Uniform estimates for the Fourier transform of surface carried measures in R 3 and an application to Fourier restriction
aa r X i v : . [ m a t h . C A ] O c t UNIFORM ESTIMATES FOR THE FOURIER TRANSFORM OFSURFACE CARRIED MEASURES IN R AND AN APPLICATIONTO FOURIER RESTRICTION.
ISROIL A.IKROMOV AND DETLEF M ¨ULLER
Abstract.
Let S be a hypersurface in R which is the graph of a smooth, finite typefunction φ, and let µ = ρ dσ be a surface carried measure on S, where dσ denotesthe surface element on S and ρ a smooth density with suffiently small support. Wederive uniform estimates for the Fourier transform ˆ µ of µ, which are sharp exceptfor the case where the principal face of the Newton polyhedron of φ, when expressedin adapted coordinates, is unbounded. As an application, we prove a sharp L p - L Fourier restriction theorem for S in the case where the original coordinates areadapted to φ. This improves on earlier joint work with M. Kempe.
Contents
1. Introduction 12. Uniform estimates for oscillatory integrals with finite type phase functionsof two variables 82.1. The case where the coordinates are adapted to φ, or where h = 2 102.2. The case of non-adapted coordinates: the contribution of regions awayfrom the principal root jet 122.3. The contribution of the homogenous domain D λ containing the principalroot jet 163. Sharpness of the uniform estimates 213.1. The case where the principal face is a compact edge 213.2. The case where the principal face is a vertex 244. Fourier restriction in the case of adapted coordinates. 335. Appendix: Proof of Lemma 1.5 36References 381. Introduction
The goal of this article is to improve on two results from our previous article [12]concerning uniform estimates for two-dimensional oscillatory integrals with smooth,
Mathematical Subject Classification.
Key words and phrases.
Oscillatory integral, Newton diagram, Fourier restriction.We acknowledge the support for this work by the Deutsche Forschungsgemeinschaft.
I. A. IKROMOV AND D. M ¨ULLER finite type phase functions, and L p - L Fourier restriction for smooth, finite type hy-persurfaces S in R which are locally the graph of a function φ in adapted coordinates.More precisely, we shall identify in Theorems 1.1 and 1.3 exactly when the logarith-mic factor in the estimate (1.6) for the Fourier transform of a surface carried measureof S in Theorem 1.9 of [12] will be present, with the exception of the case where theprincipal face of the Newton polyhedron of φ, when expressed in adapted coordinates,is unbounded and φ is non-analytic. Examples by A. Iosevich and E. Sawyer show thata different behavior can indeed occur in the latter case. Moreover, we shall show inTheorem 1.7 that the restriction theorem Corollary 1.10 of that article can be improvedas follows:Assume that S is represented near a point x as the graph of a function φ ( x , x ) , andthat, after translation of coordinates, x = 0 . Then, in the case where the coordinates( x , x ) are adapted to φ after applying a linear change of coordinates, the restrictionestimate holds true also at the endpoint p ′ = 2 h ( φ ) + 2 , where h ( φ ) denotes the heightof φ in the sense of Varchenko.If the coordinates ( x , x ) are not adapted to φ, then we will show in a sequel to thisarticle that the restriction estimate can be extended to an even wider range of p ’s.We shall build on the results and techniques developed in [11] and [12], which will beour main references, also for cross-references to earlier and related work. Let us firstrecall some basic notions from [11], which essentially go back to A. N. Varchenko [21].Let φ be a smooth real-valued function defined on a neighborhood of the origin in R with φ (0 ,
0) = 0 , ∇ φ (0 ,
0) = 0 , and consider the associated Taylor series φ ( x , x ) ∼ ∞ X j,k =0 c jk x j x k of φ centered at the origin. The set T ( φ ) := { ( j, k ) ∈ N : c jk = 1 j ! k ! ∂ j ∂ k φ (0 , = 0 } will be called the Taylor support of φ at (0 , . We shall always assume that T ( φ ) = ∅ , i.e., that the function φ is of finite type at the origin. The Newton polyhedron N ( φ )of φ at the origin is defined to be the convex hull of the union of all the quadrants( j, k ) + R in R , with ( j, k ) ∈ T ( φ ) . The associated
Newton diagram N d ( φ ) in thesense of Varchenko [21] is the union of all compact faces of the Newton polyhedron;here, by a face, we shall mean an edge or a vertex.We shall use coordinates ( t , t ) for points in the plane containing the Newton poly-hedron, in order to distinguish this plane from the ( x , x ) - plane.The Newton distance , or shorter distance d = d ( φ ) between the Newton polyhedronand the origin in the sense of Varchenko is given by the coordinate d of the point ( d, d )at which the bi-sectrix t = t intersects the boundary of the Newton polyhedron. NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 3
The principal face π ( φ ) of the Newton polyhedron of φ is the face of minimal dimen-sion containing the point ( d, d ). Deviating from the notation in [21], we shall call theseries φ pr ( x , x ) := X ( j,k ) ∈ π ( φ ) c jk x j x k the principal part of φ. In case that π ( φ ) is compact, φ pr is a mixed homogeneouspolynomial; otherwise, we shall consider φ pr as a formal power series.Note that the distance between the Newton polyhedron and the origin depends onthe chosen local coordinate system in which φ is expressed. By a local coordinate systemat the origin we shall mean a smooth coordinate system defined near the origin whichpreserves 0 . The height of the smooth function φ is defined by h ( φ ) := sup { d x } , where the supremum is taken over all local coordinate systems x = ( x , x ) at theorigin, and where d x is the distance between the Newton polyhedron and the origin inthe coordinates x .A given coordinate system x is said to be adapted to φ if h ( φ ) = d x . In [11] weproved that one can always find an adapted local coordinate system in two dimensions,thus generalizing the fundamental work by Varchenko [21] who worked in the settingof real-analytic functions φ (see also [17] for another proof in the analytic case).Following [21] (with as slight modification), we next define what we like to call Varchenko’s exponent ν ( φ ) ∈ { , } as follows (this number had been identified byVarchenko in [21] as what Karpushkin calls the ”multiplicity of the oscillation of φ at(0 , y near the origin such that theprincipal face π ( φ a ) of φ, when expressed by the function φ a in the new coordinates(i.e. φ ( x ) = φ a ( y )), is a vertex, and if h ( φ ) ≥ , then we put ν ( φ ) := 1; otherwise, weput ν ( φ ) := 0 . As has been shown by Varchenko in [21], the number ν ( φ ) arises as the exponent of alogarithmic factor in the principal part of the asymptotic expansion of two-dimensionaloscillatory integrals with real analytic phase functions φ. Analogously, we can prove the following uniform estimate for two-dimensional oscil-latory integrals with smooth, finite type phase functions φ, which improves on Theorem11.1 in [12]. Theorem 1.1.
Let φ be a smooth, real-valued phase function of finite type, definednear the origin, as before, and let h := h ( φ ) , ν := ν ( φ ) . Then there exist a neighborhood Ω ⊂ R of the origin and a constant C such that for every η ∈ C ∞ (Ω) the followingestimate holds true for every ξ ∈ R :(1.1) (cid:12)(cid:12)(cid:12) Z R e i ( ξ φ ( x ,x )+ ξ x + ξ x ) η ( x ) dx (cid:12)(cid:12)(cid:12) ≤ C k η k C ( R ) (log(2 + | ξ | )) ν (1 + | ξ | ) − /h . I. A. IKROMOV AND D. M ¨ULLER
Remarks 1.2. (a) For some special classes of hypersurfaces, related results have beenderived by L. Erd¨os and M. Salmhofer in [6], which, however, are not necessarilyuniform in all directions. For estimates with ξ = ξ = 0 , we refer to the recent workof M. Greenblatt [7].(b) For real analytic phase functions φ, if we restrict ourselves to the direction where ξ = ξ = 0 , then the asymptotic expansion of the corresponding oscillatory integralsin [21] shows that the estimate (1.1) is essentially sharp as an estimate in terms of | ξ | . (c) For real analytic phase functions, our result is covered by Karpushkin’s work [15],who proved the following:If φ is a real analytic function defined near the origin with φ (0 ,
0) = 0 , ∇ φ (0 ,
0) = 0 , and if r is a real analytic function with sufficiently small norm (in the space of realanalytic functions) then (cid:12)(cid:12)(cid:12)(cid:12)Z R e iλ ( φ ( x )+ r ( x )) η ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k η k C (log(2 + | λ | )) ν (2 + | λ | ) /h , λ ∈ R , provided the amplitude η is supported in a sufficiently small neighborhood of the origin.Moreover, the constant C then does not depend on the function r .(d) If h ( φ ) < , results analogous to Karpushkin’s have been obtained by J. J. Duis-termaat [10] in the smooth setting. In this case one always has ν ( φ ) = 0.(e) If h ( φ ) = 2 , and if the principal part of φ, when expressed in an adapted co-ordinate system, has a critical point of finite multiplicity at the origin (so that it isisolated), then an analogue to Karpushkin’s estimate has been established by Colin deVerdi`ere [3] in the smooth setting. Notice that if the principal part of φ has an isolatedcritical point at the origin, then the coordinate system is adapted to φ and ν ( φ ) = 0 . The next result, which improves on corresponding results by M. Greenblatt, showsin particular that, in most cases, the uniform estimates from Theorem 1.1 are sharp if( ξ , ξ ) = (0 , . Theorem 1.3.
Let us put J ± ( λ ) := Z R e ± iλφ ( x ,x ) η ( x ) dx, λ > , with φ and η as in Theorem 1.1. If the principal face π ( φ a ) of φ, when given inadapted coordinates, is a compact set (i.e., a compact edge or a vertex), then thereexists a neighborhood Ω of the origin such that for every η supported in Ω the followinglimits (1.2) lim λ → + ∞ λ /h (log λ ) ν J ± ( λ ) = c ± η (0) exist, where the constants c ± are non-zero and depend on the phase function φ only. NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 5
Remarks 1.4. (a) The proof of Theorem 1.3 reveals the following additional facts:If ν ( φ ) = 0 in the theorem, then the principal face π ( φ a ) is a compact edge, andthe constants c ± are completely determined by the principal part φ a pr of φ a . And, if ν ( φ ) = 1 , and if we work in super-adapted coordinates in the sense of Greenblatt (asexplained in Lemma 3.4), so that in particular π ( φ a ) consists of the vertex ( h, h ) , thenthe constants c ± are completely determined by the principal part φ a pr of φ a and theslopes of those compact edges of N ( φ a ) which contain this vertex.(b) An analogous result for real analytic phase functions φ has been proven by M.Greenblatt (Theorem 1.2 in [7]). For non-analytic, but smooth and finite type φ, thefollowing weaker result had been obtained in Theorem 1.6b of the same article:lim sup λ → + ∞ (cid:12)(cid:12)(cid:12) λ /h (log λ ) ν J ± ( λ ) (cid:12)(cid:12)(cid:12) > . (c) If the principal face π ( φ a ) is unbounded, then the estimate in Theorem 1.1 mayfail to be sharp, if φ is non-analytic, as the following class of examples by A. Iosevichand E. Sawyer [14] shows: If φ ( x , x ) := x + e − / | x | α , with α > , then | J ± ( λ ) | ≍ λ / log λ /α as λ → + ∞ , whereas ν ( φ ) = 0 . These examples also indicate that a precise determination of theasymptotic behavior of J ± ( λ ) may be difficult when the principal face is non-compact.(d) For real-analytic phase functions depending on more than two variables andsatisfying an appropriate non-degeneracy condition, the explicit form of the principalpart of the asymptotic expansion of the corresponding oscillatory integrals has beenobtained by J. Denef, J. Nicaise and P. Sargos [4].The existence of an adapted coordinate system in which the principal face is avertex is a priori not so easily verified, but there exists an equivalent, more accessiblecondition. In order to describe this, we first recall that if the principal face of theNewton polyhedron N ( φ ) is a compact edge, then it lies on a unique line κ t + κ t =1 , with κ , κ > . By permuting the coordinates x and x , if necessary, we shallalways assume that κ ≤ κ . We shall call this weight κ = ( κ , κ ) the principalweight associated to φ, and denote it also by κ pr . It induces dilations δ r ( x , x ) :=( r κ x , r κ x ) , r > , on R , so that the principal part φ pr of φ is κ - homogeneous ofdegree one with respect to these dilations, i.e., φ pr ( δ r ( x , x )) = rφ pr ( x , x ) for every r > , and(1.3) d = 1 κ pr1 + κ pr2 = 1 | κ pr | . Denote by
I. A. IKROMOV AND D. M ¨ULLER m ( φ pr ) := ord S φ pr the maximal order of vanishing of φ pr along the unit circle S centered at the origin.We also recall from [11] that the homogeneous distance of a κ -homogeneous polyno-mial P (such as P = φ pr ) is given by d h ( P ) := 1 / ( κ + κ ) = 1 / | κ | , and that(1.4) h ( P ) = max { m ( P ) , d h ( P ) } . According to [11], Corollary 4.3 and Corollary 2.3, the coordinates x are adapted to φ if and only if one of the following conditions is satisfied: (a) The principal face π ( φ ) of the Newton polyhedron is a compact edge, and m ( φ pr ) ≤ d ( φ ) . (b) π ( φ ) is a vertex. (c) π ( φ ) is an unbounded edge. We like to mention that in case (a) we have h ( φ ) = h ( φ pr ) = d h ( φ pr ) . Notice alsothat (a) applies whenever π ( φ ) is a compact edge and κ /κ / ∈ N ; in this case we evenhave m ( φ pr ) < d ( φ ) (cf. [11], Corollary 2.3). Lemma 1.5.
The following conditions on φ are equivalent: (a) There exists an adapted local coordinate system y at the origin such that theprincipal face π ( φ a ) is a vertex. (b) If y is any adapted local coordinate system at the origin, then either π ( φ a ) is avertex, or a compact edge and m ( φ a pr ) = d ( φ a ) . Consider for example the function φ ( x , x ) := ( x − x ) ( x − x ) . Then φ = φ pr ,π ( φ ) is a compact edge and m ( φ pr ) = 2 = d ( φ ) , so that case (b) above applies and thecoordinates x are adapted to φ. Moreover, ν ( φ ) = 1 . If we introduce new coordinates y given by y := x , y := x − x , then φ ( x ) = ˜ φ ( y ) , where ˜ φ ( y ) = y ( y + y ) . Theprincipal face of N ( ˜ φ ) is the vertex (2 , , so that also the coordinates y are adapted.In the case where the coordinates are not adapted to φ, we see that the principalface π ( φ ) is a compact edge such that(1.5) m := κ /κ ∈ N . Then, by Theorem 5.1 in [11], there exists a smooth real-valued function ψ of theform(1.6) ψ ( x ) = b x m + O ( x m +11 ) , with b = 0 , defined on a neighborhood of the origin such that an adapted coordinatesystem ( y , y ) for φ is given locally near the origin by means of the (in general non-linear) shear y := x , y := x − ψ ( x ) . In these coordinates, φ is given by(1.7) φ a ( y ) := φ ( y , y + ψ ( y )) . NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 7
As an immediate consequence of Theorem 1.1 we obtain uniform estimates for theFourier transform d ρdσ ( ξ ) = Z S e − iξ · x ρ ( x ) dσ ( x ) , ξ ∈ R , of surface carried measures on smooth, finite type hypersurfaces S in R . Here, dσ denotes the Riemannian volume element on S. If a point x on such a hypersurface S is given, which we may assume to be the originafter a translation of coordinates, and if we represent S locally near x = (0 ,
0) as thegraph x = φ ( x , x ) of smooth, finite type function φ with φ (0 ,
0) = 0 , ∇ φ (0 ,
0) = 0 asbefore, then we define the height of S at x by h ( x , S ) := h ( φ ) . This notion is invariantunder affine linear coordinate changes of the ambient space, as has been shown in [12].Similarly, we define ν ( x , S ) := ν ( φ ) . Denote by dσ the surface element of S. Then wehave the following improvement of Theorem 1.9 in [12]:
Corollary 1.6.
Let S be a smooth hypersurface of finite type in R and let x be afixed point on S. Then there exists a neighborhood U ⊂ S of the point x such that forevery ρ ∈ C ∞ ( U ) the following estimate holds true: | d ρdσ ( ξ ) | ≤ C k ρ k C ( S ) (log(2 + | ξ | )) ν ( x ,S ) (1 + | ξ | ) − /h ( x ,S ) for every ξ ∈ R . Our second result concerns Fourier restriction to S. We shall prove that the L p - L Fourier restriction theorem of Corollary 1.10 in [12] also holds true at the endpoint, if S is locally given as the graph of a function φ which is given in adapted coordinates: Theorem 1.7.
Let S be a smooth hypersurface of finite type in R , and let x be a fixedpoint on S. Assume that, possibly after a translation of coordinates, x = 0 , and that S is locally near x given as the graph x = φ ( x , x ) of a smooth, finite type function φ with φ (0 ,
0) = 0 , ∇ φ (0 ,
0) = 0 as before.We also assume that, after applying a suitable linear change of coordinates, thecoordinates ( x , x ) are adapted to φ, so that d = h, where d = d ( φ ) denotes theNewton distance of φ and h = h = h ( x , S ) = h ( φ ) its height. We then define thecritical exponent p c by (1.8) p ′ c := 2 h + 2 , where p ′ denotes the exponent conjugate to p, i.e., /p + 1 /p ′ = 1 . Then there exists a neighborhood U ⊂ S of the point x such that for every non-negative density ρ ∈ C ∞ ( U ) the Fourier restriction estimate (1.9) (cid:16) Z S | b f | ρ dσ (cid:17) / ≤ C p k f k L p ( R ) , f ∈ S ( R ) , holds true for every p such that (1.10) 1 ≤ p ≤ p c . Moreover, if ρ ( x ) = 0 , then the condition (1.10) on p is also necessary for thevalidity of (1.9) . I. A. IKROMOV AND D. M ¨ULLER
The second statement had already been proven in [12], Section 12, so that we shallonly have to prove the first statement, for the endpoint p = p c . Remarks 1.8. (a) The case where the coordinates ( x , x ) are not adapted to φ willbe treated in a subsequent article. It has turned out that in this case the restric-tion estimate (1.9) is valid in a wider range of p ′ s, with a critical exponent which isstrictly bigger than p c and which can be determined explicitly by means of Varchenko’salgorithm (cf. [11]) for the construction of adapted coordinates.(b) If the surface S is of finite line type and convex, and if the restriction property(1.9) holds true also in the endpoint p c = (2 h + 2) / (2 h + 1) , then it has been shown byA. Iosevich in [13] that necessarily the Fourier transform of ρ dσ must decay of order O ( | ξ | − /h ) as | ξ | → + ∞ (it can easily be shown by means of Schulz’ [18] decompositionof convex smooth functions of finite line type), i.e., ν ( x , S ) = 0 . Conversely, the decayrate O ( | ξ | − /h ) immediately implies the restriction estimate (1.9) also for the endpoint p = p c ; this is an immediate consequence of A. Greenleaf’s work in [8].However, if ν ( x , S ) = 1 , which can only happen in the non-convex case, the loga-rithmic factor in (1.1) is necessary, so that one cannot apply Greenleaf’s result directly.Restriction theorems for the Fourier transform have a long history by now, startingwith the seminal work by E.M. Stein, and P. Tomas, for the case of the Euclideansphere (see, e.g., [19]). Some restriction estimates for analytic hypersurfaces in R have been obtained by A. Magyar [16], whose results were sharp for particular classesof hypersurfaces given as graphs of functions in adapted coordinates, with the exceptionof the endpoint.2. Uniform estimates for oscillatory integrals with finite type phasefunctions of two variables
In this section we shall give a proof of Theorem 1.1. We shall closely follow the proofof Theorem 11.1 in [12], which did already provide the uniform estimates in Theorem1.1, except for a logarithmic factor which is not really needed in many cases, as weshall see.The reader is strongly recommended to have [12] at hand when reading this article,since we shall make use of the notation and many results from [12] without repeatingall of them here.By decomposing R into its four quadrants, we may reduce ourselves to the estima-tion of oscillatory integrals of the form J ( ξ ) := Z ( R + ) e i ( ξ φ ( x ,x )+ ξ x + ξ x ) η ( x , x ) dx. Notice also that we may assume in the sequel that(2.1) | ξ | + | ξ | ≤ δ | ξ | , hence | ξ | ∼ | ξ | , NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 9 where 0 < δ ≪ | ξ | + | ξ | >δ | ξ | the estimate (1.1) follows by an integration by parts, if Ω is chosen small enough.Of course, we may in addition always assume that | ξ | ≥ . If χ is any integrable function defined on Ω , we shall put J χ ( ξ ) := Z ( R + ) e i ( ξ φ ( x ,x )+ ξ x + ξ x ) η ( x , x ) χ ( x ) dx. The case where h ( φ ) < | ξ | ) , even for a wider class of phase functions), so let us assume from now onthat h := h ( φ ) ≥ . Moreover, if h = 2 , then we shall make use of the following special property: Lemma 2.1. If h ( φ ) = 2 , then, after applying a suitable linear change of coordinates,we may assume that one of the following conditions are satisfied: (i) The coordinates are adapted to φ. (ii) The coordinates are not adapted to φ, but h ( φ pr ) = h ( φ ) . In this case, we have ν ( φ ) = 0 and m ( φ pr ) = 2 . Note that in general we only have h ( φ pr ) ≥ h ( φ ) , and the inequality may be strict. Proof.
Let us assume that the coordinates x are not adapted to φ. Then the principalface π ( φ ) is a compact edge and m ( φ pr ) > d ( φ ) = d x . In particular, the principal part φ pr of φ is a polynomial which is κ -homogeneous of degree 1 , where we may assumethat 0 < κ ≤ κ , so that m := m = κ /κ ≥ φ pr can be written as φ pr ( x , x ) = cx α x β Y l ( x − c l x m ) n l , where the c l ’s are the non-trivial distinct complex roots of the polynomial t φ pr (1 , t )and the n l ’s are their multiplicities. Moreover, there exists an l such that m ( φ pr ) = n l and such that c l is real. Notice also that α ≤ , β ≤ , since otherwise the coordinateswere adapted.Assume first that κ = κ . Then m = 1 , and applying the first step in Varchenko’salgorithm (see [11], or Subsection 2.5 in [12]), we see that we can transform φ into ˜ φ by means of the linear change of variables y = x , y = x − c l x such that eitherthe coordinates y are adapted to ˜ φ, hence ˜ φ = φ a , or they are not, but then ˜ κ < ˜ κ (where ˜ φ pr is assumed to be ˜ κ -homogeneous of degree 1).After applying a suitable linear change of coordinates, we are thus reduced to thesituation where κ < κ , hence m ≥ . Let us denote by ( A , B ) and ( A , B ) the twovertices of π ( φ ) , and assume that A < A . Recall from [11], displays (3.2) and (3.3),that A = α, B = β + N, A = α + mN, B = β, and that(2.2) d x = α + m ( β + N )1 + m , with N := P l n l . Recall also that the point ( A , B ) , with A < B , will be a vertex ofall the Newton diagrams that arise when running Varchenko’s algorithm on φ, so thatwe must have β + N = B ≥ , since h ( φ ) = 2 . Then (2.2) implies that d x ≥ m m > , so that n l = m ( φ ) ≥ . Since d x ≤ h ( φ ) = 2 , (2.2) implies that β + N ≤ m ≤ . But, if we had β + N = 3 , then the conditions d x ≤ m ≥ α = 0 , m = 2 , hence d x = 2 , andso the coordinates x would be adapted, contradicting our assumption.Therefore, we must have β + N = 2 . Then β = 0 , N = n l = 2 and α < , and thusthe change of coordinates y := x , y := x − c l x m transforms the principal part φ pr into f φ pr ( y ) = cy α y . This implies h ( φ pr ) = 2 = h ( φ ) . Notice also that in this case the principal face of the Newton polyhedron of φ, whenexpressed in adapted coordinates, must be the unbounded half-line with left endpoint( α, , so that ν ( φ ) = 0 . Q.E.D.We recall the following lemma, which is a (not quite straight-forward) consequenceof van der Corput’s lemma and whose formulation goes back to J. E. Bj¨ork (see [9])and G. I. Arhipov [1].
Lemma 2.2.
Assume that f is a smooth real valued function defined on an interval I ⊂ R which is of polynomial type m ≥ m ∈ N ) , i.e., there are positive constants c , c > such that c ≤ m X j =2 | f ( j ) ( s ) | ≤ c for every s ∈ I. Then for λ ∈ R , (cid:12)(cid:12)(cid:12) Z I e iλf ( s ) g ( s ) ds (cid:12)(cid:12)(cid:12) ≤ C k g k C ( I ) (1 + | λ | ) − /m , where the constant C depends only on the constants c and c . The case where the coordinates are adapted to φ, or where h = 2 . Weshall begin with the easiest case where either the coordinates x are adapted to φ, or h = 2 and condition (ii) in Lemma 2.1 is satisfied.Recall from [11] that if κ = ( κ , κ ) is any weight with 0 < κ ≤ κ such that the line L κ := { ( t , t ) ∈ R : κ t + κ t = 1 } is a supporting line to the Newton polyhedron N ( φ ) of φ, then the κ -principal part of φφ κ ( x , x ) := X ( j,k ) ∈ L κ c jk x j x k NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 11 is a non-trivial polynomial which is κ -homogeneous of degree 1 . By definition, we thenhave φ ( x , x ) = φ κ ( x , x ) + terms of higher κ -degree(see [11] for the precise meaning of this notion).We claim that we can choose a weight κ with 0 < κ ≤ κ < L κ is asupporting line to the Newton polyhedron of φ and1 | κ | = d h ( φ κ ) ≤ h ( φ κ ) = h. Indeed, in case that the coordinates are adapted to φ, this has been shown in [12],Lemma 2.4. And, if the coordinates are not adapted to φ but h ( φ pr ) = h ( φ ) , then theprincipal face is a compact edge, and we can choose for κ the principal weight, so that φ κ = φ pr . Notice that we have κ < , since ∇ φ (0 ,
0) = 0 . Let us denote by δ r the dilation by the factor r > κ, i.e., δ r ( x , x ) = ( r κ x , r κ x ) . In analogy with the proof of Theorem 11.1 in [12] we fix a suitable smooth cut-offfunction χ on R supported in an annulus D such that the functions χ k := χ ◦ δ k forma partition of unity, and then decompose J ( ξ ) = ∞ X k = k J k ( ξ ) , where J k ( ξ ) := Z ( R + ) e i ( ξ φ ( x )+ ξ x + ξ x ) η ( x ) χ k ( x ) dx = 2 − k | κ | Z ( R + ) e i (cid:16) − k ξ φ k ( x )+2 − kκ ξ x +2 − kκ ξ x (cid:17) η ( δ − k ( x )) χ ( x ) dx, with φ k ( x ) := 2 k φ ( δ − k x ) = φ κ ( x ) + error term . We claim that given any point x ∈ D, we can find a unit vector e ∈ R and some j ∈ N with 2 ≤ j ≤ h ( φ κ ) = h such that ∂ je φ κ ( x ) = 0 . Indeed, if the coordinates are adapted to φ, then this has been shown in Section 7of [12], and if they are not adapted to φ, then the same is true whenever x does notlie on the principal root of φ κ , as shown in Section 8 of [12]. However, if x does lie anthe principal root of φ κ , then according to Lemma 2.1 we may choose j = 2 . For k ≥ k sufficiently large we can thus apply Lemma 2.2 to the integration alonglines parallel to the direction e in the integral defining J k ( ξ ) near the point x . ApplyingFubini’s theorem and a partition of unity argument, we thus obtain | J k ( ξ ) | ≤ C k η k C ( R ) − k | κ | (1 + 2 − k | ξ | ) − /j ≤ C k η k C ( R ) − k | κ | (1 + 2 − k | ξ | ) − /m , (2.3)where m denotes the maximal j that arises in this context. Summation in k then yields the following estimates:(2.4) | J ( ξ ) | ≤ C k η k C ( R ) (1 + | ξ | ) − /m , if m | κ | > , log(2 + | ξ | ) (1 + | ξ | ) − /m , if m | κ | = 1 , (1 + | ξ | ) −| κ | , if m | κ | < . Now, if h ( φ ) = 2 and if the coordinates are not adapted to φ, then m = m ( φ pr ) >d ( φ ) = 1 / | κ | , so the first case in (2.4) applies. This implies (1.1), in view of Lemma2.1 (ii).Next, assume that the coordinates are adapted. If the principal face π ( φ ) is acompact edge, then φ κ = φ pr , hence 1 / | κ | = d ( φ ) = h, and moreover m ≤ h. Thisimplies | κ | m ≤ . Since in this case, by Lemma 1.5, ν ( φ ) = 1 if and only if m = m ( φ pr ) = h ( φ ) , i.e., if and only if m | κ | = 1 , we again obtain estimate (1.1).If π ( φ ) is unbounded, then m = h and 1 / | κ | < h, so that the first case in (2.4)applies and we again verify (1.1).Finally, if π ( φ ) is a vertex, then 1 / | κ | = h = m, so that the second case in (2.4)applies and we obtain (1.1) also in this case.2.2. The case of non-adapted coordinates: the contribution of regions awayfrom the principal root jet.
Assume next that the coordinates x are not adaptedto φ and that h > ψ which defines an adapted coordinate system(2.5) y := x , y := x − ψ ( x )for the function φ near the origin. In these coordinates, φ is given by φ a ( y ) := φ ( y , y + ψ ( y )) . In the case where Varchenko’s algorithm stops after a finite number of steps becausethe principal face π ( φ a ) is a compact edge and m ( φ a pr ) = d ( φ a ) , we meet the following convention :We assume that we have then run the algorithm one further step (as in the proofof the implication (b) ⇒ (a) in Lemma 1.5), so that we may assume that π ( φ a ) is avertex, i.e., the point ( h, h ) . Under this convention, π ( φ a ) will be a vertex whenever ν ( φ ) = 1 . Consider the Taylor series(2.6) ψ ( x ) ≈ X l ≥ b l x m l of ψ, where the b l are assumed to be non-zero. After applying a linear change ofcoordinates, if necessary, we may and shall assume that b = 0 and that the m l ∈ N form a strictly increasing sequence2 ≤ m < m < · · · . NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 13
Suppose that the vertices of the Newton diagram N d ( φ a ) of φ a are the points( A l , B l ) , l = 0 , . . . , n, so that the Newton polyhedron N ( φ a ) is the convex hull ofthe set S l (( A l , B l ) + R ) , where A l < A l +1 for every l ≥ . Let L l := { ( t , t ) ∈ N : κ l t + κ l t = 1 } denote the line passing through the points( A l − , B l − ) and ( A l , B l ) , and let a l := κ l /κ l . The a l can be identified as the distinctleading exponents of all the roots of φ a in case that φ a is analytic (see Section 3 of[12]), and the cluster of roots whose leading exponent in their Puiseux series expansionis given by a l is associated to the edge γ l := [( A l − , B l − ) , ( A l , B l )] of N ( φ a ) . As in Subsection 8.2 of [12], we choose the integer l ≥ a < · · · < a l − ≤ m < a l < · · · < a l < a l +1 < · · · < a n . As has been shown in Section 3 of [12], the vertex ( A l − , B l − ) lies strictly abovethe bisectrix, i.e., A l − < B l − , since the original coordinates x were assumed to benon-adapted.Distinguishing the cases listed below, we single out a particular edge by fixing thecorresponding index λ ≥ l as in Section 3 of [12]: Cases: (a) In case (a), where the principal face of φ a is a compact edge, we choose λ sothat the edge γ λ = [( A λ − , B λ − ) , ( A λ , B λ )] is the principal face π ( φ a ) of theNewton polyhedron of φ a . (b) In case (b), where π ( φ a ) is the vertex ( h, h ) , we choose λ so that ( h, h ) =( A λ , B λ ) . Then λ ≥ , and ( h, h ) is the right endpoint of the compact edge γ λ . (c) Consider finally case (c), in which the principal face π ( φ a ) is unbounded, namelya half-line given by t ≥ ν and t = h, where ν < h. Here, we distinguish twosub-cases:(c1) If the point ( ν , h ) is the right endpoint of a compact edge of N ( φ a ), thenwe choose again λ ≥ γ λ . (c2) Otherwise, ( ν , h ) = ( A , B ) is the only vertex of N ( φ a ) , i.e., N ( φ a ) =( ν , h ) + R . In this case, there is no non-trivial root r, hence n = 0 . In the cases (a), (b) and (c1), let us put(2.7) a := a λ = κ λ κ λ . We shall assume in the sequel that φ is analytic, since the general case can be reducedto this case as in [12].In a first step, we now decompose J ( ξ ) = J − ρ ( ξ ) + J ρ ( ξ ) , where ρ is the cut-offfunction introduced in Subsection 8.1 of [12] which localizes to a narrow κ -homogeneousneighborhood of the form(2.8) | x − b x m | ≤ ε x m of the curve x = b x m . Lemma 2.3.
Let ε > . If the neighborhood Ω of the point (0 , is chosen sufficientlysmall, then J − ρ ( ξ ) satisfies estimate (1.1) .Moreover, if N ( φ a ) is of the form ( ν , h ) + R , with ν < h, (case (c2) above), thenthe same statement holds true for J ( ξ ) in place of J − ρ ( ξ ) . Proof.
The oscillatory integral J − ρ ( ξ ) can be estimated in a similar way as in the caseof adapted coordinates by means of Lemma 2.2, and no logarithmic factor is needed.The reason for this is that any root of φ pr which does not agree with the principal root x = b x m has multiplicity strictly less then d ( φ ) , as can be seen from Corollary 2.3in [11], so that the third case of (2.4) applies.Moreover, if N ( φ a ) = ( ν , h ) + R , with ν < h, then we recall from the proof ofLemma 8.1 in [12] that φ κ ( x ) = cx ν ( x − b x m ) h , which implies h ( φ κ ) = h ( φ aκ ) = h, and we see that in this case we can again apply Lemma 2.2 to the x -integrationin order to see that also J ρ ( ξ ) satisfies (1.1), without logarithmic factor, since here1 / | κ | = d h ( φ ) < h. This proves also the second statement in the lemma. Q.E.D.We may and shall therefore from now on assume that the Newton polyhedron of φ a has at least one compact edge ”lying above” the principal face, i.e., that one of thecases (a), (b) or (c1) applies. There remains J ρ ( ξ ) to be considered.In a next step we shall narrow down the domain (2.8) to a neighborhood of theprincipal root jet of the form(2.9) | x − ψ ( x ) | ≤ N λ x a λ , where N λ is a constant to be chosen later. This domain is κ λ -homogeneous in theadapted coordinates y. More precisely, we fix a cut-off function ρ ∈ C ∞ ( R ) supportedin a neighborhood of the origin such that ρ = 1 near the origin, and put ρ λ ( x , x ) := ρ (cid:16) x − ψ ( x ) N λ x a (cid:17) . Proposition 2.4.
Let N λ > . If the neighborhood Ω of the point (0 , is chosensufficiently small, then the oscillatory integral J − ρ λ ( ξ ) satisfies estimate (1.1) .Moreover, if the principal face π ( φ a ) is a vertex or unbounded, then the same holdstrue for J ( ξ ) in place of J − ρ λ ( ξ ) . Proof.
To prove the first statement in the proposition, we decompose the difference setof the domains (2.8) and (2.9) in a similar way as in Subsection 8.2 of [12] into domains D l := { ( x , x ) : ε l x a l < | x − ψ ( x ) | ≤ N l x a l } , l = l , . . . , λ − , which are κ l -homogeneous in the adapted coordinates y given by (2.5), and intermediatedomains E l := { ( x , x ) : N l +1 x a l +1 < | x − ψ ( x ) | ≤ ε l x a l } , l = l , . . . , λ − , NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 15 and E l − := { ( x , x ) : N l x a l < | x − ψ ( x ) | ≤ ε x m } . Here, the ε l > N l > l < λ (and the one in [12]), we shall denote this domain by D λ , i.e., D λ := { ( x , x ) : | x − ψ ( x ) | ≤ N λ x a λ } . The localizations to these domains will be performed by means of cut-off functions ρ l ( x , x ) := ρ (cid:16) x − ψ ( x ) N l x a l (cid:17) − ρ (cid:16) x − ψ ( x ) ε l x a l (cid:17) , l = l , . . . , λ − ,τ l ( x , x ) := ρ (cid:16) x − ψ ( x ) ε l x a l (cid:17) (1 − ρ ) (cid:16) x − ψ ( x ) N l +1 x a l +1 (cid:17) , l = l , . . . , λ − , and τ l − ( x , x ) := ρ (cid:16) x − ψ ( x ) ε x m (cid:17) (1 − ρ ) (cid:16) x − ψ ( x ) N l x a l (cid:17) , respectively by ρ λ for the domain D λ . Here, in each instance ρ ∈ C ∞ ( R ) is a suitable cut-off function supported in theinterval [ − ,
1] such that ρ = 1 near the origin. Accordingly, we decompose J ρ ( ξ ) − J ρ λ ( ξ ) = λ − X l = l J ρ l ( ξ ) + λ − X l = l − J τ l ( ξ ) . The first part of Proposition 2.4 will be verified if we show that each of the oscillatoryintegrals J ρ l ( ξ ) and J τ l ( ξ ) arising in this sum satisfies estimate (1.1).Now, the estimates of Section 11 in [12] show that the J τ l ( ξ ) satisfy estimate (1.1),even without logarithmic factor, so we only need to consider the J ρ l ( ξ ) . Estimation of J ρ l ( ξ ) for l ≤ l ≤ λ − . Applying the change of coordinates (2.5),performing a dyadic decomposition and re-scaling similarly as in the case of adaptedcoordinates, only with the weight κ replaced by the weight κ l , we find that J ρ l ( ξ ) = ∞ X k = k J k ( ξ ) , where J k ( ξ ) = 2 − k | κ l | Z ( R + ) e i (cid:16) − k ξ φ k ( y )+2 − kκl ξ y +2 − kκl ξ y +2 − kκl ξ ψ k ( y ) (cid:17) ρ al ( y ) η a ( δ l − k y ) χ ( y ) dy, with ψ k ( y ) := 2 kκ l ψ (2 − kκ l y ) , φ k ( y ) := φ aκ l ( y ) + 2 k φ ar ( δ l − k y ) , and ρ al ( y ) := ρ (cid:16) y N l y a l (cid:17) − ρ (cid:16) y ε l y a l (cid:17) . Here, δ lr denotes the dilation by r > κ l , and we have againdecomposed φ a = φ aκ l + φ ar , where φ ar depends in fact also on l and consists of terms of κ l -degree higher than 1 . Since l ≤ λ − J k ( ξ ) by means of Lemma 2.2, appliedto the y -integration, by using Corollary 3.2 (i) in [12], and obtain | J k ( ξ ) | ≤ C k η k C ( R ) − k | κ l | (1 + 2 − k | ξ | ) − /d h ( φ aκl ) ≤ C k η k C ( R ) − k | κ l | (1 + 2 − k | ξ | ) − /h (2.10)since d h ( φ aκ l ) < h. This also implies 1 = | κ l | d h ( φ aκ l ) | < | κ l | h, so that a comparison with(2.4) shows that summation over k yields | J ρ l ( ξ ) | ≤ C k η k C ( R ) (1 + | ξ | ) − /h . We next turn to the second statement in Proposition 2.4. We have to show that also J ρ λ ( ξ ) satisfies estimate (1.1). However, if the principal face π ( φ a ) is a vertex (case(b)) or unbounded (case (c1)) then Corollary 3.2 (ii) in [12] allows us to argue exactlyas before in order to see that (2.10) also holds for l = λ. And, in case (b) we have | κ λ | h = 1 , whereas in case (c1) we have | κ λ | h > , so that a comparison with (2.4)shows that estimate (1.1) is indeed valid for J ρ λ ( ξ ) . Q.E.D.2.3.
The contribution of the homogenous domain D λ containing the principalroot jet. In view of Proposition 2.4, we may and shall from now on assume that theprincipal face of N ( φ a ) is a compact edge (case (a)). What remains to be estimatedis the contribution of the domain (2.9) to J ( ξ ) , i.e., we are left with the oscillatoryintegral J ρ λ ( ξ ) . This will require different arguments then those used in [12]. We arealso assuming that x > . Recall also that according to our convention(2.11) m ( φ a pr ) < d ( φ a ) = h, so that ν ( φ ) = 0 . This means that we have to prove that J ρ λ ( ξ ) satisfies (1.1), withoutthe presence of a logarithmic factor.2.3.1. Preliminary reductions.
Following [12], Section 9, it will be convenient at thispoint to defray our notation by writing φ in place of φ a and η in place of η a , κ in placeof κ λ , δ r in place of δ λr , etc.. With some slight abuse of notation, we shall denote J ρ λ ( ξ )by J ( ξ ) . After applying the change of coordinates (2.5) , this means that from now on we shallhave to consider oscillatory integrals J ( ξ ) := Z R e i (cid:16) ξ φ ( x )+ ξ x + ξ ( x + ψ ( x )) (cid:17) ρ (cid:16) x N x a (cid:17) η ( x ) dx, NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 17 where a = κ /κ > m , and where N is a given, possibly large positive number. Noticethat the integration takes place only over the domain(2.12) | x | ≤ N x a . We shall write m := m , so that ψ can be factored as ψ ( x ) = x m σ ( x ) , with a smoothfunction σ satisfying σ (0) = 0 . J ( ξ ) can thus be written as an oscillatory integral(2.13) J ( ξ ) = Z R e iF ( x,ξ ) ρ (cid:16) x N x a (cid:17) η ( x ) dx, with a phase function F ( x, ξ ) := ξ φ ( x ) + ξ x + ξ x m σ ( x ) + ξ x depending on ξ ∈ R . The coordinates x are now adapted to φ. We shall again decom-pose φ ( x ) = φ κ + φ r , where φ κ consists of terms of κ -degree strictly bigger then 1 , the κ -degree of φ κ . In order to estimate J ( ξ ) , in a first step we shall decompose the domain (2.12) intosmaller, κ -homogeneous sub-domains. To this end, given any point c ∈ [0 , N ] , wedefine J c ( ξ ) := Z R e iF ( x,ξ ) ρ (cid:16) x − cx a ε x a (cid:17) η ( x ) dx, where ε > ρ here is possiblydifferent from the one in (2.13)).In order to prove that J ( ξ ) satisfies estimate (1.1), it will be sufficient to show thatfor every c ∈ [0 , N ] there exists an ε > J c ( ξ ) satisfies (1.1), as can beseen easily be means of a partition of unity argument.We therefore assume that c is fixed. Then we take again a smooth cut-off function χ which is supported in an annulus D such that ∞ X k = k χ ( δ k ( x )) = 1 for every x ∈ supp η \ { } . Notice that we can assume that k is a sufficiently large positive integer by choosingthe support of η sufficiently small. Then we have J c ( ξ ) = ∞ X k = k J k ( ξ ) , where J k ( ξ ) := Z R e iF ( x,ξ ) ρ (cid:16) x − cx a ε x a (cid:17) η ( x ) χ ( δ k ( x )) dx. After the change of variables x δ − k ( x ) we obtain (2.14) J k ( ξ ) = 2 −| κ | k Z e i − k ξ F k ( x,s ) ρ (cid:16) x − cx a ε x a (cid:17) η ( δ − k ( x )) χ ( x ) dx, where F k ( x, s ) := φ κ ( x ) + 2 k φ r ( δ − k ( x )) + s x + S x m σ (2 − κ k x ) + s x ,s := 2 (1 − κ ) k ξ ξ , s := 2 (1 − κ ) k ξ ξ , S := 2 ( κ − mκ ) k s ,s := ( s , s , S ) . Note that 2 ≤ m < a = κ /κ and k ≫ , so that | S | ≫ | s | . Observe also that thereexists a compact interval I such that x ∼ I, so that for every ( x , x ) in thesupport of the integrand of J k ( ξ ) as given by (2.14), we have x ∈ I and | x − cx a | . ε . Recall also from (2.1) that we are assuming that | ξ | ∼ | ξ | . Estimation of the oscillatory integrals J k ( ξ ) . In order to estimate J k ( ξ ) , we shalldistinguish several cases depending on the size of the parameters s , s and S . Recallhere that ξ is a function of ξ , s , s and S . Case 1. | S | ≥ M for some sufficiently large constant M ≫ . In this case we canapply Lemma 2.2 to the x -integration and obtain(2.15) | J k ( ξ ) | ≤ C − k | κ | k η k C (1 + 2 − k | ξ | ) / . Case 2. | S | < M, where M is chosen as in Case 1. Then | s | ≪ , provided wehave chosen k sufficiently large.If we assume that there is some j ≥ ∂ j φ κ (1 , c ) = 0 , then we claim that(2.17) | J k ( ξ ) | ≤ C − k | κ | k η k C (1 + 2 − k | ξ | ) /j . Indeed, by the homogeneity of φ κ , if we choose ε sufficiently small, then ∂ j φ κ ( x ) = 0at every point x in the support of the integrand of J k ( ξ ) , so that the estimate followsfor j ≥ x -integration. Notice thatthe term 2 k φ r ( δ − k ( x )) can be viewed as a perturbation term. Similarly, if j = 1 , theestimate follows by an integration by parts with respect to x . We observe that if (2.16) holds for some ≤ j < h, then by (2.15) , (2.17) and (2.4) we obtain the desired estimate (1.1) , even without a logarithmic factor, since h > . We may and shall therefore henceforth assume that(2.18) ∂ j φ κ (1 , c ) = 0 for 1 ≤ j < h. NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 19
Recall that we are assuming that the principal face of N ( φ ) is a compact edge, sothat φ κ = φ pr and h = 1 / | κ | . Assume first that c = 0 . Then necessarily φ pr (1 , = 0 , for otherwise φ pr wouldhave a root of multiplicity at least h at (1 , , which would contradict (2.11).Assuming without loss of generality that φ pr (1 ,
0) = 1 , we can then write (compare[12], Subsection 9.1) φ pr ( x , x ) = x B Q ( x , x ) + x n , where Q is a κ -homogeneous polynomial such that Q (1 , = 0 , and where B ≥ h > . Recall that | S | < M, so that | s | ≪ . We now distinguish two subcases:
Case 2.a. | S | < M, and | s | ≥ N for some sufficiently large constant N ≫ . Then an integration by parts in x leads to the estimate | J k ( ξ ) | ≤ C − k | κ | k η k C − k | ξ | , whichin return implies (1.1), even without logarithmic factor. Case 2.b. | S | < M, hence | s | ≪ , and | s | < N, where N is chosen as in Case2.a.We shall show that, given any point ( s , S ) ∈ [ − M, M ] × [ − N, N ] and any point x ∈ I, there exist a neighborhood U of ( s , S ) , a neighborhood V of x and some ω > /h such that we have an estimate of the form(2.19) | J k ( ξ ) | ≤ C − k | κ | k η k C (1 + 2 − k | ξ | ) ω for every ( s , S ) ∈ U, provided the function χ in the definition of J k ( ξ ) is supportedin V and ε and k are chosen sufficiently small, respectively large. The same type ofestimate will then hold also for every ( s , S ) ∈ [ − M, M ] × [ − N, N ] and for the originalfunction χ in the definition of J k ( ξ ) , as can be seen by means of a partition of unityargument. Summing over all k, this will clearly imply the estimate (1.1), even withoutlogarithmic factor.To this end, first notice that for ( s , S ) ∈ U and k sufficiently large, the function F k ( x, s ) can be viewed as a small C ∞ - perturbation of the function F pr ( x ) := x B Q ( x , x ) + s x + S σ (0) x m + x n . Thus, if ∇ F pr (0 , x ) = 0 , then we obtain (2.19), with ω = 1 , simply by integration byparts.Assume therefore that (0 , x ) is a critical point of F pr . Then x is a critical point ofthe polynomial function g ( x ) := s x + S σ (0) x m + x n , which comprises all terms of F pr depending on the variable x only. Note that 2 ≤ m < n, since n = 1 /κ > κ /κ > m. It is then easy to see that g ′′ and g ′′′ cannot vanish simultaneously at the given point x ∈ I, so that there are positive constants c , c > V of x such that c ≤ X j =2 | g ( j ) ( x ) | ≤ c for every x ∈ V. This implies an analogous estimate for the partial derivatives ∂ jx F k ( x , x , s ) of F k , uniformly for ( s , S ) ∈ U and x satisfying (2.12), provided we choose U and ε sufficiently small. Applying the van der Corput type estimate in Lemma 2.2, we thusobtain the estimate (2.19) with ω = 1 / , so that we are done provided h > . Noticealso that if g ′′ ( x ) = 0 , then by the same type of argument we see that (2.19) holdstrue with ω = 1 / > /h. We may thus finally assume that 2 < h ≤ , and that g ′ ( x ) = g ′′ ( x ) = 0 . In thiscase we have 1 κ + κ = h ≤ κ κ > m ≥ , so that 1 /κ < / . Note that B ≤ /κ is a positive integer, and h ≤ B < / , so that either B = 4 or B = 3. We translate the critical point ( x ,
0) of F pr to the origin by considering thefunction F ♯ pr ( y ) := F pr ( x + y , y ) − g ( x ) = y B Q ( x + y , y ) + 16 g (3) ( x ) y + . . . . It is easy to see that this function has height h ♯ := h ( F ♯ pr ) given by h ♯ = / / = 12 / , if B = 4 , and h ♯ = / / = 3 / , if B = 3 . In both cases, F ♯ pr has height h ♯ < F ♯ pr is of type E and D , respectively). We can therefore again applyDuistermaat’s results in [10] to the oscillatory integral J k ( ξ ) and obtain the estimate(2.19), with ω = 1 /h ♯ > /h. Note here that the estimates in [10] are stable undersmall perturbations.
Assume finally that c > . Then, by Corollary 3.2 (iii) in [12], our assumption(2.18) implies that necessarily a = κ /κ ∈ N . We can then reduce this case to the previous case c = 0 by performing anotherchange of variables x x + cx a in the integral defining J k ( ξ ) . Indeed, this is equivalent to replacing the function ψ in our previous argument by ψ ♯ ( x ) := ψ ( x ) + cx a , and assuming that c = 0 . Denote by φ ♯ ( x , x ) := φ ( x , x + cx a )the corresponding phase function. Then the coordinates ( x , x ) are adapted to φ ♯ too,as can be seen as follows:Lemma 3.1 in [11] shows that ( φ pr ) ♯ ( x , x ) := φ pr ( x , x + cx a ) is again a κ -homogeneous polynomial whose principal face intersects the bi-sectrix, and m ( φ pr ) = m (( φ pr ) ♯ ) . Therefore ( φ pr ) ♯ must be the principal part of φ ♯ . This completes the proof of Theorem 1.1.
NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 21 Sharpness of the uniform estimates
In this section, we shall give a proof of Theorem 1.3. Observe that we may assumefor this purpose that the coordinates ( x , x ) are adapted to φ, so that d := d ( φ ) = h. We shall only consider the asymptotic behavior of J + ( λ ) , since the result for J − ( λ )follows from the one for J + ( λ ) by means of complex conjugation. Remarks 3.1. If h < , then the phase function φ has a critical point at the originwith finite Milnor number, and can thus be reduced to a polynomial phase functionby means of a smooth local change of coordinates at the origin (see [2]). Therefore,in this case we could apply the classical results for analytic phase functions by A.N.Varchenko [21]. However, we will give a more elementary argument which does notrely on this classification of singularities.Notice also that if h = 1 , then the phase function has a non-degenerate critical pointat the origin in our adapted coordinates, and we could apply the method of stationaryphase in order to prove the existence of the limits in Theorem 1.3 (see [19]).We shall,however, proceed somewhat differently also in this case.3.1. The case where the principal face is a compact edge.
We begin with thesimplest case where the principal face π ( φ ) is a compact edge. Arguing as in Subsection2.2, we may then assume in addition that m ( φ pr ) < d, since otherwise a suitable local change of coordinates would reduce us to the situationwhere the principal face is a vertex.Then there is a unique weight κ such that π ( φ ) is lying on the line given by theequation κ t + κ t = 1 . Without loss of generality we may assume that 0 < κ ≤ κ .Recall also that then φ pr = φ κ and d = d h ( φ κ ) = 1 / | κ | , and that if we decompose φ ( x ) = φ κ ( x ) + φ r ( x ) , then φ r is an error term whose Newton polyhedron is contained in the set { ( k , k ) ∈ Z : κ k + κ k > } . In a first step, we shall reduce ourselves to the situation where the amplitude a isconstant on a neighborhood of the origin. To this end, if Ω is an open neighborhoodof the origin in R , let us introduce the subspace of amplitude functions˙ C (Ω) := { a ∈ C (Ω) : a (0 ,
0) = 0 } . If a ∈ ˙ C (Ω) and if F is a smooth, real-valued phase function on Ω , we consider theoscillatory integral J ( λ, F, a ) := Z e iλ ( φ κ ( x )+ F ( x )) a ( x ) dx, λ > . Proposition 3.2.
There exists a positive number ε such that for any smooth function F ∈ C ∞ ( R ) with N ( F ) ⊂ { ( k , k ) ∈ Z : κ k + κ k > } there exists a neighborhood Ω of the origin so that for any a ∈ ˙ C (Ω) the following estimate | J ( λ, F, a ) | ≤ C ( F ) k a k C (Ω) λ /d + ε holds true, with a constant C ( F ) depending only on the C N (Ω) norm of F, for somesufficiently large number N .Proof. If a ∈ ˙ C (Ω) , then a can be written as a ( x , x ) = x a ( x , x ) + x a ( x , x ) , with functions a , a ∈ C (Ω) whose C -norms can be controlled by the C -norm of a. Consequently for the oscillatory integral we have J ( λ, F, a ) = J ( λ, F, x a ) + J ( λ, F, x a ) . We shall therefore estimate J ( λ, F, x a ) ( J ( λ, F, x a ) can be treated in a similarway). As before, we choose a suitable smooth cut-off function χ on R supported inan annulus D such that the functions χ k := χ ◦ δ k form a partition of unity, and thendecompose J ( λ, F, x a ) = ∞ X k = k J ( λ, F, x a χ k ) . Here, δ r denotes again the dilation by the factor r > κ. Recall that by choosing Ω sufficiently small we may assume that k is a sufficientlylarge number. After re-scaling, we may re-write J k ( λ ) := J ( λ, F, x a χ k )as J k ( λ ) = 2 − ( | κ | + κ ) k Z e iλ − k (Φ κ +2 k F ( δ k ( x ))) x a ( δ − k ( x )) χ ( x ) dx. If λ − k ≤ M (with a fixed positive number M ), a trivial estimate for the integral J k ( λ ) gives | J k ( λ ) | ≤ C k a k C (Ω) − ( | κ | + κ ) k , hence(3.20) X λ − k ≤ M | J k ( λ ) | ≤ C M k a k C (Ω) λ | κ | + κ , if we assume without loss of generality that Ω is a ball.Assume next that λ − k > M. Since k k F ◦ δ − k k C m (Ω) → k → + ∞ , by choosingΩ sufficiently small we may assume that k k F ◦ δ − k k C m (Ω) is sufficiently small.Now, if m ( φ κ ) ≥ , we put m := m ( φ κ ) . Then 1 ≤ m < d. Notice that if x ∈ D is such that ∇ φ κ ( x ) = 0 , then, by Euler’s homogeneity relation, also φ ( x ) = 0 . Therefore, by applying Lemma 2.2, respectively an integration by parts, and assumingthat M is sufficiently big, we see that NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 23 | J k ( λ ) | ≤ C ( F ) k a k C (Ω) − ( | κ | + κ ) k (1 + 2 − k λ ) − /m . Summing in k, this implies(3.21) X λ − k >M | J k ( λ ) | ≤ C ( F ) k a k C (Ω) (1 + λ ) − /m , if m ( | κ | + κ ) > , log(2 + λ ) (1 + λ ) − /m , if m ( | κ | + κ ) = 1 , (1 + λ ) − ( | κ | + κ ) , if m ( | κ | + κ ) < . If we put ε := min { κ , /m − /d } , we see that (3.20) and (3.21) imply that | J ( λ, F, x a ) | ≤ C ( F ) k a k C (Ω) λ /d + ε , for every positive number ε < ε . Similar estimates hold true for J ( λ, F, x a ) , onlywith κ replaced by κ . Since κ ≤ κ , we see that we can use the same range of ε ’salso in this case and obtain the desired estimate in Proposition 3.2.There remains the case where m ( φ κ ) = 0 . Here, φ κ does not vanish away from theorigin, and thus ∇ φ κ ( x ) = 0 for every x ∈ D. Thus, choosing m = 1 here andapplying one integration by parts, we again obtain estimate (3.21), and can concludeas before, if d > . Finally, if d = 1 (notice that necessarily d ≥ , since ∇ φ (0 ,
0) = 0), applying twointegrations by parts to J k ( λ ) , we obtain X λ − k >M | J k ( λ ) | ≤ C ( F ) k a k C (Ω) λ − ( | κ | + κ ) , where | κ | = 1 . Thus, we can choose ε := κ in this case. Q.E.D.Let us now consider the oscillatory integral J + ( λ ) := Z R e iλφ ( x ) η ( x ) dx, where η ∈ C ∞ (Ω) . Choose a smooth bump function χ supported in Ω which is iden-tically 1 on a neighborhood of the origin. Then, if we choose Ω sufficiently small,Proposition 3.2 implies that the oscillatory integrals J + ( λ ) and η (0 , J ( λ ) , with J ( λ ) := Z R e iλφ ( x ) χ ( x ) dx, differ by a term of decay rate O ( λ − /d − ε ) . This shows that, in order to prove Theorem1.3 in this case, it suffices to prove that the limit(3.22) lim λ → + ∞ λ /d J ( λ ) = c exists and that c = 0 . To this end, put δ := ε/ , with ε > P and Q by Q ( x ) := X | α |≤ /δ +3 ∂ α φ (0) α ! x α =: φ κ ( x ) + P ( x ) . Notice that all the derivatives of the function e iλ ( φ ( x ) − Q ( x )) up to order 3 are uniformlybounded with respect to λ on the set where λ δ | x | <
1. We therefore decompose J ( λ ) = Z e iλφ ( x ) χ ( x ) χ ( λ δ x ) dx + Z e iλφ ( x ) χ ( x )(1 − χ ( λ δ x )) dx. Due to Proposition 3.2 (with F := P ), the second summand has decay rate of order O ( λ − /d − ε +3 δ ) = O ( λ − /d − δ ) as λ → + ∞ , if Ω is supposed to be chosen sufficientlysmall. In order to prove (3.22), we may therefore replace J ( λ ) by the first summand, J ( λ ) , which we again decompose as J ( λ ) = Z e iλφ ( x ) χ ( x ) χ ( λ δ x ) dx = Z e iλQ ( x ) χ ( x ) dx + Z e iλ ( φ κ ( x )+ P ( x )) χ ( x ) (cid:16) χ ( λ δ x ) e iλ ( φ ( x ) − Q ( x )) − (cid:17) dx. Again, by applying Proposition 3.2, we see that the second summand has decay rate O ( λ − /d − δ ) as λ → + ∞ , and thus we are reduced to proving that the limitlim λ → + ∞ λ /d Z e iλQ ( x ) χ ( x ) dx = c exists and that c = 0 . But, Q ( x ) = φ κ ( x ) + P ( x ) is a polynomial, with principal part φ κ , and therefore this statement follows from the classical results for analytic phasefunctions due to Varchenko [21] (see also [7]).3.2. The case where the principal face is a vertex.
Assume now that π ( φ ) = { ( d, d ) } is a vertex, so that in particular d is a positive integer. After multiplying thephase function with a suitable real constant (this can be implemented by means ofa suitable scaling in λ and, possibly, complex conjugation of J + ( λ )), we may assumewithout loss of generality that the principal part of φ is given by φ pr ( x ) = x d x d . We may also assume that the coordinates ( x , x ) are super-adapted, in the senseof Greenblatt [7]. Then, if d = h = 1 , according to Lemma 1.0 in [7], the criticalpoint of φ at the origin is non-degenerate, and thus the statement of Theorem 1.3 is awell-known consequence of the method of stationary phase.Let us therefore henceforth assume that d = h ≥ NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 25
Two compact edges.
First, assume that the Newton polyhedron N ( φ ) has twocompact edges containing the vertex ( d, d ) as one of their endpoints, say γ a , lying”above” the bi-sectrix and on the line given by κ a t + κ a t = 1 , and γ b , lying ”below”the principal face and on the line given by κ b t + κ b t = 1 . Notice that then(3.23) a := κ a κ a < κ b κ b =: b. Lemma 3.3.
The function φ can be written as φ ( x , x ) = x d x d + φ a ( x , x ) + φ b ( x , x ) , where φ a ( x , x ) = x d ˜ φ a ( x , x ) and φ b ( x , x ) = x d ˜ φ b ( x , x ) , with smooth functions ˜ φ a and ˜ φ b . The proof of Lemma 3.3 is straightforward. Notice also that we have x d x d + φ a ( x , x ) = φ κ a ( x , x ) + φ a,r and x d x d + φ b ( x , x ) = φ κ b ( x , x ) + φ b,r , where φ κ a is κ a := ( κ a , κ a )-homogeneous of degree 1 , and φ a,r consists of terms of κ a -degree higher than 1 , and where the analogous statements holds true for φ κ b and φ b,r . Lemma 3.4.
After applying a suitable smooth local change of coordinates at the origin,we may assume that the functions x φ κ a ( x , ± and x φ κ b ( ± , x ) have no rootof multiplicity greater or equal to d, respectively.Proof. We may assume that b ≥ , for otherwise, after interchanging the coordinates x and x , we will have b ≥ a ≥ . Then, the proof of Theorem 7.1 in [7] for the existence of “super-adapted coordinates”shows that, after applying a suitable local change of coordinates at the origin, we mayassume that φ κ b ( ± , x ) has no non-zero root of order greater or equal to d (of course,the edge γ b may have changed and even have become unbounded, but this would bea case to be considered later). We also remark that the change of coordinates in [7]is such that the edge γ a remains to be an edge of the Newton diagram in the newcoordinates.According to Proposition 2.2 in [11], we can then write, say for x > ,φ κ b ( x , x ) = x α x β Y l ( x q − c l x p ) n l , for suitable integers α, β ≥ p, q ≥ p/q = b, where c l ∈ C \ { } and n l ∈ N \ { } . Since we are assuming that ( d, d ) is the upper vertex of the edge γ b , wesee that α = d and β + ( P l n l ) q = d. Therefore, necessarily β < d, which shows that φ κ b ( ± , x ) that also x = 0 is no root of order greater or equal to d. We now turn to φ κ a . If a ≤ , after interchanging again the coordinates x and x , hence also the edges γ a and γ b , we may assume that x φ κ a ( x , ±
1) has no root ofmultiplicity greater or equal to d, and that a ≥ . Applying then the previous argument again to γ b , we see that in addition we may assume that φ κ b ( ± , x ) has no root ofmultiplicity greater or equal to d, and are done.Assume finally that a > . Then we can accordingly write φ κ a ( x , x ) = x α x β Y l ( x q − c l x p ) n l , where now p/q = a. Since ( d, d ) is the lower vertex of γ a , we see that β = d and α + ( P l n l ) p = d, hence α < d. Moreover, if a / ∈ N , then Corollary 2.3 in [11] showsthat n l < d for every l, which shows that φ κ a ( x , ±
1) has no root of multiplicity greateror equal to d. And, if a ∈ N , then q = 1 and p = a > , hence n l < n l p ≤ d, so that again n l < d, and we can conclude as before. Q.E.D.Let us assume in the sequel that the adapted coordinates are chosen so that theconclusions in Lemma 3.4 do apply, and consider again the oscillatory integral J + ( λ ) := Z R e iλφ ( x ) η ( x ) dx. Note that in this case we have to prove thatlim λ → + ∞ λ /d log λ J + ( λ ) = c η (0) , where c = 0 . With χ as before, let us consider the oscillatory integrals J ( λ ) := Z e iλφ ( x ) ( η ( x ) − η (0) χ ( x )) dx and J ( λ ) := Z R e iλφ ( x ) χ ( x ) dx. We then have the following substitute for Proposition 3.2, which allows to reduce toproving that the following limit(3.24) lim λ → + ∞ λ /d log λ J ( λ ) = c exists and is non-zero. Lemma 3.5. If Ω is chosen sufficiently small, then the following estimate | J ( λ ) | ≤ C k η k C (Ω) λ /d holds true. NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 27
Proof.
Permuting the coordinates x , x , if necessary, we may choose a weight κ =( κ , κ ) with 0 < κ ≤ κ , such that the line given by κ t + κ t = 1 is a supportingline to N ( φ ) which contains only the point ( d, d ) of N ( φ ) . Arguing now in the sameway is in the proof of Proposition 3.2, with m := d, we obtain the desired estimate.Q.E.D.Choose a smooth cut-off function χ ∈ C ∞ ( R ) supported in a sufficiently smallneighborhood of the origin. In order to prove (3.24), let us decompose J ( λ ) = J ( λ ) + J ∞ ( λ ) , where J ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) χ (cid:18) x ε | x | a (cid:19) χ (cid:18) x ε | x | /b (cid:19) dx, (3.25) J ∞ ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) (cid:18) − χ (cid:18) x ε | x | a (cid:19) χ (cid:18) x ε | x | /b (cid:19)(cid:19) dx, (3.26)where ε > Lemma 3.6.
Let ε > . Then, if Ω is chosen sufficiently small, the following estimate | J ∞ ( λ ) | ≤ C k η k C (Ω) λ /d holds true.Proof. We decompose J ∞ ( λ ) = J a ( λ ) + J b ( λ ) , where J a ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) (cid:18) − χ (cid:18) x ε | x | a (cid:19)(cid:19) dx,J b ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) (cid:18) − χ (cid:18) x ε | x | /b (cid:19)(cid:19) χ (cid:18) x ε | x | a (cid:19) dx, and show that both terms separately satisfy the estimate in Lemma 3.6.We begin with J a ( λ ) . Using the dilations δ r associated to the weight κ a , we dyadicallydecompose J a ( λ ) = P ∞ k = k J k ( λ ) in a similar way as in the proof of Lemma 3.2. Here,after re-scaling, J k ( λ ) is given by J k ( λ ) = 2 −| κ a | k Z e iλ − k (Φ κa +2 k φ r ( δ k ( x ))) χ ( δ − k ( x )) (cid:18) − χ (cid:18) x ε | x | a (cid:19)(cid:19) χ ( x , x ) dx, where | κ a | = 1 /d. Notice that | x | . ε . | x | . x , x ) in the support of the integrand. Let m denote the maximal order ofvanishing of φ κ a transversal to its roots on this domain. Then m < d, since, accordingto Lemma 3.4, we are assuming that φ κ a ( x , ±
1) has no root of order greater or equal to d. Consequently, we have m | κ a | < . Arguing as in the proof of Lemma 3.2 in order toestimate the J k ( λ ) , and summing in k, we then find that | J a ( λ ) | ≤ Cλ −| κ a | = Cλ − /d . Finally, J b ( λ ) can be estimated in a very similar way, making use of the dilations as-sociated to the weight κ b in place of κ a . Note that the additional factor χ (cid:16) x / ( ε | x | a ) (cid:17) appearing in the integral defining J b ( λ ) is under control because of (3.23). Q.E.D.The proof of (3.24) is thus reduced to proving the next Lemma 3.7.
The following limit lim λ → + ∞ λ /d log λ J ( λ ) exists and is non-zero. Moreover, it does not depend on the choice of ε. Proof.
Let us first assume that the integer d ≥ χ is an even function and that χ is radial, so that in particular χ ( x , x ) = χ ( ± x , ± x ) . This implies that, if we decompose the integral defining J ( λ ) into the four integralsover each of the quadrants of R , then, after an obvious change of coordinates, allfour of them will have the same amplitude, as well as the same principal part x d x d fortheir phases. Since we shall see that the leading term in the asymptotic expansion of J ( λ ) will only depend on the principal part of the phase function, we may thus reduceourselves to considering the integral J +0 ( λ ) over the positive quadrant only.Notice that by (3.23) b − a > . In the integral for J +0 ( λ ) we apply the change ofvariables x = x a y , x = y b − a y , and denote by ˜ φ the phase function when expressed in the coordinates y, i.e., ˜ φ ( y ) = φ ( x ) . Observe that this change of coordinates is of class C on the closed positive quadrant,and of class C ∞ away from the coordinate axes, and that it leads to the following formof the phase function ˜ φ :˜ φ ( y , y ) = y d (1+ a )1 y d (1+ b ) b − a (1 + ρ ( y δ , y δ )) , where ρ ( z , z ) is a smooth function with ρ (0 ,
0) = 0 , and where δ = 1 /p > N ( ˜ φ ) = ( d, d ) + R under thischange of variables, and since x = y b − a y , x = y a y ab − a , it is clear that if f is any smooth function of x which is flat at the origin, i.e., whichvanishes to infinity oder at the origin, then ˜ f , defined by ˜ f ( y ) = f ( x ) , can be factoredas ˜ f ( y ) = y d (1+ a )1 y d (1+ b ) b − a g ( y ) , where also g is smooth and flat at the origin. NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 29
The oscillatory integral J +0 ( λ ) then transforms into J +0 ( λ ) = Z e iλ ˜ φ ( y ,y ) y a y ab − a χ (cid:16) y ε (cid:17) χ (cid:16) y ε (cid:17) ˜ χ ( y , y ) dy, where ˜ χ is of class C on the closed positive quadrant, and of class C ∞ away from thecoordinate axes, and ˜ χ (0 ,
0) = 1 . Observe next that if M is any positive constant, then the contribution to the integral J +0 ( λ ) by the sub-domain where λy d (1+ a )1 ≤ M is trivially of order O ( λ − /d ) as λ → + ∞ . We may therefore consider the oscillatory integral I ( λ ) := Z λy d (1+ a )1 >M Z e iλ ˜ φ ( y ,y ) y a y ab − a χ (cid:16) y ε (cid:17) χ (cid:16) y ε (cid:17) ˜ χ ( y , y ) dy dy = Z λy d (1+ a )1 >M y a χ (cid:16) y ε (cid:17) I int ( λ, y ) dy in place of J +0 ( λ ) , where M is a fixed, sufficiently large positive number.Assuming that ε > z := y (1 + ρ ( y δ , y δ )) b − ad (1+ b ) to the inner integral I int ( λ, y ) := Z e iλ ˜ φ ( y ,y ) χ (cid:16) y ε (cid:17) ˜ χ ( y , y ) y ab − a dy , which leads to I int ( λ, y ) = Z e iλy d (1+ a )1 z d (1+ b ) b − a χ (cid:16) z (1 + ˜ ρ ( y , z )) ε (cid:17) ˜ χ , ( y , z ) z ab − a dz , where ˜ ρ and ˜ χ , have similar properties as ρ and ˜ χ , respectively. Changing variablesin this integral to t := z bb − a , and applying some classical results on one-dimensionaloscillatory integrals with critical points of order d (see A. Erde’lyi [5], Section 2.9), wethus obtain I int ( λ, y ) = b − a b (cid:16) C d ( λy d (1+ a )1 ) /d + R ( λ, y ) (cid:17) , where C d = 0 is given explicitly by(3.27) C d := Γ(1 /d ) d e πi d , and where the remainder term satisfies an estimate | R ( λ, y ) | ≤ C ′ d ( λy d (1+ a )1 ) /d + δ , where δ > C ′ d can be chosen indepen-dently of a and b (we mention this here for later use). The latter estimate impliesthat (cid:12)(cid:12)(cid:12) Z λy d (1+ a )1 >M y a χ (cid:16) y ε (cid:17) R ( λ, y ) dy (cid:12)(cid:12)(cid:12) ≤ C λ /d , whereas the corresponding integral over the principal part of I int ( λ, y ) behaves asymp-totically like c log λ/λ /d , as required. Explicitly, our argument shows that(3.28) lim λ → + ∞ λ /d log λ J ( λ ) = 4 b − a b C d , if d is even.Finally, if d is odd, a very similar reasoning shows that(3.29) lim λ → + ∞ λ /d log λ J ( λ ) = 2 b − a b ( C d + C d ) . Q.E.D.We have thus proved the theorem in the case where the Newton polyhedron N ( φ )has two compact edges containing the vertex ( d, d ) as one of their endpoints.Assume therefore next that at least one of the two edges containing the vertex ( d, d )is unbounded. We shall then argue in a similar way as in the previous case, however,by approximating the unbounded faces by compact line segments which have ( d, d ) asone of their vertices and which lie on supporting lines to N ( φ ) whose angle with theunbounded face tend to zero.3.2.2. Two unbounded edges.
Assume next that both edges containing ( d, d ) are un-bounded, i.e., that N ( φ ) = ( d, d ) + R . Let us then choose numbers a, b such that0 < a < < b, where later we shall let a tend to 0 and b to ∞ . We associate to a and b weights κ a := (cid:16) a ) d , a (1 + a ) d (cid:17) , κ b := (cid:16) b ) d , b (1 + b ) d (cid:17) . Then the supporting lines mentioned before will be given by κ a t + κ a = 1 and κ b t + κ b = 1 , respectively, and the identities (3.23) remain valid.We can then proceed as in the previous case, reducing to the asymptotic analysis of J ( λ ) , which in return is decomposed into J ( λ ) and J ∞ ( λ ) , given by (3.25) and (3.26),respectively. We further decompose J ∞ ( λ ) = J a ( λ ) + J b ( λ ) as in the proof of Lemma3.6.In place of this lemma, we here have NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 31
Lemma 3.8.
Let ε > . Then, if Ω is chosen sufficiently small, the following estimates | J a ( λ ) | ≤ A d (cid:16) λ /a (cid:17) λ − /d , (3.30) | J b ( λ ) | ≤ A d (cid:16) λ b (cid:17) λ − /d (3.31) hold true, with a constant A d which does not depend on a and b, but only on d .Proof. We shall prove the estimate for J a ( λ ); the proof of the corresponding estimatefor J b ( λ ) is obtained by the same kind of reasoning, essentially just by interchangingthe roles of the variables x , x in the argument. Assuming that ε is chosen sufficientlysmall, we may decompose J a ( λ ) = J a ( λ ) + J ∞ a ( λ ) , where J a ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) χ (cid:18) εx | x | a (cid:19) (cid:18) − χ (cid:18) x ε | x | a (cid:19)(cid:19) dx and J ∞ a ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) (cid:18) − χ (cid:18) εx | x | a (cid:19)(cid:19) dx. Notice that the integrand of J a ( λ ) is supported where ε | x | a . | x | . ε | x | a , and the integrand of J ∞ a ( λ ) is supported where1 ε | x | a . | x | . Using a dyadic decomposition of J a ( λ ) by means of the dilations δ r associated tothe weight κ a , we can estimate J a ( λ ) in the same way as we did estimate J a ( λ ) in theproof of Lemma 3.6. Notice to this end that the corresponding integrals J k ( λ ) will beperformed here over a domain where ε /a . | x | . ε . | x | . . And, since now we have φ κ a ( x , x ) = x d x d , there is no root of multiplicity d or greaterof φ κ a on this domain, hence we obtain the estimate | J a ( λ ) | ≤ C d λ − /d . As for J ∞ a ( λ ) , observe first that there is another smooth, even bump function ˜ χ which is identically 1 near the origin such that 1 − χ ( εx / | x | a ) = ˜ χ (cid:0) x / ( ε /a | x | /a ) (cid:1) . Moreover, even though this function will depend on a, we may assume that its deriva-tives are uniformly bounded for 0 < a < . We accordingly re-write J ∞ a ( λ ) := Z e iλφ ( x ,x ) χ ( x , x ) ˜ χ (cid:18) x ε /a | x | /a (cid:19) dx. Decomposing the integral into the contributions by the four quadrants, we reduce ourconsiderations to estimating the integral I ( λ ) := Z ∞ Z ∞ e iλφ ( x ,x ) χ ( x , x ) ˜ χ x ε /a x /a ! dx dx . Observe next that the phase function can be written as φ ( x , x ) = x d x d a ( x , x ) + d − X ν =0 ( x ν ϕ ν ( x ) + x ν ψ ν ( x )) , where the functions ϕ ν , ψ ν are smooth and flat at the origin and where a is a smoothfunction such that a (0 ,
0) = 1. This shows that the change of variables x := x /a y , x := y will transform the phase function φ into a phase function ˜ φ of the form˜ φ ( y , y ) = y d (1+1 /a )2 (cid:16) y d ˜ a ( y , y ) + d − X ν =0 y ν ˜ ϕ ν ( y ) (cid:17) =: y d (1+1 /a )2 ψ ( y , y ) , where the functions ˜ ϕ ν are again smooth and flat at the origin and where ˜ a is smoothwith ˜ a (0 ,
0) = 1 . Accordingly, we re-write I ( λ ) = Z ∞ Z ∞ e iλy d (1+1 /a )2 ψ ( y ,y ) ˜ χ ( y , y ) ˜ χ (cid:16) y ε /a (cid:17) dy y /a dy . Notice that if M is any fixed positive constant, than the contribution to I ( λ ) by theregion where λy d (1+1 /a )2 ≤ M is trivially bounded by C M λ − /d , with a constant C M which does not depend on a, so that we may assume that λy d (1+1 /a )2 > M in the innerintegral, where M is a sufficiently large constant.In order to estimate the inner integral, observe that the C M -norm of ψ as a functionof y and y may be very large as a → , due the type of change of coordinates that weapplied. However, for y fixed, the d ’th derivative of ψ with respect to y is boundedfrom below by a fixed constant not depending on y and a. Indeed, by choosing Ωsufficiently small, it is easy to see that we may assume that ∂ d ψ ( y , y ) ≥ a (0 , d ! / d ! / . We may thus apply van der Corput’s estimate in order to estimate the inner integralwith respect to y by C ( λy d (1+1 /a )2 ) − /d , with a constant C which does not depend on a, and then perform the integration in y , to find that | I ( λ ) | ≤ A d (cid:16) λ /a (cid:17) λ − /d , as required. Q.E.D. NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 33
We are thus left with the main term J ( λ ) , which can be treated exactly as in theproof of Lemma 3.7, so that the conclusion of this lemma holds true. In particular,the limit relations (3.28) and (3.29) hold true. Letting a → b → ∞ , we finallyderive from those limit relations in combination with Lemma 3.8 that indeedlim λ → + ∞ λ /d log λ J ( λ ) = c, where c is given by 4 C d , if d is even, and by 2( C d + C d ) , if d is odd. This provesTheorem 1.3 also in this case.3.2.3. A compact and an unbounded edge.
Finally, if one of the edges containing ( d, d ) iscompact and the other one is unbounded, then let us assume without loss of generalitythat the edge lying above the bi-sectrix is compact, and the one below is unbounded.Then we define a := κ a /κ a associated to the upper, compact edge as in Subsection3.2.1, and approximate the lower, horizontal edge by a compact line segment of slope1 /b as in Subsection 3.2.2, and consider what will happen to the integrals J a ( λ ) , J b ( λ )and J ( λ ) , defined in the same way as before, when b → + ∞ . Applying the same kind of reasoning as before, one then finds that J a ( λ ) = O ( λ − /d )as λ → + ∞ , that J b ( λ ) satisfies estimate (3.31) from Lemma 3.8, and that the maincontribution is again given by J ( λ ) , which can be treated as before by Lemma 3.6.We can then conclude as in the previous case by letting b → + ∞ . This completes the proof of Theorem 1.3. Q.E.D.4.
Fourier restriction in the case of adapted coordinates.
Let us turn to proving the restriction estimate (1.9) in Theorem 1.7. We may assumethat x = (0 , , and that the hypersurface S is given as the graph x = φ ( x , x ) ofa smooth, finite type function φ in adapted coordinates ( x , x ) , which is defined inan open neighborhood Ω of the origin such that φ (0 ,
0) = 0 , ∇ φ (0 ,
0) = 0 . Recall thatthen ν ( x , S ) = ν ( φ ) and h ( x , S ) = h = d ( φ ) . If ν ( φ ) = 0 , then by A. Greenleaf’s work [8] (compare also [19], Ch. VIII, 5.15(b)), the L p ( R )- L ( S ) restriction theorem for the Fourier transform in Theorem 1.7is an immediate consequence of the uniform estimate in Corollary 1.6 for the Fouriertransform of the surface carried measure ρdσ of the hypersurface S. We shall therefore assume in the sequel that ν ( φ ) = 1. This implies in particularthat h = h ( φ ) ≥ . Note that in this case Greenleaf’s theorem misses the endpoint p = p c = (2 h + 2) / (2 h + 1) , on which we shall concentrate in the sequel. As we shall see,this endpoint can nevertheless be obtained if we invoke tools from Littlewood-Paleytheory. Our approach has some resemblance to Stein’s proof in [19], Ch. VIII, 5.16, ofStrichartz’ estimates for the Fourier restriction to quadratic surfaces from [20].We shall denote by µ the surface carried measure ρdσ from Theorem 1.7. By de-composing R again into its four quadrants, we may assume without loss of generality that µ is of the form h µ, f i = Z ( R +) f ( x ′ , φ ( x ′ )) η ( x ′ ) dx ′ , f ∈ C ( R ) , where η ( x ′ ) := ρ ( x ′ , φ ( x ′ )) p |∇ φ ( x ′ ) | is smooth and has its support in a sufficientlysmall neighborhood Ω of the origin. In the sequel, we shall split the coordinates in R as x = ( x ′ , x ) ∈ R × R . If χ is an integrable funtion defined on Ω , we put µ χ := ( χ ⊗ µ, i.e., h µ χ , f i = Z ( R +) f ( x ′ , φ ( x ′ )) η ( x ′ ) χ ( x ′ ) dx ′ . Observe that then(4.1) c µ χ ( − ξ ) = J χ ( ξ ) , ξ ∈ R , with J χ ( ξ ) defined as in Section 2.We next choose a weight κ with 0 < κ ≤ κ such that the line L κ is a supportingline to the Newton polyhedron N ( φ ) and so that1 | κ | = d h ( φ κ ) = h ( φ κ ) = h. This is possible, since according to Lemma 1.5 the principal face π ( φ ) of N ( φ ) iseither a vertex, or a compact edge such that m ( φ pr ) = d ( φ ) . In the first case, we have φ κ ( x , x ) = cx h x h , and in the second φ κ = φ pr , so that in both cases(4.2) m ( φ κ ) = h. The corresponding dilations will be denoted by δ r . Fixing a suitable smooth cut-offfunction χ ≥ R supported in an annulus D such that the functions χ k := χ ◦ δ k form a partition of unity, we then decompose the measure µ as µ = X k ≥ k µ k , (4.3)where µ k := µ χ k . Let us extend the dilations δ r to R by putting δ er ( x ′ , x ) := ( r κ x , r κ x , rx ) . We re-scale the measure µ k by defining µ , ( k ) := 2 − k µ k ◦ δ e − k , i.e.,(4.4) h µ , ( k ) , f i = 2 | κ | k h µ k , f ◦ δ e k i = Z ( R +) f ( x ′ , φ k ( x ′ )) η ( δ − k x ′ ) χ ( x ′ ) dx ′ , with φ k ( x ) := 2 k φ ( δ − k x ) = φ κ ( x )+error terms . This shows that the measures µ , ( k ) aresupported on the smooth hypersurfaces S k defined as the graph of φ k , their total vari-ations are uniformly bounded, i.e., sup k k µ , ( k ) k < ∞ , and that they are approachingthe surface carried measure µ , ( ∞ ) on S defined by h µ , ( ∞ ) , f i := Z ( R +) f ( x ′ , φ ( x ′ )) η (0) χ ( x ′ ) dx ′ NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 35 as k → ∞ . We claim that there is a constant C such that(4.5) | [ µ , ( k ) ( ξ ) | ≤ C (1 + | ξ | ) − /h for every ξ ∈ R , k ≥ k . Indeed, we may again assume that | ξ | + | ξ | ≤ δ | ξ | , where 0 < δ ≪ | ξ | + | ξ | > δ | ξ | the estimate (4.5) follows by an integrationby parts, if Ω is chosen small enough, i.e., k sufficiently large.We may thus in particular assume that | ξ | ∼ | ξ | . Note that (4.1) and (4.4) showthat [ µ , ( k ) ( − ξ ) = 2 | κ | k J χ k ( δ e k ξ ) . Therefore, in view of (4.2) the estimate (2.3) for J k ( ξ ) = J χ k ( ξ ) in Subsection 2.1implies in our case that | [ µ , ( k ) ( ξ ) | ≤ C (1 + 2 − k | δ e k ξ | ) − /h , which yields (4.5) if | ξ | ∼ | ξ | . According to Theorem 1 in [8], the estimates in (4.5) imply the restriction estimates (cid:18)Z | ˆ f ( x ) | dµ , ( k ) ( x ) (cid:19) / ≤ C k f k p , f ∈ S ( R ) , (4.6)with p = (2 h + 2) / (2 h + 1) , and the proof in [8] reveals that the constant C can bechosen independently of k. Let us re-scale these estimates, by putting f ( r ) ( x ) := r | κ | / f ( δ er x ) , r > , for any function f on R . Then b f ( r ) = r −| κ | / − \ f ◦ δ er − , and (4.6) implies Z | ˆ f ( x ) | dµ k ( x ) = Z | b f (2 − k ) ( x ) | dµ , ( k ) ( x ) ≤ C ( | κ | / k k f ◦ δ e k k p , hence Z | ˆ f ( x ) | dµ k ( x ) ≤ C k f k p , (4.7)with a constant C which does not depend in k. Fix a cut-off function ˜ χ ∈ C ∞ ( R ) supported in an annulus centered at the originsuch that ˜ χ = 1 on the support of χ, and define dyadic decomposition operators ∆ ′ k by d ∆ ′ k f ( x ) := ˜ χ ( δ k x ′ ) ˆ f ( x ′ , x ) . Then R | ˆ f ( x ) | dµ k ( x ) = R | d ∆ ′ k f ( x ) | dµ k ( x ) , so that (4.7) yields in fact that Z | ˆ f ( x ) | dµ k ( x ) ≤ C k d ∆ ′ k f k p , for any k ≥ k . In combination with Minkowski’s inequality, this implies (cid:18)Z | ˆ f ( x ) | dµ ( x ) (cid:19) / = X k ≥ k Z | ˆ f ( x ) | dµ k ( x ) ! / ≤ X k ≥ k k ∆ ′ k f k p ! / = C X k ≥ k (cid:18)Z | ∆ ′ k f ( x ) | p dx (cid:19) /p ! p/ /p ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ≥ k | ∆ ′ k f ( x ) | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) , since p < Appendix: Proof of Lemma 1.5
To prove Lemma 1.5, we shall apply the techniques and results from [11], in particularthe reasoning in the proof of Lemma 3.2 of that article.In order to show that (a) implies (b), we may assume without loss of generality thatthe coordinates x are adapted to φ, and that the principal face π ( φ ) is a vertex, say π ( φ ) = { ( ℓ, ℓ ) } , i.e., φ pr ( x , x ) = cx ℓ x ℓ . Assume that y is another adapted coordinate system, say x = F ( y ) , where F is a local,smooth diffeomorphism at the origin, and write ˜ φ ( y ) := φ ( F ( y )) . Possibly after permuting the coordinates x and x , we may choose a weight κ =( κ , κ ) with 0 < κ ≤ κ in the following way: Case I.
If ( ℓ, ℓ ) is the right endpoint of a compact edge γ of the Newton diagramof φ, then we choose the unique weight κ so that γ lies on the line L κ := { ( t , t ) ∈ R : κ t + κ t = 1 } (which is then a supporting line to N ( φ )). Case II.
Otherwise, i.e., if N ( φ ) is contained in the half-plane t ≥ ℓ, then wechoose κ so that the vertex ( ℓ, ℓ ) is the unique point of the Newton polyhedron N ( φ )contained in the supporting line L κ . Permuting the coordinates y and y , if necessary, we may assume without lossof generality that ( x , x ) = ( F ( y , y ) , F ( y , y )) satisfies ∂F j (0 , ∂y j = 0 for j = 1 , F , F in the form(5.1) F ( y , y ) = y ψ ( y , y ) + η ( y ) , F ( y , y ) = y ψ ( y , y ) + η ( y ) , where ψ , ψ , η , η are smooth functions satisfying ψ (0 , = 0 , ψ (0 , = 0 , η (0) = η (0) = 0 . We may further assume that ψ (0 ,
0) = ψ (0 ,
0) = 1 . Denote by k j the order of vanishingof η j at 0 , j = 1 , . Then clearly k j ≥ . Notice that in Case II, we may and shall assume that κ /κ > k . NIFORM ESTIMATES AND A SHARP RESTRICTION THEOREM 37
We first recall some observation from [11]. If F κ denotes the κ -principal part of F, then ˜ φ ( y , y ) = φ κ ◦ F κ ( y , y ) + terms of higher κ -degree,so that ˜ φ κ = φ κ ◦ F κ . Moreover, φ κ ◦ F κ is a κ -homogeneous polynomial, so that its Newton diagram N d ( ˜ φ κ )is again a compact interval (possibly a single point). In case that this interval intersectsthe bi-sectrix too, then it contains the principal face of N ( ˜ φ ) . a) The case where k > κ κ , and either κ > κ , or κ = κ and k > . In this case, one finds that F κ ( y , y ) = ( y , y ) (see [11]), hence ˜ φ κ = φ κ , so that π ( ˜ φ ) = π ( φ ) is a vertex. b) The case where k > κ κ , κ = κ and k = 1 . Then k > , k = 1 , so that F κ ( y , y ) = ( y + ay , y ) for some constant a ∈ R , hence˜ φ κ ( y , y ) = c ( y + ay ) ℓ y ℓ . From a view at the Newton diagram of this polynomial, wesee that π ( ˜ φ ) = π ( φ ) is a vertex. c) The case where k < κ κ . As in the proof of Lemma 3.2 in [11], we then introduce a second weight µ := (1 , k ) , and choose d > L µ := { ( t , t ) ∈ R : t + k t = d } is the supportingline to the Newton polyhedron N ( φ ) . It has been shown in the proof of Lemma 3.2 in[11] (Case c)) that the principal face of N ( ˜ φ ) then lies on the line L µ . Noticing that theline L µ is steeper than the line L κ , we see that Case I cannot arise here, since otherwisewe would have d y < d x , contradicting our assumption that also the coordinates y areadapted. And, in Case II, we see that ( ℓ, ℓ ) will be the only point of N ( φ ) containedin L µ , so that φ µ = φ pr . This shows that ˜ φ µ = φ µ ◦ F µ = φ pr ◦ F µ . Moreover, the µ -principal part of F is given by F µ ( y , y ) = ( y , y + a y k ) , if k > , and by F µ ( y , y ) = ( y + a y , y + a y ) , if k = 1 , with a = 0 if and only if k = 1 . In the first case, we obtain ˜ φ µ = cy ℓ ( y + a y k ) ℓ , so that π ( ˜ φ ) = π ( φ ) is again avertex. A similar reasoning applies in the second case, if a = 0 or a = 0 . And, if a = 0 = a , we find that ˜ φ µ = c ( y + a y ) ℓ ( y + a y ) ℓ . This means that the principalface of N ( ˜ φ ) is a compact edge passing through the point ( ℓ, ℓ ) , and clearly m ( ˜ φ pr ) = ℓ, so that m ( ˜ φ pr ) = ℓ = d ( ˜ φ ) . e) The case where k = κ κ . Observe that k κ > κ , unless κ = κ and k = 1 , since κ /κ ≤ k κ > κ , we then see that φ κ ( y , y ) = ( y , y + a y k ) , hence˜ φ κ ( y , y ) = cy ℓ ( y + a y k ) ℓ . This shows that again π ( ˜ φ ) = π ( φ ) is a vertex.Finally, assume that κ = κ and k = 1 , so that also k = 1 . Then φ κ is of theform φ κ ( y , y ) = ( y + a y , y + a y ) , hence ˜ φ κ ( y , y ) = c ( y + a y ) ℓ ( y + a y ) ℓ . As before, this means that the principal face of N ( ˜ φ ) is a compact edge passing throughthe point ( ℓ, ℓ ) , and we have m ( ˜ φ pr ) = ℓ = d ( ˜ φ ) . There remains to show that (b) implies (a). To this end, we may assume withoutloss of generality that y = x, i.e., that x is an adapted coordinate system, and that π ( φ ) is a compact edge and m ( φ pr ) = d ( φ ) . We shall denote the latter by d. Let usdenote by ( A , B ) and ( A , B ) the two vertices of π ( φ ) , and assume that A < A . According to [11], displays (3.2) and (3.3), we can then write the principal part of φ as φ pr ( x , x ) = cx α x β Y l ( x − c l x m ) n l , where the c l ’s are the non-trivial distinct complex roots of the polynomial φ pr (1 , x )and the n l ’s are their multiplicities. Moreover, there exists an l such that d = n l andsuch that c l is real.We then apply the change of coordinates y := x , y := x − c l x m , which preservesthe mixed homogeneity of φ pr and transforms this polynomial into a polynomial ofthe same form, cx ˜ α x ˜ β Q l ( x − ˜ c l x m ) n l , but now with ˜ β = d. The vertices of the cor-responding Newton diagram are given by ( A , B ) and ( ˜ A , ˜ B ) and lie on the sameline as ( A , B ) and ( A , B ) (see [11]), where obviously ˜ B = ˜ β = d. This shows that( ˜ A , ˜ B ) = ( d, d ) , and consequently the principal face of the Newton polyhedron of ˜ φ is given by the vertex ( d, d ) . Q.E.D.
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Department of Mathematics, Samarkand State University, University Boulevard15, 140104, Samarkand, Uzbekistan
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