Fourier transforms of irregular mixed homogeneous hypersurface measures
aa r X i v : . [ m a t h . C A ] N ov Fourier Transforms of Irregular Mixed HomogeneousHypersurface Measures
Michael Greenblatt
May 23, 2017
Abstract
With the help of Van der Corput lemmas, decay estimates are proven forFourier transforms of mixed homogeneous hypersurface measures with den-sities that can be quite irregular. The primary results are local in nature,but can be extended to global theorems in an appropriate sense. The esti-mates are sharp for a certain range of indices in the theorems.MSC codes: 42B20 (primary), 42B10 (secondary).Keywords: Fourier transform, oscillatory integral.
1. Background and Theorem Statements .We start with the following definition.
Definition 1.1.
Suppose a , ..., a n are positive numbers. A function f : R n → R is mixedhomogeneous of degree ( a , ..., a n ) if for all t > x , ..., x n ) ∈ R n we have f ( t a x , ..., t an x n ) = tf ( x , ..., x n ) (1 . f ( x ) be a functionon R n that is mixed homogeneous of degree ( a , ..., a n ) and we let g ( x , ..., x n ) be a locallyintegrable nonnegative function that is mixed homogeneous of degree ( ρa , ..., ρa n ) on R n − { } for some ρ ∈ R (which may be negative). We let φ ( x , ..., x n ) be an integrablefunction supported on the unit ball such that | φ ( x , ..., x n ) | ≤ g ( x , ..., x n ) and (cid:12)(cid:12)(cid:12)(cid:12) n X i =1 x i a i ∂φ∂x i ( x , ..., x n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ g ( x , ..., x n ) (1 . φ ( x , ..., x n ) satisfying (1 .
2) would be φ ( x , ..., x n ) = α ( x , ..., x n ) g ( x , ..., x n ), where α is smooth and supported in the unit ball. Note thatthe case where φ ( x , ..., x n ) is itself smooth is included in the situation where ρ = 0 and g ( x , ..., x n ) = 1.This research was supported in part by NSF grant DMS-1001070.1e examine the Fourier transform of the surface measure of the graph of f ( x , ..., x n ),weighted by φ ( x , ..., x n ), given by T ( λ , λ , ..., λ n ) = Z R n e iλ f ( x ,...,x n )+ iλ x + ... + iλ n x n φ ( x , ..., x n ) dx ... dx n (1 . − λ , − λ , ..., − λ n ), butto simplify notation we will consider T ( λ , λ , ..., λ n ) as written here. Note that f ( x ) and g ( x ) (and therefore) φ ( x ) can be arbitrarily irregular in directions other than the mixed-radial direction; this is because our theorems (including the sharp estimates) will be provedwith the use of Van der Corput lemmas in the mixed-radial direction only. Hence f ( x ), g ( x ), and φ ( x ) can be badly behaved in other directions.It should be pointed out that there is an extensive literature concerning thedecay rate of Fourier transforms of hypersurface measures. When φ ( x ) is smooth and theHessian of f ( x ) has full rank everywhere, there are well-known asymptotic formulas forthe decay rate. We refer to chapter 8 of [S] for details. Whenever the Hessian of thephase has positive rank, one can also obtain decay estimates generalizing the full rankcase. We refer to [L] as an example of this. It has been quite difficult to prove sharpestimates for general hypersurfaces (hence the restriction to mixed homogeneous surfaceshere). Many papers in this area have considered other classes of hypersurfaces such as theconvex hypersurfaces considered in [BrNW] [CDMaM] [CMa] [NSeW] [Sc], or the case ofscalar oscillatory integrals (the situation where λ = .... = λ n = 0). We mention [ChKNo][PY] [GPT] [Gr] [V] as examples of the latter.Our first theorem regarding T ( λ , λ , ..., λ n ) is as follows. Some cases of it followfrom Theorem 14 of [ISa]. Theorem 1.1.
Let f and g be as above and let v denote the number of distinct values inthe set { a , ..., a n } . Assume that a i > a i = 1 for all i . a) Let dµ g denote the measure | g ( x , ..., x n ) | dx ... dx n . Suppose ǫ > B f,g,ǫ >
0, for all t > µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } ) ≤ B f,g,ǫ t ǫ .If ǫ = v +1 there exists a C f,g,ǫ > | T ( λ , λ , ..., λ n ) | ≤ C f,g,ǫ (1 + | λ | ) − min( ǫ, v +1 ) (1 . ǫ = v +1 , then (1 .
4) holds except with an additional factor of ln(2 + | λ | ) on the right.Conversely, suppose 0 < η < φ ( x , ..., x n ) supported in the unit ballsatisfying (1 .
2) one has an estimate | T ( λ , λ , ..., λ n ) | ≤ C f,g,η (1 + | λ | ) − η (1 . < ǫ < η one has an estimate of the form µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } ) ≤ B f,g,ǫ t ǫ (1 . ǫ for which such a B f,g,ǫ exists is at most v +1 , the exponentin (1 .
4) is sharp. b) Suppose g ( x , ..., x n ) ∈ L p ([ − , n ), where 1 ≤ p ≤ ∞ . Let p ′ be the exponentconjugate to p ; that is, p ′ + p = 1. If p ′ = v + 1 then for some C ′ f,g,p > | T ( λ , λ , ..., λ n ) | ≤ C ′ f,g,p (1 + | λ ′ | ) − min( p ′ , v +1 ) (1 . a )Here | λ ′ | denotes the magnitude of the vector λ ′ = ( λ , ..., λ n ). If p ′ = v + 1 the sameestimate holds with an additional factor of (ln(2 + | λ ′ | )) v +1 on the right-hand side. c) Suppose g ( x , ..., x n ) ∈ L p ([ − , n ) for some 1 ≤ p ≤ ∞ , and let p ′ be the conjugateexponent as in part b). If p ′ = v then for some C ′′ f,g,p > | T (0 , λ , ..., λ n ) | ≤ C ′′ f,g,p (1 + | λ ′ | ) − min( p ′ , v ) (1 . b )If p ′ = v the same estimate holds with an additional factor of (ln(2 + | λ ′ | )) v .Theorem 1.1 can be interpreted as follows. Suppose a i > i . The graphof f ( x , ..., x n ) has a well-defined normal vector at the origin, given by (0 , ..., , φ ( x , ..., x n ), in terms of the component of λ in this normal direction.The exponent is given by the growth rate in t of this weighted measure of the portion ofthe surface with | x n +1 | < t . Part b) gives estimates for tangential components of λ , whilepart c) estimates the Fourier transform of the weight function φ ( x , ..., x n ) itself similarlyto part b).When v = 1 (the homogeneous case) one can get an exponent as high as inTheorem 1.1 a) and b). In this situation, with additional regularity assumptions on f ( x )and g ( x ), one can use damping functions in conjunction with the methods of [SoS] to prove L p boundedness of maximal averages associated with the hypersurfaces when p >
2. Werefer to [ISa] for theorems of this kind. It might be interesting to explore what theoremsof this nature are possible in the setting of this paper.In the setting of Theorem 1.1a), one might ask if there are situations where B f,g,ǫ exists for some ǫ ≥ v +1 and where the stronger estimate | T ( λ , λ , ..., λ n ) | ≤ C f,g,ǫ (1 + | λ | ) − ǫ still holds. If one imposes extra regularity conditions on f ( x , ..., x n )and φ ( x , ..., x n ) in directions other than along the curves t → ( c t a , ..., c n t an ) determinedby the mixed homogeneity, it is not too hard to find situations where this is the case. Werefer to [ISa] for examples of this. But even in the absence of such regularity, it is conceiv-able that the worst decay estimates one gets on these curves average out over ( c , ..., c n )so that one still obtains the stronger estimate. It would be interesting to try to determinein what situations this occurs, if any.Although parts b) and c) of Theorem 1.1 are only sometimes sharp, when ahomogeneity a i appears only once, one has sharp bounds on | T ( λ , λ , ..., λ n ) | of the form3 | λ i | − ǫ that are analogous to those given in Theorem 1.1 a). These bounds are given bythe following theorem; note that part c) gives a sharpness statement for both parts a) andb) . Theorem 1.2.
Suppose the homogeneity a i is such that a j = a i for all j = i . Let themeasure dµ g be as in Theorem 1.1. Suppose ǫ i > D f,g,ǫ i >
0, forall t > µ g ( { ( x , ..., x n ) ∈ [ − , n : | x i | < t } ) ≤ D f,g,ǫ i t ǫ i . a) If ǫ i = v +1 there exists a E f,g,ǫ i > | T ( λ , λ , ..., λ n ) | ≤ E f,g,ǫ i (1 + | λ i | ) − min( ǫ i , v +1 ) (1 . a )If ǫ i = v +1 , then (1 . a ) holds except with an additional factor of ln(2 + | λ i | ) on the right. b) If ǫ i = v there exists a F f,g,ǫ i > | T (0 , λ , ..., λ n ) | ≤ F f,g,ǫ i (1 + | λ i | ) − min( ǫ i , v ) (1 . b )If ǫ i = v , then (1 . b ) holds except with an additional factor of ln(2 + | λ i | ) on the right. c) If 0 < η < φ ( x , ..., x n ) satisfying (1 .
2) one has an estimate | T (0 , λ , ..., λ n ) | ≤ F f,g,η (1 + | λ i | ) − η (1 . < ǫ < η one has an estimate of the form µ g ( { ( x , ..., x n ) ∈ [ − , n : | x i | < t } ) ≤ E f,g,ǫ t ǫ ———————————————————————————————————————One can combine parts a) and b) of Theorem 1.1 to obtain decay estimates interms of powers of | λ | , where | λ | is the magnitude of the full vector ( λ , ..., λ n ). Forexample, as an immediate consequence of Theorem 1.1 a) and b) we have the following. Corollary 1.3.
Suppose ǫ < v +1 < p ′ and for some B f,g,ǫ >
0, for all t > µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } ) ≤ B f,g,ǫ t ǫ . Then for some C f,g,ǫ > | T ( λ , λ , ..., λ n ) | ≤ C f,g,ǫ (1 + | λ | ) − ǫ (1 . ǫ for which such a B f,g,ǫ exists is given by δ ≤ v +1 , then theexponent in the right-hand side of (1 .
10) cannot be taken to be greater than δ .Next, we will have a global analogue of the local result of Corollary 1.3 for smooth φ ( x , ..., x n ). To ensure that all integrals are well-defined, our global analogue is as follows.While φ ( x , ..., x n ) will still be compactly supported, its support can be arbitrarily large,4nd we will obtain uniform bounds on | T ( λ , λ , ..., λ n ) | over all φ ( x , ..., x n ) that satisfythe following decay conditions. We stipulate that for some k > P ni =1 1 a i − ǫ and some A > | φ ( x , ..., x n ) | ≤ A (1 + n X i =1 | x i | a i ) − k , (cid:12)(cid:12)(cid:12)(cid:12) n X i =1 x i a i ∂φ∂x i ( x , ..., x n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ A (1 + n X i =1 | x i | a i ) − k (1 . Theorem 1.4.
Suppose (1 .
11) holds and suppose 0 < ǫ < v +1 is such that for someconstant C f,ǫ the Lebesgue measure of the set { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } is bounded by C f,ǫ t ǫ for all t >
0. Then for some D f,A,ǫ > | T ( λ , λ , ..., λ n ) | ≤ D f,A,ǫ (1 + | λ | ) − ǫ (1 . |{ x ∈ [ − , n : | f ( x ) | < t }| ∼ t ǫ for small t ; if φ is nonnegative with φ (0 , ..., = 0 and is supported on a small enoughneighborhood of the origin, then the proof of the sharpness statement of Theorem 1.1a) atthe end of this section shows that the exponent ǫ cannot be improved for this φ ( x ). Examples.Example 1.
Suppose f ( x , ..., x n ) is a bounded function on [ − , n such that for some C, r >
0, for each t > |{ ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t }| < Ct r (ab-solute values here denotes Lebesgue measure). Let g ( x , ..., x n ) = | f ( x , ..., x n ) | ρ for some ρ > − r , and let φ ( x , ..., x n ) = α ( x , ..., x n ) g ( x , ..., x n ) = α ( x , ..., x n ) | f ( x , ..., x n ) | ρ ,where α ( x , ..., x n ) is a smooth compactly supported function on ( − , n . Note thatgeometrically, a damping function | f ( x , ..., x n ) | ρ represents | x n +1 | ρ when the surface isviewed in R n +1 .Then we have that µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } ) is given by Z { ( x ,...,x n ) ∈ [ − , n : | f ( x ,...,x n ) | 0, then onecan take p = ∞ . If ρ < 0, then one can take p to be any number less than r | ρ | .5s for Theorem 1.2, note that one has that µ g ( { ( x , ..., x n ) ∈ [ − , n : | x i | < t } ) is givenby Z { ( x ,...,x n ) ∈ [ − , n : | x i | Suppose we are in the homogeneous case; that is, there is an a > a = 1,such that a i = a for all i . Then v = 1, which gives the best possible exponents in Theorem1.1; one can get a decay rate up to (1 + | λ | ) − and (1 + | λ ′ | ) − in parts a) and b)respectively. Suppose g ( x , ..., x n ) = | x α ...x α n n | for some nonnegative integers α i , so thatfor example φ ( x , ..., x n ) can be of the form α ( x , ..., x n ) x α ...x α n n where α ( x , ..., x n ) is asmooth compactly supported function on ( − , n .Then µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } ) is equal to Z { ( x ,...,x n ) ∈ [ − , n : | f ( x ,...,x n ) | Suppose we are in the opposite situation from Example 2, and each a i is different, so that v = n . Then for a given i , Theorem 1.2 provides estimates of theform | T ( λ ) | ≤ F f,g,ǫ i (1 + | λ i | ) − min( ǫ i , n +1 ) (when a positive ǫ i exists) and one has anoverall estimate | T ( λ ) | ≤ F f,g,ǫ (1 + | λ | ) − ǫ where ǫ = min( ǫ , ..., ǫ n , n +1 ). If for some i theexponent ǫ i cannot be taken to be greater than n +1 , then by Theorem 1.2c) the minimumover i of the supremal ǫ i gives the supremum of the exponents η for which we have anestimate of the form | T ( λ ) | ≤ F f,g,η (1 + | λ | ) − η . Proof of sharpness statements in Theorem 1.1 and 1.2. The sharpness statements in Theorem 1.1a) and Theorem 1.2c) can be proved inrelatively short order. We focus our attention on the situation in Theorem 1.1a). Suppose0 < η < . 5) holds for all φ ( x , ..., x n ) supported in the unit ball satisfying61 . α ( x ) be a bump function on R whose Fourier transform is nonnegative, com-pactly supported, and equal to 1 on a neighborhood of the origin, and let N be a largepositive number. If 0 < ǫ < η , then (1 . 5) implies that for some constant A f,g,ǫ one has Z R | T ( λ , , ..., || λ | ǫ − α ( N λ ) dλ < A f,g,ǫ (1 . . 3) for T ( λ , ..., λ n ), this implies that (cid:12)(cid:12)(cid:12)(cid:12) Z R n +1 e iλ f ( x ,...,x n ) φ ( x , ..., x n ) | λ | ǫ − α ( N λ ) dλ dx ... dx n (cid:12)(cid:12)(cid:12)(cid:12) < A f,g,ǫ (1 . λ first in (1 . (cid:12)(cid:12)(cid:12)(cid:12) Z R n β N ( f ( x , ..., x n )) φ ( x , ..., x n ) dx ... dx n (cid:12)(cid:12)(cid:12)(cid:12) < A ′ f,g,ǫ (1 . β N ( y ) is the convolution of | y | − ǫ with N ˆ α ( yN ). Taking φ ( x ) = γ ( x ) g ( x ) for somenonnegative bump function γ ( x ) equal to one on a neighborhood of the origin we get (cid:12)(cid:12)(cid:12)(cid:12) Z R n β N ( f ( x )) γ ( x ) g ( x ) dx ... dx n (cid:12)(cid:12)(cid:12)(cid:12) < A ′′ f,g,ǫ (1 . N → ∞ gives Z R n | f ( x ) | − ǫ γ ( x ) g ( x ) dx ... dx n < A ′′′ f,g,ǫ (1 . γ ( x ) is equal to 1 on a neighborhood of the origin, mixed homogeneity gives Z [ − , n | f ( x ) | − ǫ g ( x ) dx ... dx n < ∞ (1 . | f ( x ) | − ǫ is in L ([ − , n ) with respect to the measure dµ g . Hence it is inweak L , and we have the existence of a constant G f,g,ǫ such that µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | − ǫ > t } ) ≤ G f,g,ǫ t (1 . t by t − ǫ , (1 . 24) gives the sharpness statement (1 . 6) as needed. This gives thesharpness statement in Theorem 1.1a).Theorem 1.2c) is proved in exactly the same way, replacing the λ variable bythe λ i variable and the function f ( x , ..., x n ) by x i . 2. Proofs of Theorems 1.1, 1.2, and 1.4. 7e start with the well-known Van der Corput lemma (see p. 334 of [S]). Lemma 2.1. Suppose h ( x ) is a C k function on the interval [ a, b ] with | h ( k ) ( x ) | > A on[ a, b ] for some A > 0. Let φ ( x ) be C on [ a, b ]. If k ≥ c k dependingonly on k such that (cid:12)(cid:12)(cid:12)(cid:12) Z ba e ih ( x ) φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ c k A − k (cid:18) | φ ( b ) | + Z ba | φ ′ ( x ) | dx (cid:19) (2 . k = 1, the same is true if we also assume that h ( x ) is C and h ′ ( x ) is monotone on [ a, b ].We will also make use of the following variant of Lemma 2.1 for k = 1. Lemma 2.2. Suppose the hypotheses of Lemma 2.1 hold with k = 1, except insteadof assuming that h ′ ( x ) is monotone on [ a, b ] we assume that | h ′′ ( x ) | < B ( b − a ) A for someconstant B > 0. Then we have (cid:12)(cid:12)(cid:12)(cid:12) Z ba e ih ( x ) φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ A − (cid:18) Z ba | φ ′ ( x ) | dx + ( B + 2) sup [ a,b ] | φ ( x ) | (cid:19) (2 . Proof. We write e ih ( x ) = ( h ′ ( x ) e ih ( x ) ) h ′ ( x ) and integrate by parts in the integral beingestimated, integrating the h ′ ( x ) e ih ( x ) factor to e ih ( x ) and differentiating φ ( x ) h ′ ( x ) . We obtain Z ba e ih ( x ) φ ( x ) dx = e ih ( b ) φ ( b ) h ′ ( b ) − e ih ( a ) φ ( a ) h ′ ( a ) − Z ba e ih ( x ) ddx (cid:18) φ ( x ) h ′ ( x ) (cid:19) dx = e ih ( b ) φ ( b ) h ′ ( b ) − e ih ( a ) φ ( a ) h ′ ( a ) − Z ba e ih ( x ) φ ′ ( x ) h ′ ( x ) dx + Z ba e ih ( x ) φ ( x ) h ′′ ( x )( h ′ ( x )) dx (2 . | h ′ ( x ) | > A ensures that each of the two boundary terms is bounded inabsolute value by A − sup [ a,b ] | φ ( x ) | . As for the first integral term, taking absolute valuesof the integrand and inserting | h ′ ( x ) | > A gives that this term is bounded in absolutevalue by A − R ba | φ ′ ( x ) | dx . In the second integral term, we use that | h ′′ ( x ) | < B ( b − a ) A and | h ′ ( x ) | > A , resulting in (cid:12)(cid:12)(cid:12)(cid:12) φ ( x ) h ′′ ( x )( h ′ ( x )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ B A − ( b − a ) | φ ( x ) |≤ B A − ( b − a ) sup [ a,b ] | φ ( x ) | (2 . BA − sup [ a,b ] | φ ( x ) | . Adding the bounds forthe different terms gives us the bounds on the right-hand side of (2 . 2) and we are done. Beginning of the Proof of Theorem 1.1. 8e will prove the bounds of Theorem 1.1 for | T ′ ( λ , λ , ..., λ n ) | = Z [0 , n e iλ f ( x ,...,x n )+ iλ x + ... + iλ n x n φ ( x , ..., x n ) dx ... dx n (2 . . 3) follow from changingvariables to ( ± x , ..., ± x n ) and then using the same argument. We change variables in(2 . 5) via ( x , ..., x n ) = ( y a , ..., y an n ), obtaining Z [0 , n e iλ f ( y a ,...,y ann )+ iλ y a + ... + iλ n y ann × a ...a n y a − ...y an − n φ ( y a , ..., y an n ) dy ... dy n (2 . F ( y , ..., y n ) = f ( y a , ..., y an n ) ψ ( y , ..., y n ) = 1 a ...a n y a − ...y an − n φ ( y a , ..., y an n )¯ g ( y , ..., y n ) = 1 a ...a n y a − ...y an − n g ( y a , ..., y an n ) (2 . g ( x , ..., x n ) is as in (1 . F is homogeneous of degree one, ¯ g is homogeneous ofdegree P ni =1 ( a i − 1) + ρ , where ρ is as in the definition of g ( x , ..., x n ), and a direct calcu-lation reveals ψ ( y , ..., y n ) satisfies (1 . 2) with each a i = 1 and where g ( x , ..., x n ) replacedby a constant multiple of ¯ g ( y , ..., y n ) which we henceforth refer to as ˜ g ( y , ..., y n ).Thisconstant depends only on the a i . We next rewrite (2 . 6) as Z [0 , n e iλ F ( y ,...,y n )+ iλ y a + ... + iλ n y ann ψ ( y , ..., y n ) dy ... dy n (2 . S + n denote { ( v , ..., v n ) ∈ S n : v i > i } , where S n denotes the unit sphere. Inpolar coordinates, (2 . 6) becomes a constant depending only on n times Z S + n Z √ n e iλ F ( rv ,...,rv n )+ iλ ( rv ) a + ... + iλ n ( rv n ) an r n − ψ ( rv , ..., rv n ) dr dv ... dv n (2 . √ n due to the integrand being supported on [0 , n . Notethat the condition (1 . 2) when a i = 1 for all i translates into | ψ ( rv , ..., rv n ) | ≤ ˜ g ( rv , ..., rv n ) | ∂ r ψ ( rv , ..., rv n ) | ≤ r ˜ g ( ry , ..., ry n ) (2 . a )Since ˜ g is homogeneous of degree t = P ni =1 ( a i − 1) + ρ , (2 . a ) can further be rewrittenas | ψ ( rv , ..., rv n ) | ≤ ˜ g ( v , ..., v n ) r t | ∂ r ψ ( rv , ..., rv n ) | ≤ ˜ g ( v , ..., v n ) r t − (2 . b )9ext, we enumerate the v elements of the set { a , ..., a n } as { b , ..., b v } , and we denote thecoefficient P { k : ak = b i } λ k v ak k of r b i in the phase function of (2 . 9) by µ i . Using also that F is homogeneous of degree one, (2 . 9) can therefore be rewritten as Z S + n Z √ n e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr dv ... dv n (2 . a i = 1, we also have that no b i = 1. We next dividethe inner integral of (2 . 11) dyadically, writing it as X m ≥ log( −√ n ) Z − m − m − e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr (2 . P ( r ) denote the phase function λ F ( v , ..., v n ) r + µ r b + ... + µ v r b v in (2 . b = 1 so that the power of r in the first term of P ( r ) is on the same footingas the power of r in the other terms of P ( r ). Let z denote the v + 1 by 1 column vectorwhose i th entry is r i ∂ ir P ( r ). Let M denote the v + 1 by v + 1 matrix whose ij entry is b j − ( b j − − ... ( b j − − i + 1), and let w denote the v + 1 by 1 column vector whose firstentry is λ F ( v , ..., v n ) r and whose i th entry for i > µ i − r b i − . Then we have theidentity z = Mw (2 . M is invertible since after elementary row operations it can be converted intothe Vandermonde matrix whose i th row is given by ( b i , ..., b in ), and all the b j are distinct.As a result, there is a constant c depending only on the b i such that there is always some i for which | z i | ≥ c | w | (2 . a )More explicitly, there is a constant c ′ depending only on the b j such that for any r thereis some i with 1 ≤ i ≤ v + 1 such that | ∂ ir ( λ F ( v , ..., v n ) r + µ r b + ... + µ v r b v ) | ≥ c ′ r i (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) (2 . b )Furthermore, by directly bounding each term, one immediately has that there is a constant c ′′ depending on the b j such that | ∂ i +1 r ( λ F ( v , ..., v n ) r + µ r b + ... + µ v r b v ) | ≤ c ′′ r i +1 (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) (2 . . b ) implies that there is some c > b j such that for eachgiven r , for r ∈ [(1 − c ) r , (1 + c ) r ] one has | ∂ ir ( λ F ( v , ..., v n ) r + µ r b + ... + µ v r b v ) | ≥ c ′ r i (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) (2 . . . 16) implies that a given term of (2 . 12) may be writtenas the sum of at most c terms on which (2 . 16) holds for a single i . To this end, we write Z − m − m − e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr = c X k =0 Z I k e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr (2 . I k denote intervals on which (2 . 16) holds for a single i on the interval I k . Next, weuse (2 . 16) to apply the Van der Corput lemma on each term of (2 . i > i = 1, using (2 . b ) to bound | ψ ( rv , ..., rv n ) | and its r derivative. The result is (cid:12)(cid:12)(cid:12)(cid:12) Z I k e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cr (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) − i r n − | ˜ g ( v , ..., v n ) | r t +10 (2 . C depends on the b j and v . It should be pointed out that the bound (2 . 18) for aterm of (2 . 12) also follows from the corollary to the proposition of section 3 of [RS], butwe include the somewhat related argument here for completeness.Since I k is c of a dyadic interval, the right-hand side of (2 . 18) is in turn bounded by C ′ Z I k (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) − i r n − | ˜ g ( v , ..., v n ) | r t dr (2 . . b ) one has Z − m − m − e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr ≤ C ′′ Z − m − m − r n − | ˜ g ( v , ..., v n ) | r t dr (2 . . 19) and (2 . 20) to conclude that Z − m − m − e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr C Z − m − m − min (cid:18) , (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) − i (cid:19) r n − | ˜ g ( v , ..., v n ) | r t dr (2 . i is as large as possible, namely i = v + 1. Hence (2 . 21) is boundedby ≤ C Z − m − m − min (cid:18) , (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) − v +1 (cid:19) r n − | ˜ g ( v , ..., v n ) | r t dr (2 . k , we get Z − m − m − e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr ≤ C ′ Z − m − m − (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) − v +1 r n − | ˜ g ( v , ..., v n ) | r t dr (2 . . 23) over all m and insert it back into (2 . (cid:12)(cid:12)(cid:12)(cid:12) Z S + n Z r ≥ e iλ F ( v ,...,v n ) r + iµ r b + ... + iµ v r bv r n − ψ ( rv , ..., rv n ) dr dv ... dv n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z ( rv ,...,rv n ) ∈ [0 , n min (cid:18) , (cid:18) | λ F ( v , ..., v n ) r | + v X l =1 | µ l r b l | (cid:19) − v +1 (cid:19) × r n − | ˜ g ( v , ..., v n ) | r t dr dv ... dv n (2 . . 24) is that for a given ( v , ..., v n ) themaximum possible r might be in the middle of a dyadic interval and thus adding over all m and integrating does not exactly give (2 . . 24) will still hold.We next convert back from polar into rectangular coordinates in the right-hand side of(2 . (cid:12)(cid:12)(cid:12)(cid:12) Z [0 , n e iλ F ( y ,...,y n )+ iλ y a + ... + iλ n y ann ψ ( y , ..., y n ) dy ... dy n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z [0 , n min (cid:18) , (cid:18) | λ F ( y , ..., y n ) | + v X l =1 (cid:12)(cid:12) X { m : b m = b l } λ m y b l m (cid:12)(cid:12)(cid:19) − v +1 (cid:19) ×| ˜ g ( y , ..., y n ) | dy ... dy n (2 . x -coordinates of (2 . 5) gives (cid:12)(cid:12)(cid:12)(cid:12) Z [0 , n e iλ f ( x ,...,x n )+ iλ x + ... + iλ n x n φ ( x , ..., x n ) dx ... dx n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z [0 , n min (cid:18) , (cid:18) | λ f ( x , ..., x n ) | + v X l =1 (cid:12)(cid:12) X { m : a m = a l } λ m x m (cid:12)(cid:12)(cid:19) − v +1 (cid:19) ×| g ( x , ..., x n ) | dx ... dx n (2 . | λ f ( x , ..., x n ) | in the minimum in (2 . 26) and for part b) we will use the | P { m : a m = a i } λ m x m (cid:12)(cid:12) terms. Partc) is essentially a repeat of part b), except that with the absence of a λ f ( x , ..., x n ) termin the phase we will be able to replace v + 1 by v . We start with part a). Proof of part a) of Theorem 1.1. From now until the end of the proof of part a) of Theorem 1.1 we use the notation C to denote a constant that may depend on f, g, and ǫ as in the statement of Theorem1.1a). Let dµ g be the measure | g ( x , ..., x n ) | dx ... dx n as in the statement of Theorem 1.1.Letting T ′ ( λ , ..., λ n ) denote the integral in the left-hand side (2 . 26) as before, by (2 . | T ′ ( λ , ..., λ n ) | ≤ C Z [0 , n min(1 , | λ f ( x , ..., x n ) | − v +1 ) dµ g (2 . µ g ( { ( x , ..., x n ) ∈ [0 , n : | f ( x , ..., x n ) | < | λ | − } )+ | λ | − v +1 Z { ( x ,...,x n ) ∈ [0 , n : | f ( x ,...,x n ) | > | λ | } | f ( x , ..., x n ) | − v +1 dµ g (2 . | f ( x , ..., x n ) | − , the integral in (2 . 28) is equal to1 v + 1 Z ∞| λ | − t − v +1 − µ g ( { ( x , ..., x n ) ∈ [0 , n : | λ | − < | f ( x , ..., x n ) | < t } ) dt (2 . ǫ satisfies an estimate µ g ( { ( x , ..., x n ) ∈ [ − , n : | f ( x , ..., x n ) | < t } ) ≤ B f,g,ǫ t ǫ ,then (2 . 29) is bounded by C Z ∞| λ | − t − v +1 − min(1 , t ǫ ) dt (2 . µ g is a bounded measure on [0 , n . Given(2 . 30) and the fact that first term in (2 . 28) is bounded by C | λ | − ǫ , we conclude that | T ′ ( λ , ..., λ n ) | ≤ C | λ | − ǫ + C | λ | − v +1 Z ∞| λ | − t − v +1 − min(1 , t ǫ ) dt (2 . ǫ < v +1 , we use min(1 , t ǫ ) ≤ t ǫ in (2 . 31) and obtain that | T ′ ( λ , ..., λ n ) | ≤ C | λ | − ǫ , thedesired estimate (1 . ǫ = v +1 we gain an additional logarithmic factor. If ǫ > v +1 , weuse 1 in the minimum if t > 1, and t ǫ in the minimum if t ≤ 1. In this case we get | T ′ ( λ , ..., λ n ) | ≤ C | λ | − ǫ + C | λ | − v +1 Z | λ | − t ǫ − v +1 − dt + C | λ | − v +1 Z ∞ t − v +1 − dt (2 . | λ | − v +1 , again the desired estimate for part a) ofTheorem 1.1. Since the sharpness aspect of part a) of Theorem 1.1 was proved at the endof section 1, we have completed the proof part a) of Theorem 1.1. Proof of part b) of Theorem 1.1. From now until the end of the proof of part b) of Theorem 1.1 we use the notation C to denote a constant that may depend on f, g, and p as in the statement of Theorem1.1b). Note that part b) of Theorem 1.1 is immediate if p = 1 due to the integrabilityof g , so we assume that p > 1. Recall that | T ′ ( λ , ...λ n ) | is the left-hand side of (2 . λ , ..., λ n ), we proceed as follows. Let l ≥ | λ l | ≥ n | ( λ , ..., λ n ) | . Then (2 . 26) implies that | T ′ ( λ , ...λ n ) | ≤ C Z [0 , n min (cid:0) , (cid:12)(cid:12) X { m : a m = a l } λ m x m (cid:12)(cid:12) − v +1 (cid:1) | g ( x , ..., x n ) | dx ... dx n (2 . < p ≤ ∞ we therefore have | T ′ ( λ , ...λ n ) | ≤ C || g || p (cid:18) Z [0 , n min (cid:0) , (cid:12)(cid:12) X { m : a m = a l } λ m x m (cid:12)(cid:12) − v +1 (cid:1) p ′ dx ... dx n (cid:19) p ′ (2 . p + p ′ = 1. For the fixed l in (2 . λ ′′ be the n -dimensional vector whose m thentry is λ m if a m = a l and is zero otherwise. Note that a l was chosen so that | λ ′′ | ≥ n | λ ′ | .Let v be the unit vector in the direction of λ ′′ . We perform the integration in (2 . 34) firstin the directions perpendicular to v and then in the v direction. We obtain Z [0 , n (cid:0) min (cid:0) , (cid:12)(cid:12) X { m : a m = a l } λ m x m (cid:12)(cid:12) − v +1 (cid:1)(cid:1) p ′ dx ... dx n ≤ C Z √ n (cid:0) min (cid:0) , ( | λ ′′ | t ) − v +1 (cid:1)(cid:1) p ′ dt (2 . ≤ C | λ ′′ | − + C | λ ′′ | − p ′ v +1 Z √ n | λ ′′ | − t − p ′ v +1 dt (2 . p ′ < v + 1, the above is bounded by C | λ ′′ | − p ′ v +1 . If p ′ > v + 1 it is bounded by C | λ ′′ | − ,and if p ′ = v + 1 it is bounded by C | λ ′′ | − ln | λ ′′ | . Put together, (2 . 36) is bounded by14 | λ ′′ | − min( p ′ v +1 , unless p ′ = v + 1 when one has an additional logarithmic factor. Since | λ ′′ | is within a factor of √ n of | λ ′ | , we can replace C | λ ′′ | − min( p ′ v +1 , by C | λ ′ | − min( p ′ v +1 , here. Inserting this back into (2 . p ′ = v + 1 we see that | T ′ ( λ , ...λ n ) | ≤ C | λ ′ | − min( v +1 , p ′ ) || g || p (2 . p ′ = v + 1 there is an additional factor of | ln | λ ′ || p ′ . This gives (1 . a ) for p > 1. (Onecan have 2 + | λ ′ | instead of | λ ′ | in (1 . a ) simply because the absolute value of the integrandof T ′ ( λ , ...λ n ) is integrable). This completes the proof of part b) of Theorem 1.1. Proof of part c) of Theorem 1.1. We repeat the argument from (2 . 5) to (2 . λ = 0. At this point, wereplace M by the v by v matrix M ′ given by deleting the first column and v + 1st row of M . We then proceed as before. Since there is no longer a λ F ( v , ..., v n ) r term in (2 . v derivatives to make the argument work. Thus the remainder of the proof ofpart c) of the theorem proceeds exactly as part b), with v + 1 replaced by v and we obtainthe estimate (1 . b ). This completes the proof of part c) of Theorem 1.1 and therefore thewhole theorem. Proof of Theorem 1.2. When a i is a homogeneity appearing only once, (2 . 33) becomes | T ′ ( λ , ...λ n ) | ≤ C Z [0 , n min (cid:0) , | λ i x i | − v +1 (cid:1) | g ( x , ..., x n ) | dx ... dx n (2 . . 38) is exactly (2 . 27) with λ replaced by λ i and f ( x , ..., x n ) replaced by x i .The exact sequence of steps going from (2 . 27) to (2 . 32) therefore gives the estimate (1 . a )in place of (1 . 4) as desired. This gives part a) of Theorem 1.2. As for part b), as in part c)of the proof of Theorem 1.1, the absence of a λ F ( v , ..., v n ) r term in (2 . 12) when λ = 0means that one needs only v derivatives in the uses of the Van der Corput lemma that ledto (2 . . 38) to | T ′ (0 , λ ...λ n ) | ≤ C Z [0 , n min (cid:0) , | λ i x i | − v (cid:1) | g ( x , ..., x n ) | dx ... dx n (2 . . 27) to (2 . 32) give (1 . b ), which gives us part b) of Theorem1.2. Since we dealt with the sharpness statement of part c) at the end of the last section,we are done with the proof of Theorem 1.2. Proof of Theorem 1.4. ψ ( x , ..., x n ) be a smooth nonnegative function supported on the unit ball andequal to 1 on a neighborhood of the origin, and let ψ ( x , ..., x n ) = ψ (2 − a x , ... − an x n ) − ψ ( x , ..., x n ). We rewrite the integral (1 . 3) defining T ( λ , ..., λ n ) as Z R n e iλ f ( x ,...,x n )+ iλ x + ... + iλ n x n φ ( x , ..., x n ) ψ ( x , ..., x n ) dx + K X j =0 Z R n e iλ f ( x ,...,x n )+ iλ x + ... + iλ n x n φ ( x , ..., x n ) ψ (2 − ja x , ..., − jan x n ) dx ... dx n (2 . K denotes a constant depending on the size of the support of φ ( x , ..., x n ) and ourestimates will not depend on K . Note that the first term of (2 . 40) satisfies the conditionsof Corollary 1.3 with g ( x ) = 1, so the term is bounded in absolute value by C (1 + | λ | ) − ǫ as needed. Hence we may devote our attention to bounding the sum in (2 . . 40) to ( y , ..., y n ) = (2 − ja x , ..., − jan x n ), and usingthe mixed homogeneity of f the sum becomes K X j =0 Z R n e i j λ f ( y ,...,y n )+ iλ ja y + ... + iλ n jan y n (2 j P ni =1 1 ai ) φ (2 ja y , ..., jan y n ) × ψ ( y , ..., y n ) dy ... dy n (2 . ψ ( y , ..., y n ) factor the integrand in (2 . 41) is supported on an annuluscontained in the unit ball but not intersecting a smaller ball centered at the origin. We nowapply Corollary 1.3 to each term in (2 . 41) and add the resulting estimates. In order to dothis, we determine for each j th term a number M j for which the function p j ( y , ..., y n ) =(2 j P ni =1 1 ai ) φ (2 ja y , ..., jan y n ) ψ ( y , ..., y n ) satisfies (1 . 2) with g ( x ) = M j . As a result,by Corollary 1.3 the corresponding term of the sum in (2 . 41) will be bounded in absolutevalue by CM j − jǫ . We will see that P j M j − jǫ is uniformly bounded when (1 . 11) holds,and therefore Theorem 1.4 follows.A direct computation using (1 . 11) reveals that | p j ( y , ..., y n ) | , n X i =1 (cid:12)(cid:12) y j ∂p j ∂y ( y , ..., y n ) (cid:12)(cid:12) ≤ CA (2 j P ni =1 1 ai )(2 − jk )( n X i =1 | y i | a i ) − k (2 . C here just depends on the function ψ . Since ( y , ..., y n ) is in the annuluscentered at the origin, the ( P ni =1 | y i | a i ) − k factor is bounded, so we have | p j ( y , ..., y n ) | , n X i =1 (cid:12)(cid:12) y j ∂p j ∂y ( y , ..., y n ) (cid:12)(cid:12) ≤ C ′ A j (cid:0) P ni =1 1 ai − k (cid:1) (2 . p j ( x ) satisfies (1 . 2) with g ( x ) = C ′′ A j ( P ni =1 1 ai − k ) . So as long as k > P ni =1 1 a i − ǫ ,which we are assuming, then for ζ = k − P ni =1 1 a i + ǫ > 0, Theorem 1.1 a) gives M j − jǫ ≤ C ′′′ j (cid:0) P ni =1 1 ai − k (cid:1) × − jǫ = C − jζ (2 . P j M j − jǫ is uniformly bounded as needed. This completes the proof of Theorem1.4. References. 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