Two problems associated with convex finite type domains

Publ. Mat. 46 (2002), 153–177 TWO PROBLEMS ASSOCIATED WITH CONVEX FINITE TYPE DOMAINS Alexander Iosevich, Eric Sawyer and Andreas Seeger Abstract 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 Lpestimates for p>4, generalizing the Carleson-Sj¨olin 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. 1. Introduction Let Ω be a convex domain in Rdwith smooth boundary. We assume that ∂Ωisoffinite 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-Sj¨olin theorem concerning Lpconvergence 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. 1.1. Riesz means. We assume that the origin belongs to the interior of Ω. Let ρ:Rd→ [0,∞) homogeneous of degree 1 be the Minkowski functional associated to Ω; i.e. ρis homogeneous of degree one, so that ρ(ξ)=1ifξ∈∂Ω. The 2000 Mathematics Subject Classification. 42B, 11H. Key words. Lattice points, convex bodies, finite type, maximal Riesz means. Research supported in part by NSF grants. 154 A. Iosevich, E. Sawyer, A. Seeger boundary Σρ:= ∂Ω is then the unit sphere for the generalized distance function ρ. The Bochner-Riesz operator associated to ρis defined by  Sλ,ρf(ξ)=(1−ρ(ξ))λ + f(ξ);(1.1) 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 Lpboundedness of the Bochner-Riesz operator implies Lpconvergence of the Riesz means F−1[(1 −ρ/t)λ + f]to the limit fif f∈Lpand t→∞. A necessary condition for Lpboundedness is λ>λ(p)=d|1/p −1/2|−1/2.(1.2) Indeed in view of the compact support of the multiplier it is necessary for Lpboundedness that the inverse Fourier transform of (1 −ρ)λ +belongs to Lp. Using standard asymptotic expansions one can show (working near points on Σρwhere the curvature does not vanish) that (1.2) is necessary for F−1[(1 −ρ)λ +]∈Lp. It is known [9], [29] that the validity of an L2restriction theorem for the Fourier transform implies the Lpboundedness of the Bochner-Riesz operator. Since Σρis of finite type, say ≤n, it follows from [3] that the Fourier transform of  dσ(ξ) of a smooth density carried by Σρis O(|ξ|−µ) for some µwith µ≥(d−1)/n. Using the appropriate versions of the Stein-Tomas restriction theorem [10] one can show that Lpboundedness 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 xd=d−1 i=1 xn 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 Σρis of finite line type. Then Sλ,ρ is bounded on Lp(Rd). It is conjectured that Lpboundedness holds for the same range of exponents as for the sphere. The conjecture for the sphere is that Lp boundedness should hold for λ>λ(p) for p<2d/(d+ 1). This is currently known only in two dimensions, see [4]. Sj¨olin [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 Σρdoes not vanish, we refer to Bourgain [1] and for background to [29]. Our proof of Theorem 1.1 uses a variant of C´ordoba’s geometrical proof [6] of the Carleson-Sj¨olin theorem and rescaling. Two Problems Associated with Convex Domains 155 1.2. 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 HP(Σ) the affine tangent plane at P∈Σ, and let B(P,δ)={y∈Σ : dist(y,HP(Σ)) <δ}.(1.3) Then | dσ(ξ)|≤C|B(P+,|ξ|−1)|+|B(P−,|ξ|−1)| (1.4) where P±are the points on Σ for which ξis 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 Ed−1so that Φ(0) = ∇Φ(0) = 0. We say that a vector vin Ed−1has contact of order at least n+1if Φ(sv)=O(sn+1)ifs→0. The sets Sn={v∈En:vhas contact of order at least n+1}(1.5) are linear subspaces of Ed−1and there are even integers m1,...,m kso that m1<···<m k,1≤k≤d−1, and m0:= m1−1≥1 and 0=Smk···Sm0:= Ed−1;(1.6) moreover the sequence is maximal, in the sense that Sn=Smkif mk−1< n≤mk. Define (1.7) ai=mjif d−1−dim Smj−1<i≤d−1−dim Smj, j=1,...,k. The d−1-tuple a=(a1,...,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 xd=d−1 i=1 |xi|aiwhere the aiare even integers, with ai≤ai+1, 1≤i≤d−2. In this case the multitype is (a1,...,a d−1) and the subspaces Smabove are Sm= span({ei:ai>m}) (and Sm={0}if m≥ad−1). 156 A. Iosevich, E. Sawyer, A. Seeger We now fix P∈Σ, choose a unit normal nPand parametrize Σ near P as a graph over its tangent plane at P. Thus the parametrization is given by v=Γ(v)→ P+v+Φ(v)nP (1.8) for v∈TPΣ, and Φ is a convex function vanishing of second order at the origin. We perform the above construction for Φ(v) defined on Ed−1=TPΣ and obtain a flag of subspaces 0=Smk P···Sm0 P=TPΣ.(1.9) Let Wjbe the orthogonal complement of Smj Pin Smj−1 P,j=1,...,k, then TPΣ=W1⊕···⊕Wk.(1.10) We denote by ΠP jthe orthonormal projection on TPΣtoWj. We also have a similar decomposition and projections ΠP jto W∗ jon T∗ PΣ, here we let W∗ jthe space of linear functionals on Wjextended by 0 on the orthogonal complement of Wj. We can extend these projections to linear maps on T∗ PRd≃(Rd)∗by defining ΠP jnP=0. On T∗ PΣ we define a nonisotropic distance function ρ∗by ρ∗(η)= k  j=1 |ΠP jη| mj mj−1;(1.11) here |·| denotes the Euclidean distance in Wj.Ifξ∈T∗ PRdis taken from a suitable conic neighborhood of nPand ΠPdenotes the projection to T∗ PΣ we define ΘP(ξ)=ρ∗ΠPξ ξ,nP.(1.12) Finally we set for l≤d−2 νl(P)= d−1  i=l a−1 i= k  j=1 dim Smj−1 P−dim Smj P mj (1.13) and write ν(P)≡ν1(P). An alternative description of ν(P) (see [16]) is ν(P) = sup{q: dist(·,H PΣ) ∈Lq(Σ)};(1.14) in fact for q=ν(P) the function dist(·,H PΣ)−1belongs to the space Lq,∞(Σ). Our result for the Fourier transform of surface carried measure is Two Problems Associated with Convex Domains 157 Proposition 1.2. Let P∈∂Ω. Then there is a neighborhood Uof P and a conic neighborhood Vof {±nP}in Rdso that for all χ∈C∞ 0(U) and all ξ∈Vwith |ξ|≥1we have | χdσ(ξ)|χCNmin{|ξ|−ν,|ξ|−1 2−ν2[ΘP(ξ)]ν−ν2−1 2}; here χCN= maxα≤Nχ(α)L∞(U)and Nis sufficiently large. In this statement N>d+mkwill suffice. Note the proposition is an improvement over previous results only in the case where all the principal curvatures vanish (and thus a1>2). 1.3. A lattice point estimate. Let NΩ(t) = card(tΩ∩Zd).(1.15) It is well known (and elementary) that NΩ(t) is asymptotic to tdvol(Ω) as t→∞and that the error term EΩ(t)=N(t)−tdvol(Ω)(1.16) is O(td−1). 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 EΩ(t)= O(td−1−α d−α). 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 Kr¨atzel [18] and Huxley [14], and in particular the papers by Kr¨atzel and Nowak [20] and recent improvements by W. M¨uller [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: xk 1+xk 2≤1}where k≥4 is even. See also [23] for more refined results. Generalizations to domains of the form Ω = {x:xk 1+···+xk d≤1}are in [24, II], [19]. Here we give a version for general convex bodies with finite type boundary in higher dimensions. Let ν(P)=ν1(P) and ν2(P)asin (1.13) above. 158 A. Iosevich, E. Sawyer, A. Seeger Theorem 1.3. Let ν= min P∈∂Ων(P),µ=1 2+ min P∈∂Ων2(P). Then there is a constant Cdepending on Ωso that |EΩ(t)|≤CΩ(1 + td−1−ν+td−1−µ d−µ).(1.17) Specifically, if Γis the set of all points P∈∂Ωat which all principal curvatures vanish then EΩ(t)= P∈Γ td−1−ν(P)GP(t)+O(td−1−µ d−µ)(1.18) where GP(t)is bounded as t→∞. If the normal line determined by nP coincides with Reifor some i∈{1,...,d}then lim supt→∞ |GP(t)|>0. We note that the number µ/(d−µ) is greater then (2d−1)−1since µ>1/2. In particular if the Gauß curvature only vanishes at one point at the surface and if ν<µ/(d−µ), then there is A∈SO(d)so that lim supt→∞ tν−d+1|EAΩ(t)|is positive (for other model cases compare [19], [23]). (1.18) over P∈Γ is finite since Γ is a discrete subset of ∂Ω (as noted in [16], cf. the proof of Lemma 2.2 below). We remark that it is well known that for almost all rotations A∈SO(d) the error terms EAΩ(t) improve, see [5], [31], [32], [23], [15], and [2]. 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,Bwe write ABif there is an absolute positive constant Cso that A≤CB. We write A≈Bif ABand BA. Acknowledgement. We thank the referee for pointing out some misprints and for making a suggestion concerning the exposition. 2. An estimate for Fourier transforms of surface carried measures We begin by reviewing some facts about classes of convex functions in [3], [26], [16], [17]. Let BT⊂Rndenote the open ball of radius Tcentered at 0; it is always assumed that T≤1. Two Problems Associated with Convex Domains 159 Fix a flag Vof subspaces 0 = Vk··· V 0of Ed−1, with V0= Ed−1, and let m=(m1,...,m k)beak-tuple of even positive integers with m1<··· <m k. For 0 <b≤M,N∈Z+,N>m k, let Sd−1 T(b, M, V,m,N) be the class of all CN(BT) functions gwith the property that g(0) = ∇g(0) = 0 d2 (dt)2g(x+tθ)t=0 ≥0 for all θ∈Sd−2,x∈BT max 2≤j≤mld dtj g(x+tθ)t=0≥bfor all θ∈Sd−2∩Vl−1,x∈BT max |α|≤N∂ ∂xα g(x)≤Mfor all x∈BT. (2.1) Here Sd−2denotes the unit sphere in Ed. We also define a(V,m)= (a1(V,m),...,a l(V,m)) by ai(V,m)=mj(V,m)ifd−1−dim Vj−1<i≤d−1−dim Vj,(2.2) in analogy to (1.7). Now if P∈Σ (with Σ = ∂Ω as in the introduction) and Ed−1=TPΣ then let Vj=Smj P⊂TPΣ as in (1.5). Let Φ be as in (1.8). Then there is T>0 and a neigborhood Uof 0 so that for all w∈Uthe functions y→ Φ(w+y)−Ψ(w)−y,∇wΦ(w)belong to Sn T(b, M, V,m,N); moreover there are positive constants c0,C0,C1so that B(w,δ)={y:|Φ(y)−Φ(w)−∇ wΦ(w),y−w| ≤ δ}(2.3) belongs to BTif δ≤c0Tmkand satisfies meas(B(w,δ)) ≤Cδν;(2.4) see Proposition 2.1 in [17]. Lemma 2.1. Suppose that Φ∈Sd−1 T(b, M, V,m,N)and suppose that a=(a1,...,a d−1)is the multitype at the origin. Let Ψw(y)=Φ(y)− Φ(w)−∇ wΦ(w),y −wand let a(w)=(a1(w),...,a d−1(w)) be the multitype of Ψwat the origin. Then there is a neighborhood Uof the origin so that ai(w)≤aifor i=1,...,d−1and all w∈U. 160 A. Iosevich, E. Sawyer, A. Seeger Proof: Let Smibe as in (1.5) and let 6>dim Smi. Recall that Sn= Smj−1for mj<n≤mj−1. Using continuity and compactness arguments together with the definition of the spaces Smiwe see that there is a neighborhood U⊂ Uof the origin so that for every w∈ U, every y∈U and every 6-tuple of orthonormal vectors {u1,...,u }   i=1  s≤mj(ui,∇y)sΨw(y)≥b0>0.(2.5) The result of the lemma follows quickly from the definition of the multitype. We now let Σ denote the graph of Φ. On T0Σ=Rd−1we define a nonisotropic distance function ρby ρ(y)= k  j=1 |Πjy|mj;(2.6) 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 Φbe a convex smooth function defined in a neighborhood of the origin in Rd−1, so that Φ(0) = ∇Φ(0) = 0.LetVbe the flag of subspaces {Smj}defined as in (1.5).Letabe the multitype of Φnear 0,B(w,δ)as in (2.3) and ρas in (2.6).Letν=d−1 i=1 a−1 i, ν2=d−1 i=2 a−1 i. Then there is a neighborhood Uof the origin and δ0>0so that for all 0<δ≤δ0and all w∈U meas(B(w,δ)) ≤Cδα[ρ(w)]ν−α,ν≤α≤1 2+ν2. Proof: We may assume that a1>2 since otherwise the theorem follows already from the estimate (2.4). Let {u1,...,u d−1}an orthonormal basis of Rd−1so that Smj= span{ui,d−1−dim Smj<i≤d−1}(2.7) for j=0,...,k−1. By performing a rotation we may assume that the ui are the standard coordinate vectors. Two Problems Associated with Convex Domains 161 Define dilations Atby Atx=(t1 a1x,...,t 1 ad−1x).(2.8) According to [26], [16] we may split Φ(x)=Q(x)+R(x) where Qis a convex polynomial satisfying Q(Atx)=tQ(x)(2.9) and 0<|Q(x)|≤C1|x||∇Q(x)|≤C2|x|2 i,j  ∂2Q ∂xi∂xj (x),(2.10) and the remainder term Rsatisfies s−1∂|α| ∂xαR(Asx)s1/m (2.11) for |x|≤Tand all multiindices α=(α1,...,α d−1) with |α|≤N. Since Qis positive away from the origin and homogeneous with respect to dilations (At) we have that Q(y)≈ρ(y) where ρis as in (2.6); in fact ρ(y)≈d−1 i=1 |y,ui|ai. Set Φ(y)=2 Φ(A2−y) and note that Φ(y)=Q(y)+R(y), where Rand its derivatives tend to zero uniformly on compact sets, as 6→∞. Denote by a(w)=(a1(w),...,a d−1(w)) the multitype of Qat w. Then a(0) = aand by Lemma 2.1 there is M>0 so that ai(w)≤aifor 0≤ρ(w)≤2−M+2 and, by (2.10/11), a1(w) = 2 for 0 <ρ(w)≤2−M+2; note that nothing is said about the position of the spaces Sm(w). Now for any point wthere 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 h→ Qx(h)=Q(x+h)−Q(x)−∇Q(x),h belong to a class Sd−1 T(w)(b(w),M(w),V(w),m(w),N) so that ai(V(w), m(w)) ≥aiand a1(V(w),m(w)) = 2. By the metric property of the nonisotropic balls B(w,δ) there are constants C2C11 and δ11 so that B(y,δ)⊂{x:C−1 1ρ(y)≤ρ(x)≤C1ρ(y)}if ρ(y)≥C2δ;(2.12) we may assume that C1≥22M+4. 168 A. Iosevich, E. Sawyer, A. Seeger where Φ ≡ΦPis convex, vanishes of second order at the origin of Rd−1 and has multitype a(P) there; χ0is smooth, compactly supported and equal to one in a neighborhood of the origin. By the convexity P,nP= P,ei!= 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 Fi,P (2πtκei)=e−2πıtκP,eiκ−νc0(P)eπi 2νsign(κ)+O(κ−ν−η) where c0(P)>0 and ηis the reciprocal of the least common multiple of a1,...,a n.Thus I(t)=c0(P)π−1 κ>0 |κ|−ν−1sin 2πκtP,ei−π/(2ν)+O(κ−ν−1−η). 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 supt→∞ |GP(t)|>0. Remark. For almost all rotations the estimates for the error term improve. There is r>2 so that |EAΩ(t)|≤C(A)td−1−d−1 d+1 log1/r(2 + t) (indeed Cis in Lq(SO(d)) for q<r). As in [2] this is proved using a result on the maximal function M(θ) = sup r>0 r(d+1)/2|χΩ(rθ)| which was shown by Svensson [30]tobeinLq0(Sd−1) for some q0>2 (under our assumption of finite line type, see also [25] for a similar result with additional real analyticity assumption). Indeed, let Rε,A(t)= k=0 χΩ(2πtAk) ζ(2πεtk) and Mj(A) = sup 2j≤t≤2j+1 |Rεj,A(t)|,with εj=2 −2jd/(d+1) then for q≤q0 MjLq(SO(d)) ≤2jd  k=0 (1 + |εj2j|k|)−N(2j|k|)−(d+1)/2 ×|M(Ak |k|)|qdA1/q 2j(d−1−d−1 d+1 )MLq(Sd−1) Two Problems Associated with Convex Domains 169 by the (standard) choice of εj. But |EAΩ(t)|t−(d−1−d−1 d+1 )log−1/r(2 + t) 1+  j>0 |Mj(A)2−j(d−1−d−1 d+1 )(1 + j)−1/r|q  1/q which is in Lq(SO(d)) for r<q 0. We remark that the methods in W. M¨uller’s paper [22] could be used to improve the above bound to |EAΩ(t)|≤C(A)td−1−d−1 d+1 −βwhere β=β(Ω) >0 and Cis finite almost everywhere. 4. Bochner-Riesz multipliers - the case of one nonvanishing principal curvature In this section we shall prove a general theorem concerning multipliers of Bochner-Riesz type associated to surfaces with at least one nonvanishing principal curvature. Then, in the subsequent section, we shall deduce Theorem 1.1 by rescaling arguments. In what follows Mpwill be the space of Fourier multipliers on Lp(Rd); mMpis the operator norm of the operator Tmdefined by  Tmf(ξ)= m(ξ) f(ξ). We split variables in Rdas ξ=(  ξ,ξd) and in the statement of the proposition we further split  ξ=(ξ1,ξ)∈R×Rd−2. The proof of the following result uses the ideas from the two-dimensional case, see [9], [6]. Proposition 4.1. Let ε>0,N≥d+1+2/ε, and let g∈CN(Rd−1). Suppose that there is a cube Ucentered at the origin and a>0so that ∂2g ∂ξ2 1 (ξ1,ξ)≥a in U.Letχbe supported in Uand let φbe a smooth function supported in (1/2,2).Let0<δ1and mδ(ξ)=χ(ξ)φ(δ−1(ξd−g(ξ1,ξ))). Then mδM4≤Cεδ−d−2 4−ε, where Cεdepends only on a,ε,U, the CN(U)norms of the functions g, χand the Cd+1 norm of φ. 170 A. Iosevich, E. Sawyer, A. Seeger Proof: We may assume that Uis the unit cube, and that the support of χhas small diameter. We decompose mδ=kmδ,k where k= (k2,...,k d−1) ranges over (d−2)-tuples of integers ki≤Cδ−1/2and mδ,k(ξ)=mδ(ξ) d−1  i=2 ψ(δ−1/2ξi−ki) for suitable ψ∈C∞ 0satisfying ∞ n=−∞ ψ(s−n) = 1, so that supp ψ⊂ [−1,1]. Let  ψ∈C∞ 0([−2,2]) so that  ψis equal to 1 on the support of ψ. Denote by Tkthe convolution operator with Fourier multiplier mδ,k and by Rkthe convolution operator with Fourier multiplier  ψ(δ−1/2ξ− k). Note that RkLp→Lp≤C,1≤p≤∞. Then for 2 ≤p≤∞  k Rkgkp  kgkp p1/p which follows for p=∞from Minkowski’s inequality and for p=2by orthogonality; for 2 <p<∞one uses interpolation. Since Tk=RkTkRk it follows that k TkL4→L4 ≤Cδ−(d−2)/4sup k TkL4→L4 and therefore it suffices to show that TkL4→L4δ−ε.(4.1) The estimate (4.1) is proved using arguments in [6] which we will sketch. For ν∈Zwe define operators Tk,ν and Sνby  Sνf(ξ)=  ψ(δ−1/2ξ1− ν) and  Tk,νf(ξ)=ψ(δ−1/2ξ1−ν) Tkf(ξ). Then Tk=νTk,νSνfwhere the sum is extended over integers νwith |ν|δ−1/2since we assume that the support of χis small. Now  ν Tk,νSνf 2 4 = ν,ν (Tk,νSνf)(Tk,νSνf)2 ≤ :2δ1/21  (ν,ν): |ν−ν|≈2 (Tk,νSνf)(Tk,νSνf)2 . (4.2) Two Problems Associated with Convex Domains 171 It can be checked that the family of functions (Tk,νSνf)(Tk,νSνf) has an orthogonality property which implies that  (ν,ν) |ν−ν|≈2 (Tk,νSνf)(Tk,νSνf)2  ν |Tk,νSνf|21/2 2 4 .(4.3) The proof of (4.3) is based on an idea of C. Fefferman [9]; in higher dimensions one uses the following Lemma 4.2. Suppose that a∈Rd−2,|a|1, and the vectors  ξ,η, ζ, ωsatisfy (i) ξ+η− ζ−ω=0, (ii) ξ1>ζ 1>0,η1<ω 1<0, (iii) | ξ|,|η|,| ζ|,|ω|∈[2−1δ1/2,2+1δ1/2], (iv) ξ,η,ζand ωbelong to the cube of sidelength 4δ1/2centered at a. Then g( ξ)+g(η)−g( ζ)−g(ω)≥c2δ1/2|ξ1−ζ1|+|η1−ω1|.(4.4) In (4.4),cdepends only on the lower bound of gξ1ξ1and the C4norm of gin supp χ. Sketch of Proof: A Taylor expansion about the origin yields g( ξ)+g(η)−g( ζ)−g(ω)=I+II +III +IV where I=1 2gξ1ξ1(0)(ξ2 1+η2 1−ζ2 1−ω2 1) II=1 2ξ1gξ1ξ(0),ξ+η1gξ1ξ(0),η−ζ1gξ1ξ(0),ζ−ω1gξ1ξ(0),ω  III=1 2ξ,g ξξ(0)ξ+η,g ξξ(0)η−ζ,g ξξ(0)ζ−ω,g ξξ(0)ω  IV =r( ξ)+r(η)−r( ζ)−r(ω) 172 A. Iosevich, E. Sawyer, A. Seeger where rvanishes of third order at the origin. (4.4) is proved by verifying I≈2δ1/2(|ξ1−ζ1|+|η1−ω1|) II ≤C2δ III ≤Cδ IV ≤C22δ(|ξ1−ζ1|+|η1−ω1|). 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  ν |Tk,νSνf|21/24 δ−εf4.(4.5) Let Γk(t)=(−∇ξg(t, δ1/2k),1) which gives a one parameter family of vectors normal to Σρ. For σ≥2 let Rk,σ be the set of all cylinders whose base is a d−2 dimensional ball of radius sand whose height is σs (any s>0), so that the axis is parallel to Γk(t) for some |t|≤1. Define the maximal function Mk,σf(x) = sup x∈R R∈Rk,σ 1 |R|R |f(y)|dy. Then arguing as in [6] and using standard estimates for the kernel of Tk,ν we see that  ν |Tk,νSνf(x)|2w(x)dx  ν |Sνf(x)|2Mk,δ−1/2w(x)dx. The Lpnorm of (ν|Sνf|2)1/2is bounded by the Lpnorm of f, for p≥2 (see [6]) and therefore we can finish our proof by using duality and showing that Mk,σf2≤Cεσεf2 (4.6) uniformly in k. If we knew that for every ξthe function t→ ξ,Γk(t)changed sign at most Mtimes then it would follow from a result by C´ordoba [7] that (4.6) holds with σεreplaced by C1M[log σ]C2. This hypothesis may not be satisfied, but we can get around this point by a simple approximation. Namely, divide [−1,1] into σε/2intervals [aj,b j] of lengths σ−ε/2. Two Problems Associated with Convex Domains 173 Let Pk,j(t) be the vector valued Taylor polynomial of degree [2/ε]of ∇ξg(·,δ1/2k) expanded about aj, and let Γk,j(t)=(−Pk,j(1),1). Then |Γk(t)−Γk,j(t)|≤Cσ−1for t∈[aj,b j]. Let Rk,σ,j be the set of all cylinders whose base is a d−2-dimensional ball of radius swhose height is σs, so that the axis is parallel to Γk,j(t) for some |t|≤1. If Mk,σ,j denotes the associated maximal operator then it is immediate that Mk,σf≤jMk,σ,jfwhere the sum contains only O(σε/2) terms. C´ordoba’s result yields the L2bound Cε[log σ]C2for each Mk,σ,j. This finishes the proof of (4.6). 5. Proof of Theorem 1.1 The L1version of the theorem is well known, and therefore by an interpolation argument one has to show the boundedness on L4/3(Rd), or, equivalently, on L4(Rd). We split (1 −ρ(ξ))λ +=h0(ρ(ξ)) + h1(ρ(ξ)) where h0is supported in {t:t≤1−J0}for suitable small J0and h1is supported in {t:t>1−2J0}. Then h0(ρ(ξ)) is a Fourier multiplier in M1; the mild singularity at the origin can be handled e.g. by an averaging argument in [8, p. 248], replacing ρby ρNfor large N. Let ξ0∈Σρ. It suffices to show that there exists a neighborhood V of ξ0(in Rd) so that h1(ρ(ξ))χis a multiplier on Rdfor λ>(d−2)/4 if χ∈C∞and supported in V. The multiplier norm is invariant under rotations and we may assume that Σρcan be parametrized as a graph ξd=G( ξ),  ξ∈Rd−1near ξ0, so that ρ(ξ)<1ifξd>G( ξ). We write χ(ξ)h1(ρ(ξ)) = χ(ξ)H(ξ)(ξd−G( ξ))λ +where H(ξ)=1−ρ(ξ) ξd−G( ξ)λ . A Taylor expansion of ρabout ξd=G( ξ) shows that His smooth on supp χ; therefore by the algebra property of Mpit suffices to show that χ(ξ)(ξd−G(ξ1,ξ))λ +belongs to M4if supp χis sufficiently close to ξ0. Let a=(a1,...,a d−1) be the multitype of Σρat ξ0, in the sense of Subsection 1.2. By an affine transformation we may assume that ξ0=0, G(0) = ∇G(0) = 0, and that G=Q+Rwhere Qand Rare as in the proof of Lemma 2.2: The function Qis mixed homogeneous of degree (a1,...,a d−1), i.e. if As( ξ)=(s1 a1ξ1,...,s 1 ad−1ξd−1) then Qsatisfies Q(As( ξ)) = sQ( ξ). The remainder term Rsatisfies s−1∂|α| ∂ξαR(As ξ)≤ CM,Ns1/m for small xand sand all multiindices α=(α1,...,α d−1) with 174 A. Iosevich, E. Sawyer, A. Seeger |α|≤N. In particular |R( ξ)|≤Q( ξ)/10 if Q( ξ)≤2−r0+2 for suitably large r0. Next we set Rr( ξ)=2 rR(A2−r ξ), so that Gr=Q+Rrtends to Gin the C∞topology, as r→∞. Since the Hessian of Qhas rank 1 where 1/4<Q( ξ)≤4 (see (2.10)), the same is true for Gr=Q+Rrif ris large; we may assume that the matrix norm of (Q+Rr) is bounded below uniformly in rif r≥r0. Let φ1be supported in (1/2,2) such that k≥0φ1(2ks) = 1 for 0 < s≤1. Then we have to show a bound for the M4norm of κj(ξ)=χ(ξ)φ1(2j(ξd−G(ξ1,ξ)))(ξd−G(ξ1,ξ))λ +. Here we may assume that χ(ξ) = 0 when Q( ξ)≥2−r0. We now perform a further decomposition in terms of G( ξ). Let η∈ C∞ 0(R) so that η(s)=1if|s|≤1/2 and η(s)=0if|s|≥1; also let η0=ηand for integer r>0 let ηr(s)=η(2−rs)−η(2−r+1s). Let κj,n(ξ)=κj(ξ)ηn(2jG( ξ)) so that κj,n is supported where |ξd−G( ξ)|≈2−jand G( ξ)≈2n−jif n≥0 and G( ξ)2−jif n= 0. Using the assumption on the support of the cutoff function χwe see that κj,n = 0 for j≤n+r0. For the pieces κj,n we employ a scaling argument (for a similar argument in two dimensions see [12]). For the scaling we use the dilations ξ→ (A2n−j( ξ),2n−jξd). Define for n>0 κj,n( ξ,ξd)=φ1(2n(ξd−Gj−n(ξ1,ξ)))(ξd−Gj−n(ξ1,ξ))λ +η1(Gj−n( ξ)); for n= 0 we use the same formula but with η1replaced by η=η0. Then κj,n(A2n−j ξ,2n−jξd)=2 (n−j)λχ(A2n−j ξ,2n−jξd)κj,n( ξ,ξd) so that κj,nMp2(n−j)λκj,nMp. It is now easy to see that the C4norm of κj,0is 2−jλ and κj,0is supported in a fixed ball with diameter independent of j. 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