Two Problems Associated With Convex Finite Type Domains

We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp $L^p$ estimates for $p>4$, generalizing the Carleson-Sjolin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.


Introduction
Let Ω be a convex domain in R d with smooth boundary.We assume that ∂Ω is of finite line type, that is, at each point each tangent line has finite order of contact.
We discuss two problems in this paper.Both problems have in common that progress can be made using some approximate scaling properties of ∂Ω.We derive an extension of the Carleson-Sjölin theorem concerning L p convergence results for Riesz means defined by a distance function associated to Ω; we assume that 1 ≤ p ≤ 4/3.We also give asymptotics for the number of integer lattice points inside large dilates of Ω; the bounds for the error terms are sharp in some cases where there exist points with all lines tangent to the boundary having high order of contact with ∂Ω.

Riesz means.
We assume that the origin belongs to the interior of Ω.Let ρ : R d → [0, ∞) homogeneous of degree 1 be the Minkowski functional associated to Ω; i.e. ρ is homogeneous of degree one, so that ρ(ξ) = 1 if ξ ∈ ∂Ω.The boundary Σ ρ := ∂Ω is then the unit sphere for the generalized distance function ρ.The Bochner-Riesz operator associated to ρ is defined by here our definition of the Fourier transform is f (ξ) = f (y)e −ı y,ξ dy.It is well known that if 1 ≤ p < ∞ the L p boundedness of the Bochner-Riesz operator implies L p convergence of the Riesz means F −1 [(1 − ρ/t) λ + f ] to the limit f if f ∈ L p and t → ∞.
A necessary condition for L p boundedness is Indeed in view of the compact support of the multiplier it is necessary for L p boundedness that the inverse Fourier transform of (1 − ρ) λ + belongs to L p .Using standard asymptotic expansions one can show (working near points on Σ ρ where the curvature does not vanish) that (1.2) is necessary for It is known [9], [29] that the validity of an L 2 restriction theorem for the Fourier transform implies the L p boundedness of the Bochner-Riesz operator.Since Σ ρ is of finite type, say ≤ n, it follows from [3] that the Fourier transform of dσ(ξ) of a smooth density carried by Σ ρ is O(|ξ| −µ ) for some µ with µ ≥ (d − 1)/n.Using the appropriate versions of the Stein-Tomas restriction theorem [10] one can show that L p boundedness holds for 1 ≤ p ≤ 2(µ + 1)/(µ + 2) and λ > λ(p) (cf.[29]).Note that 2(µ+1)/(µ+2) = (2n+2d−2)/(2n+d−1) for the example x n i with even n, so that the range obtained in this way is small for large n.Theorem 1.1.Suppose that d ≥ 2, 1 ≤ p ≤ 4/3, λ > d(1/p−1/2)−1/2, and that Σ ρ is of finite line type.Then S λ,ρ is bounded on L p (R d ).
It is conjectured that L p boundedness holds for the same range of exponents as for the sphere.The conjecture for the sphere is that L p boundedness should hold for λ > λ(p) for p < 2d/(d + 1).This is currently known only in two dimensions, see [4].Sjölin [28] extended this result to arbitrary planar domains with smooth boundary, for some variants concerning convex domains in the plane with nonsmooth boundary, see also the more recent paper by Ziesler and the third author [27].For partial results in higher dimensions, in the case that the Gauß curvature of Σ ρ does not vanish, we refer to Bourgain [1] and for background to [29].Our proof of Theorem 1.1 uses a variant of Córdoba's geometrical proof [6] of the Carleson-Sjölin theorem and rescaling.

Multitype and an estimate for the Fourier transform of surface carried measure.
A precise estimate of the Fourier transforms of surface carried measure is due to Bruna, Nagel, and Wainger [3].Let Σ = ∂Ω and H P (Σ) the affine tangent plane at P ∈ Σ, and let where P ± are the points on Σ for which ξ is a normal vector and |B| denotes the surface measure of B. For many problems it is important to know not just the size of the balls but also the distribution function of x → |B(x, δ)| and how it relates to the notions of multitype and type.We review the definition of multitype which is implicit in [26], see also [17].
Consider a smooth real valued function Φ defined in a neighborhood of the origin in a d − 1-dimensional Euclidean vector space E d−1 so that Φ(0) = ∇Φ(0) = 0. We say that a vector v in The sets are linear subspaces of E d−1 and there are even integers m 1 , . . ., m k so that moreover the sequence is maximal, in the sense that The d − 1-tuple a = (a 1 , . . ., a d−1 ) is then called the multitype of Φ at 0.
To illustrate the above definitions consider a convex body whose boundary passes through the origin and nearby is given by the equation where the a i are even integers, with a i ≤ a i+1 , 1 ≤ i ≤ d − 2. In this case the multitype is (a 1 , . . ., a d−1 ) and the subspaces S m above are S m = span({e i : We now fix P ∈ Σ, choose a unit normal n P and parametrize Σ near P as a graph over its tangent plane at P .Thus the parametrization is given by for v ∈ T P Σ, and Φ is a convex function vanishing of second order at the origin.We perform the above construction for Φ(v) defined on , j = 1, . . ., k, then We denote by Π P j the orthonormal projection on T P Σ to W j .We also have a similar decomposition and projections Π P j to W * j on T * P Σ, here we let W * j the space of linear functionals on W j extended by 0 on the orthogonal complement of W j .We can extend these projections to linear maps on and write ν(P ) ≡ ν 1 (P ).An alternative description of ν(P ) (see [16]) is ν(P ) = sup{q : dist(•, H P Σ) ∈ L q (Σ)}; (1.14) in fact for q = ν(P ) the function dist(•, H P Σ) −1 belongs to the space L q,∞ (Σ).
Our result for the Fourier transform of surface carried measure is Proposition 1.2.Let P ∈ ∂Ω.Then there is a neighborhood U of P and a conic neighborhood V of {±n P } in R d so that for all χ ∈ C ∞ 0 (U ) and all ξ ∈ V with |ξ| ≥ 1 we have and N is sufficiently large.
In this statement N > d + m k will suffice.Note the proposition is an improvement over previous results only in the case where all the principal curvatures vanish (and thus a 1 > 2).

A lattice point estimate.
Let It is well known (and elementary) that N Ω (t) is asymptotic to t d vol(Ω) as t → ∞ and that the error term ). Moreover if ∂Ω has suitable curvature properties then the error term improves; in particular if the Fourier transform of the surface measure on the boundary satisfies dσ(ξ) = O(|ξ| −α ) then the classical method (see e.g.[11], [13,Theorem 7.7.16], and [24]) yields . This estimate however is not sharp, and several authors beginning with van der Corput have obtained improvements for the case of nonvanishing Gauß curvature; see the monographs by Krätzel [18] and Huxley [14], and in particular the papers by Krätzel and Nowak [20] and recent improvements by W. Müller [22] for results on general convex bodies with nonvanishing curvature in higher dimensions.In [24, I], [25] Randol obtained better estimates for the case of convex domains in the plane with finite type boundary; these are sharp for Ω = {x : where k ≥ 4 is even.See also [23] for more refined results.Generalizations to domains of the form Ω = {x : [19].
Here we give a version for general convex bodies with finite type boundary in higher dimensions.Let ν(P ) = ν 1 (P ) and ν 2 (P ) as in (1.13) above.
Then there is a constant C depending on Ω so that Specifically, if Γ is the set of all points P ∈ ∂Ω at which all principal curvatures vanish then where G P (t) is bounded as t → ∞.If the normal line determined by n P coincides with Re i for some i ∈ {1, . . ., d} then lim sup t→∞ |G P (t)| > 0.
We shall derive the estimate for the Fourier transform in Proposition 1.2 in the next section.Section 3 contains the application to the lattice point problem.In Section 4 and Section 5 we prove results on Bochner-Riesz multipliers; here we first consider the case of one nonvanishing principal curvature and then in Section 5 the case of convex domains.

Notation. Given two quantities A, B we write A
B if there is an absolute positive constant C so that A ≤ CB.We write A ≈ B if A B and B A.
T (b, M, V, m, N) be the class of all C N (B T ) functions g with the property that Here S d−2 denotes the unit sphere in E d .We also define a(V, m) = (a 1 (V, m), . . ., a l (V, m)) by in analogy to (1.7).Now if P ∈ Σ (with Σ = ∂Ω as in the introduction) and (1.5).Let Φ be as in (1.8).Then there is T > 0 and a neigborhood U of 0 so that for all w ∈ U the functions y → T (b, M, V, m, N) and suppose that a = (a 1 , . . ., a d−1 ) is the multitype at the origin.Let Ψ w (y) = Φ(y) − Φ(w) − ∇ w Φ(w), y − w and let a(w) = (a 1 (w), . . ., a d−1 (w)) be the multitype of Ψ w at the origin.Then there is a neighborhood U of the origin so that a i (w) ≤ a i for i = 1, . . ., d − 1 and all w ∈ U.
Proof: Let S mi be as in (1.5) and let > dim S mi .Recall that S n = S mj−1 for m j < n ≤ m j−1 .Using continuity and compactness arguments together with the definition of the spaces S mi we see that there is a neighborhood U ⊂ U of the origin so that for every w ∈ U, every y ∈ U and every -tuple of orthonormal vectors {u 1 , . . ., u } i=1 s≤mj The result of the lemma follows quickly from the definition of the multitype.
We now let Σ denote the graph of Φ.On T 0 Σ = R d−1 we define a nonisotropic distance function ρ by note that that the unit ball for ρ * in (1.11) is the polar set for the unit ball for ρ.
The following lemma gives an improvement of estimates in [16] and [17].A rescaling argument is used as in those papers; the present improvement is obtained using a more careful argument for the rescaled pieces.
Lemma 2.2.Let Φ be a convex smooth function defined in a neighborhood of the origin in R d−1 , so that Φ(0) = ∇Φ(0) = 0. Let V be the flag of subspaces {S mj } defined as in (1.5).Let a be the multitype of Φ near 0, B(w, δ) as in (2.3) and ρ as in (2.6).
Then there is a neighborhood U of the origin and δ 0 > 0 so that for all 0 < δ ≤ δ 0 and all w ∈ U Proof: We may assume that a 1 > 2 since otherwise the theorem follows already from the estimate (2.4).Let {u 1 , . . ., u d−1 } an orthonormal basis of R d−1 so that for j = 0, . . ., k−1.By performing a rotation we may assume that the u i are the standard coordinate vectors.

Define dilations A t by
According to [26], [16] we may split where Q is a convex polynomial satisfying and the remainder term R satisfies for |x| ≤ T and all multiindices α = (α 1 , . . ., α d−1 ) with |α| ≤ N .Since Q is positive away from the origin and homogeneous with respect to dilations (A t ) we have that where ρ is as in (2.6); in fact ρ(y) , where R and its derivatives tend to zero uniformly on compact sets, as → ∞.
Denote by a(w) = (a 1 (w), . . ., a d−1 (w)) the multitype of Q at w. Then a(0) = a and by Lemma 2.1 there is M > 0 so that a i (w) ≤ a i for 0 ≤ ρ(w) ≤ 2 −M +2 and, by (2.10/11), a 1 (w) = 2 for 0 < ρ(w) ≤ 2 −M +2 ; note that nothing is said about the position of the spaces S m (w).Now for any point w there is an open ball U (w) of radius T (w)/4 and a flag V(w) consisting of l(w) nested subspaces and an l(w)-tuple m(w) so that for x ∈ U (w) the functions By the metric property of the nonisotropic balls B(w, δ) there are constants C 2 C 1 1 and δ 1 1 so that we may assume that C 1 ≥ 2 2M +4 .
We shall now show that there are constants c 0 > 0, C 0 > 1 so that for 2 − ≤ c 0 in the open ball of radius 2 −M centered at the origin.We may cover the compact annulus W by finitely many open balls U i with center w i ∈ W and radius T (w i )/4 so that Since Φ converges to Q in the C N -topology uniformly on compact sets, there is a positive constant c 0 so that for 2 − ≤ c 0 the functions . By the finite type property there is a δ 0 > 0 so that for γ ≤ δ 0 and x ∈ U i the caps by the analogue of (2.4) with exponent 1/2 + ν 2 ; here C is independent of .Now in order to show that (2.13) holds we assume that C −1 and observe that the image of B(y, δ) under the linear transformation ) instead and observe that in this range δ α 2 (α−ν) δ ν , provided that α ≥ ν.This together with (2.13) proves the asserted statement.
There is a neighborhood U ⊂ U of the origin and a conic neighborhood V of e d so that for ξ ∈ V

17)
Proof: We may assume that (2.7) holds and that the u i 's form the standard basis in R d−1 .Observe that then Assume that s/2 ≤ ρ(x) ≤ 2s and s is small.Then |A 1/s x| ≈ 1 and Similarly by (2.11) the remainder term R xi satisfies the same estimate so that for small x and therefore Now let x(ξ) be the unique point at which ξ is normal to the graph of Φ.By the Bruna-Nagel-Wainger estimate for the Fourier transform (1.4) and Lemma 2.2 we have that and since x(ξ) is determined by ξ i /ξ d = ±Φ xi (x(ξ)) for i = 1, . . ., d − 1, the estimate (2.17) follows.

Lattice point estimates
In this section we prove Theorem 1.3.We use a variant of the classical proof (see [24] for the two-dimensional case).Choose We work with the ε-regularization χ Ω * ζ ε of the characteristic function of Ω and define By the Poisson summation formula By the divergence theorem where and n i denotes the i th component of the outer normal vector n P .
Let Γ be the set of points P ∈ Σ at which all principal curvatures vanish.As noticed in [16] it follows from (2.10/11) that the set Γ is discrete, thus finite by compactness.For every P ∈ Γ we choose a narrow conic symmetric neighborhood V P of the normals {±n P }, a small neighborhood U P of P in Σ, and a C ∞ 0 function χ P whose restriction to Σ vanishes off U and so that χ P equals one in a neighborhood of P .We may arrange these neighborhoods so that the sets V P ∩ {ξ : |ξ| ≥ 1}, P ∈ Γ are pairwise disjoint, and that the normals to all points in a neighborhood of U P are contained in V P (thus the U P 's are disjoint too).Define F i,P (ξ) = Σ χ P (y)n i (y)e −ı y,ξ dσ(y).
If the cones V P are chosen sufficiently narrow, we have (3.4) The estimate for ξ ∈ V P follows from Proposition 1.2, and the estimate for ξ / ∈ V P follows by a simple integration by parts; namely if t → γ(t) parametrizes Σ near P then | γ (t), ξ | ≈ |ξ| for γ(t) ∈ U P and ξ / ∈ V P .Moreover by the Bruna-Nagel-Wainger estimate we have here we used the definition of Γ and the fact that χ P equals one near P .
We now estimate the remainder term R ε (t) where ε 1/t will be suitably chosen.Let dist ∞ denote the distance taken with respect to the ∞ metric in R d , or Z d .For P ∈ Γ let Let When evaluating A i P we essentially sum over integers in a tubular neighborhood of a line and by the estimate (2.4) we certainly get Next for the estimation of D i P we use the rapid decay estimate in (3.4) to obtain and for C i P we use (3.5) which yields and we claim that for λ ≥ 1 (3.10) which implies We verify (3.10).Let a = a(P ) be the multitype at P .In view of dist(k, Rn P ) ≥ 3/4 it is straightforward to check that Thus we may replace the sum in (3.10) by an integral.After performing a suitable rotation in this integral we have to show that (3.12) and therefore the integral in (3.12) is bounded by

This shows (3.10).
To finish the proof we note that where C is a constant depending only on the geometry of Ω.Thus, by taking into account the leading term in (3.1) we see that and the desired estimate follows if we choose ε = t −d/ (d−µ) .This completes the proof of (1.17).
Lower bounds: To show (1.18) we work with our choice ε = ε(t) = t −d/(d−µ) .For (1.18) we simply set which we already showed to be bounded above.However we have to verify the claim that lim sup t∈∞ |G P (t)| > 0 in the case where n P = ±e i .We now assume that n P = e i (the case n P = −e i is handled in the same way).Then define We split this sum into parts G P (t) = I(t) + II(t) where For the estimation of II we note that To examine I(t) we parametrize by our assumption on n P = e i .
where Φ ≡ Φ P is convex, vanishes of second order at the origin of R d−1 and has multitype a(P ) there; χ 0 is smooth, compactly supported and equal to one in a neighborhood of the origin.By the convexity P, n P = P, e i = 0. To examine the integral we may use an asymptotic expansion derived in [26] (stated there for κ → ∞, but the statement for κ → −∞ follows similarly).We obtain where c 0 (P ) > 0 and η is the reciprocal of the least common multiple of a 1 , . . ., a n .Thus The sum defines a periodic function which is not identically zero, by the uniqueness theorem for Fourier series.Combining this with the estimation for the error term II(t) we see that lim sup t→∞ |G P (t)| > 0.
Remark.For almost all rotations the estimates for the error term improve.There is r > 2 so that [2] this is proved using a result on the maximal function which was shown by Svensson [30] to be in L q0 (S d−1 ) for some q 0 > 2 (under our assumption of finite line type, see also [25] for a similar result with additional real analyticity assumption).Indeed, let R ε, then for q ≤ q 0 Proof: We may assume that U is the unit cube, and that the support of χ has small diameter.We decompose ) so that ψ is equal to 1 on the support of ψ.Denote by T k the convolution operator with Fourier multiplier m δ,k and by R k the convolution operator with Fourier multiplier 1/p which follows for p = ∞ from Minkowski's inequality and for p = 2 by orthogonality; for 2 < p < ∞ one uses interpolation.Since and therefore it suffices to show that The estimate (4.1) is proved using arguments in [6] which we will sketch.For ν ∈ Z we define operators T k,ν and S ν by f where the sum is extended over integers ν with |ν| δ −1/2 since we assume that the support of χ is small.Now It can be checked that the family of functions (T k,ν S ν f )(T k,ν S ν f ) has an orthogonality property which implies that The proof of (4.3) is based on an idea of C. Fefferman [9]; in higher dimensions one uses the following (iv) ξ , η , ζ and ω belong to the cube of sidelength 4δ 1/2 centered at a . Then In (4.4), c depends only on the lower bound of g ξ1ξ1 and the C 4 norm of g in supp χ.

Sketch of Proof:
A Taylor expansion about the origin yields where r vanishes of third order at the origin.(4.4) is proved by verifying The straightforward calculation is omitted; we note that formula (6.30) in [21] turns out to be useful in order to carry it out.
Proof of Proposition 2.1, cont.: By (4.3) it remains to show that ) which gives a one parameter family of vectors normal to Σ ρ .
For σ ≥ 2 let R k,σ be the set of all cylinders whose base is a d − 2 dimensional ball of radius s and whose height is σs (any s > 0), so that the axis is parallel to Γ k (t) for some |t| ≤ 1.
Define the maximal function Then arguing as in [6] and using standard estimates for the kernel of T k,ν we see that The L p norm of ( ν |S ν f | 2 ) 1/2 is bounded by the L p norm of f , for p ≥ 2 (see [6]) and therefore we can finish our proof by using duality and showing that If we knew that for every ξ the function t → ξ, Γ k (t) changed sign at most M times then it would follow from a result by Córdoba [7] that (4.6) holds with σ ε replaced by C 1 M [log σ] C2 .This hypothesis may not be satisfied, but we can get around this point by a simple approximation.Namely, divide Let P k,j (t) be the vector valued Taylor polynomial of degree [2/ε] of ∇ ξ g(•, δ 1/2 k) expanded about a j , and let Γ k,j (t) = (−P k,j (1), 1).Then Let R k,σ,j be the set of all cylinders whose base is a d − 2-dimensional ball of radius s whose height is σs, so that the axis is parallel to Γ k,j (t) for some |t| ≤ 1.If M k,σ,j denotes the associated maximal operator then it is immediate that M k,σ f ≤ j M k,σ,j f where the sum contains only O(σ ε/2 ) terms.Córdoba's result yields the L 2 bound C ε [log σ] C2 for each M k,σ,j .This finishes the proof of (4.6).

Proof of Theorem 1.1
The L 1 version of the theorem is well known, and therefore by an interpolation argument one has to show the boundedness on L 4/3 (R d ), or, equivalently, on L 4 (R d ).
Let ξ 0 ∈ Σ ρ .It suffices to show that there exists a neighborhood and supported in V .The multiplier norm is invariant under rotations and we may assume that Σ ρ can be parametrized as a graph ξ d = G( ξ), ξ ∈ R d−1 near ξ 0 , so that ρ(ξ) < 1 if ξ d > G( ξ).We write Next we set R r ( ξ) = 2 r R(A 2 −r ξ), so that G r = Q + R r tends to G in the C ∞ topology, as r → ∞.Since the Hessian of Q has rank 1 where 1/4 < Q( ξ) ≤ 4 (see (2.10)), the same is true for G r = Q + R r if r is large; we may assume that the matrix norm of (Q + R r ) is bounded below uniformly in r if r ≥ r 0 .
It is now easy to see that the C 4 norm of κ j,0 is 2 −jλ and κ j,0 is supported in a fixed ball with diameter independent of j.Therefore κ j,0 Mp 2 −jλ , 1 ≤ p ≤ ∞.
and obtain a flag of subspaces 0 = S m k P • • • S m0 P = T P Σ. (1.9)Let W j be the orthogonal complement of S