ p ´ l ¯ p `p µ ´ qp p ´ q , p µ ´ q l * . Then we have } ÿ l " k T l j,k } L p ď C E, K ,ψ,n,p ÿ l “ min " λ p µ ´ q ¨ p ´ l ¯ p `p µ ´ qp p ´ q , p µ ´ q l * ď C E, K ,ψ,n,p λ µ ´ . Now we turn to deal with ř l „ j „ k T l j,k in (5.3). We decompose T l j,k dyadically according tothe second variety y ´ α x “
0, specifically we write T l j,k f p x q “ ÿ l ż R e iλS p x,y q K p x, y q ” ´ φ ´ p x ´ y q λ n ¯ı φ l p y ´ x q¨ φ l p y ´ α x q φ k p x q φ k p y q ψ p x, y q f p y q dy : “ ÿ l T l ,l j,k f p x q . Similarly, we shall also consider the relation between l and k and divide this sum into twoparts T l j,k f p x q “ ÿ l „ k T l ,l j,k f p x q ` ÿ l " k T l ,l j,k f p x q . For each operator in the latter sum, we can see that in the support of T l ,l j,k , if we set l “ k ` M ,we have ν “ C K ´ k p n ´ q ´ M , δ « ´ k ´ M ; C K “ C E, K ,ψ µk , δ « ´ k ´ M . Repeating the above process, we can also conclude the L p estimate for ř l " k T l ,l j,k . We omitthe details. Thus we are left with ř T l ,l j,k where l „ l „ j „ k . Continue to decompose thisoperator and repeat the above process for the case l m " k p m ě q until we are left with thelast sum ÿ l s „¨¨¨„ l „ j „ k T l , ¨¨¨ ,l s j,k . p DECAY OF OSCILLATORY INTEGRAL OPERATORS 17
Notice that there are only finite operators among this sum, so we can consider only one suchoperator. In fact, in the support of T l , ¨¨¨ ,l s j,k , we have ν “ C K ´ k p n ´ q , δ « ´ k ; C K “ C E, K ,ψ µk , δ « ´ k . By convex combination between the oscillatory estimate and the size estimate, we can get thedesired result (5.2).
Case II : There is no l such that α l “ ÿ l „ j „ k T l j,k f p x q , while the other part ÿ l " j „ k T l j,k f p x q . can be dealt with by summing all local L p estimates. Regarding the orthogonal part as theinitial operator and dyadically decompose it according to the second variety. Continue theprocess until the s -th step, we now come to the operator ÿ l s „¨¨¨„ l „ j „ k T l , ¨¨¨ ,l s j,k . In the support of this operator, we can see that | y ´ α x | « C K ´ k , ¨ ¨ ¨ , | y ´ α s x | « C K ´ k , however, the singular kernel K p x, y q is possible to vanish. So we shall provide another dyadicdecomposition for this operator as follows T l , ¨¨¨ ,l s ; tj,k “ ÿ t „ l s „¨¨¨„ l „ j „ k T l , ¨¨¨ ,l s ; tj,k ` ÿ t " l s „¨¨¨„ l „ j „ k T l , ¨¨¨ ,l s ; tj,k . In the support of each operator among the latter sum, we have ν “ C K ´ k p n ´ q , δ « ´ t ; C K “ C E, K ,ψ µt , δ « ´ t . Repeating the previous process, we obtain } ÿ t " l s „¨¨¨„ l „ j „ k T l , ¨¨¨ ,l s ; tj,k } L p ď C E, K ,ψ,n,p λ µ ´ n . Now, we are left with the finite sum ř t „ l s „¨¨¨„ l „ j „ k T l , ¨¨¨ ,l s ; tj,k . By almost-orthogonality, wefocus only on one such operator. In fact, in the support of one such operator, we have ν “ C K ´ k p n ´ q , δ « ´ k ; C K “ C E, K ,ψ µk , δ « ´ k . This is same with the last part of
Case I , thus we are done with the proof. Proof of Theorem 1.1
Now, we are ready to give the proof of Theorem 1.1 since all preparation works have beenfinished. Firstly, we use Stein’s interpolation between (3.4) and (3.20) and consequently have } T Y f } L n ´ µn ´ ´ µ ď C E, K ,ψ,n,p λ ´ ´ µn } f } L n ´ µn ´ ´ µ . (6.1)Secondly, we apply Lemma 3.6, in which setting p “ a “ n ´ ´ µ , to (3.19) and (3.4) andconsequently have } T X f } L n ´ µn ´ ´ µ ď C E, K ,ψ,n,p λ ´ ´ µn } f } L n ´ µn ´ ´ µ . (6.2)Finally, (4.2), (4.1), (5.2), (6.1), (6.2) together imply (1.4) for p ă
2. The routine adjointargument shall give the corresponding result for p ą Further argument
The previous argument is based on our additional condition (AC). However, in [4] there is nosuch a condition. With the removal of (AC), we establish a result, similar to Theorem 1.1, asfollows.
Theorem 7.1.
We assume the same assumptions with Theorem 1.1 except (AC) , then } T f } L p ď C E,S,ψ,µ,n,p λ ´ ´ µn } f } L p (7.1) holds for n ´ µn ´ ´ µ ă p ă n ´ µ ´ µ . Furthermore, for the endpoint p “ n ´ µn ´ ´ µ , we also have thenearly sharp decay estimate } T f } L n ´ µn ´ ´ µ ď C E,S,ψ,µ,n,p λ ´ ´ µn | log p λ q| ´ µn ´ µ } f } L n ´ µn ´ ´ µ . (7.2)Recall the strategy to prove Theorem 1.1: we first split the operator T into three parts T X , T ∆ , T Y , the next step is to use damped estimates to deal with T X and T Y , finally we uselocal interpolation to treat T ∆ . In the second step, the success of establishing the sharp L estimates (3.3), (3.4) relies on Lemma 3.2 and Remark 3.4. Here, the removal of (AC) resultsin the failure of Remark 3.4. Instead, we make use of Remark 3.3 to give the nearly sharp L estimates as follows. Theorem 7.2.
If the Hessian of the phase function is of the form (2.1) , we have } T zY f } L ď C E, K ,ψ,n,z λ µn ´ log p λ q} f } L , (7.3) } T zX f } L ď C E, K ,ψ,n,z λ µn ´ log p λ q} f } L . (7.4) Proof.
We still use the notations of (3.10) and (3.11). If Re p z q “ , on one hand, fromRemark 3.3 we know that } D Xj } L ď C E, K ,ψ,n,z λ µn ´ , (7.5) } D Yk } L ď C E, K ,ψ,n,z λ µn ´ , (7.6) p DECAY OF OSCILLATORY INTEGRAL OPERATORS 19 where C E,A,B,µ,n,z is a constant with at most polynomial growth in z . On the other hand, wehave the trivial size estimate } D Xj } L ď C E, K ,ψ ” ´ j p n ´ q ı jµ ´ j “ ´ j p n ´ µ q , (7.7) } D Yk } L ď C E, K ,ψ ” ´ k p n ´ q ı kµ ´ k “ ´ k p n ´ µ q . (7.8)Hence (3.10), (7.5) as well as (7.7) give that } T zX } L “ } ÿ j D Xj } L ď ÿ j } D Xj } L ď C E, K ,ψ,n,z ÿ j min ! λ µn ´ , ´ j p n ´ µ q ) “ C E, K ,ψ,n,z λ µn ´ log p λ q . Similarly, for T zY , there is } T zY } L “ } ÿ k D Yk } L ď ÿ k } D Yk } L ď C E, K ,ψ,n,z ÿ k min ! λ µn ´ , ´ k p n ´ µ q ) “ C E, K ,ψ,n,z λ µn ´ log p λ q . Therefore, we finish the proof of Theorem 7.2. (cid:3)
By virtue of Theorem 3.3, the proof of Theorem 7.1 is same with that of Theorem 1.1 whichwe have shown in Section 7. Here, we omit the details for simplicity.8.
A necessary condition
In this part, we will give an example to demonstrate a necessary condition for p to preservethe sharp decay.If we assume that S p x, y q “ x n ´ y ` xy n ´ , K p x, y q “ | x ´ y | ´ µ ; λ ě , ψ p x, y q ” |p x, y q| ď . Moreover, we assume that f p y q “ χ r , s p y q , if 0 ď x ď λ , we know that | T f p x q| ě ˇˇˇˇˇż cos ˆ y n ´ ` y n ´ ˙ ˇˇˇˇ y ´ ˇˇˇˇ ´ µ dy ˇˇˇˇˇ ě ż cos ˆ n ´ ¨ ` ¨ n ´ ˙ ˇˇˇˇ ´ ˇˇˇˇ ´ µ dy ě . If the apriori estimate } T f } L p À λ ´ ´ µn } f } L p holds, we can see that λ ´ ´ µn ¨ p Á „ż R ˇˇˇˇż R e iλ p x n ´ y ` xy n ´ q | x ´ y | ´ µ ψ p x, y q χ r , s p y q dy ˇˇˇˇ p dx p ě «ż λ ˇˇˇˇż R e iλ p x n ´ y ` xy n ´ q | x ´ y | ´ µ ψ p x, y q χ r , s p y q dy ˇˇˇˇ p dx ff p ě ˜ż λ p dx ¸ p “ λ ´ p p ` . This implies λ ´ ´ µn Á λ ´ p . Since λ can be arbitrarily large, this requires p ď n ´ µ . Interchanging the roles of x and y , we can get the other necessary condition p ě nn ´ ` µ . Remark 8.1.
Observe the necessary condition, there is a gap between it with our result.
Acknowledgement:
The author would like to acknowledge financial support from JiangsuNatural Science Foundation, Grant No. BK20200308. The author thanks Zuoshunhua Shi formany helpful suggestions.
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