maximal operators associated to hypersurfaces

Sharp Lp-estimates for maximal operators associated to hypersurfaces in R3 for p>2.

Abstract.

We study the boundedness problem for maximal operators associated to smooth hypersurfaces in 3-dimensional Euclidean space. For we prove that if no affine tangent plane to passes through the origin and is analytic, then the associated maximal operator is bounded on if and only if where denotes the so-called height of the surface For non-analytic finite type we obtain the same statement with the exception of the exponent Our notion of height is closely related to A. N. Varchenko’s notion of height for functions such that can be locally represented as the graph of after a rotation of coordinates.

Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on and the -boundedness of the associated maximal operator , and a conjecture by Iosevich and Sawyer which relates the -boundedness of to an integrability condition on for the distance function to tangential hyperplanes, in dimension three.

In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnol’d in our context.

2000 Mathematical Subject Classification. 35D05 35D10 35G05
Key words and phrases. Maximal operator, Hypersurface, Oscillatory integral, Newton diagram, Oscillation index, Contact index
We acknowledge the support for this work by the Deutsche Forschungsgemeinschaft.

1. Introduction

Let be a smooth hypersurface in and let be a smooth non-negative function with compact support. Consider the associated averaging operators given by

 Atf(x):=∫Sf(x−ty)ρ(y)dσ(y),

where denotes the surface measure on The associated maximal operator is given by

 (1.1) Mf(x):=supt>0|Atf(x)|,(x∈Rn).

We remark that by testing on the characteristic function of the unit ball in it is easy to see that a necessary condition for to be bounded on is that

In 1976, E. M. Stein [27] proved that conversely, if is the Euclidean unit sphere in then the corresponding spherical maximal operator is bounded on for every The analogous result in dimension was later proved by J. Bourgain [3]. These results became the starting point for intensive studies of various classes of maximal operators associated to subvarieties. Stein’s monography [28] is an excellent reference to many of these developments. From these early works, the influence of geometric properties on the validity of -estimates of the maximal operator became evident. For instance, A. Greenleaf [9] proved that is bounded on if and provided has everywhere non-vanishing Gaussian curvature and in addition is starshaped with respect to the origin.

In contrast, the case where the Gaussian curvature vanishes at some points is still wide open, with the exception of the two-dimensional case i.e., the case of finite type curves in studied by A. Iosevich in [13]. As a partial result in higher dimensions, C. D. Sogge and E. M. Stein showed in [24] that if the Gaussian curvature of does not vanish to infinite order at any point of then is bounded on in a certain range However, the exponent given in that article is in general far from being optimal, and in dimensions sharp results are known only for particular classes of hypersurfaces.

The perhaps best understood class in higher dimensions is the class of convex hypersurfaces of finite line type (see in particular the early work in this setting by M. Cowling and G. Mauceri in [6], [5], the work by A. Nagel, A. Seeger and S. Wainger in [20], and the articles [14], [15] and [16] by A. Iosevich, E. Sawyer and A. Seeger). In [20], sharp results were for instance obtained for convex hypersurfaces which are given as the graph of a mixed homogeneous convex function Further results were based on a result due to Schulz [23](see also [31]), which states that, possibly after a rotation of coordinates, any smooth convex function of finite line type can be written in the form , where is a convex mixed homogeneous polynomial that vanishes only at the origin, and is a remainder term consisting of terms of higher homogeneous degree than the polynomial . By means of this result, Iosevich and Sawyer proved in [15] sharp -estimates for the maximal operator for For further results in the case see also [28].

As is well-known since the early work of E. M. Stein on the spherical maximal operator, the estimates of the maximal operator on Lebesgue spaces are intimately connected with the decay rate of the Fourier transform

 ˆρdσ(ξ)=∫Se−iξ⋅xρ(x)dσ(x),ξ∈Rn,

of the superficial measure i.e., to estimates of oscillatory integrals. These in return are closely related to geometric properties of the surface and have been considered by numerous authors ever since the early work by B. Riemann on this subject (see [28] for further information). Also the afore mentioned results for convex hypersurfaces of finite line type are based on such estimates. Indeed, sharp estimates for the Fourier tranform of superficial measures on have been obtained by J. Bruna , A. Nagel and S. Wainger in [4], improving on previous results by B. Randol [22] and I. Svensson [29]. They introduced a family of nonisotropic balls on , called ”caps”, by setting

 B(x,δ):={y∈S:\rm dist\,(y,x+TxS)<δ}, δ>0.

Here denotes the tangent space to at . Suppose that is normal to at the point . Then it was shown that

 |ˆρdσ(ξ)|≤C|B(x0,|ξ|−1)|,

where denotes the surface area of These estimate became fundamental also in the subsequent work on associated maximal operators.

However, such estimates fail to be true for non-convex hypersurfaces, which we shall be dealing with in this article. More precisely, we shall consider general smooth hypersurfaces in

Assume that is such a hypersurface, and let be a fixed point in We can then find a Euclidean motion of so that in the new coordinates given by this motion, we can assume that and Then, in a neighborhood of the origin, the hypersurface is given as the graph

 U∩S={(x1,x2,1+ϕ(x1,x2)):(x1,x2)∈Ω}

of a smooth function defined on an open neighborhood of and satisfying the conditions

 (1.2) ϕ(0,0)=0,∇ϕ(0,0)=0.

To we can then associate the so-called height in the sense of A. N. Varchenko [30] defined in terms of the Newton polyhedra of when represented in smooth coordinate systems near the origin (see Section 2 for details). An important property of this height is that it is invariant under local smooth changes of coordinates fixing the origin. We then define the height of at the point by This notion can easily be seen to be invariant under affine linear changes of coordinates in the ambient space (cf. Section 11) because of the invariance property of under local coordinate changes.

Now observe that unlike linear transformations, translations do not commute with dilations, which is why Euclidean motions are no admissible coordinate changes for the study of the maximal operators We shall therefore study under the following transversality assumption on

Assumption 1.1.

The affine tangent plane to through does not pass through the origin in for every Equivalently, for every so that and is transversal to for every point

Notice that this assumption allows us to find a linear change of coordinates in so that in the new coordinates can locally be represented as the graph of a function as before, and that the norm of when acting on is invariant under such a linear change of coordinates.

If is flat, i.e., if all derivatives of vanish at the origin, and if then it is well-known and easy to see that the maximal operator is -bounded if and only if so that this case is of no interest. Let us therefore always assume in the sequel that is non-flat, i.e., of finite type. Correspondingly, we shall always assume without further mentioning that the hypersurface is of finite type in the sense that every tangent plane has finite order of contact.

Theorem 1.2.

Assume that is a smooth hypersurface in satisfying Assumption 1.1, and let be a fixed point. Then there exists a neighborhood of the point such that for any the associated maximal operator is bounded on whenever

Notice that even in the case where is convex this result is stronger than the known results, which always assumed that is of finite line type.

The following Theorem shows the sharpness of this theorem.

Theorem 1.3.

Assume that the maximal operator is bounded on for some where satisfies Assumption 1.1. Then, for any point with we have Moreover, if is analytic at such a point then

As an immediate consequence of these two results, we obtain

Corollary 1.4.

Suppose is a smooth hypersurface in satisfying Assumption 1.1, and let be a fixed point. Then there exists a neighborhood of this point such that for every

This shows in particular that if is a smooth deformation by linear terms of a smooth, finite type function defined near the origin in and satisfying (1.2), then the height of at any critical point of the function is bounded by the height at for sufficiently small perturbation parameters and This proves a conjecture by V.I. Arnol’d [2] in the smooth setting at least for linear perturbations. For analytic functions of two variables, such a result has been proved for arbitrary analytic deformations by V. N. Karpushkin [17].

From these results, global results can be deduced easily. For instance, if is a compact hypersurface, then we define the height of by Corollary 1.4 shows that in fact

 h(S):=maxx∈Sh(x,S)<∞,

and from Theorems 1.2, 1.3 we obtain

Corollary 1.5.

Assume that is a smooth, compact hypersurface in satisfying Assumption 1.1, that on and that

If is analytic, then the associated maximal operator is bounded on if and only if If is only assumed to be smooth, then for we still have that the maximal operator is bounded on if and only if

Let be an affine hyperplane in Following A. Iosevich and E. Sawyer [14], we consider the distance from to In particular, if then will denote the distance from to the affine tangent plane to at the point The following result has been proved in [14] in arbitrary dimensions and without requiring Assumption 1.1.

Theorem 1.6 (Iosevich-Sawyer).

If the maximal operator is bounded on where then

 (1.3) ∫SdH(x)−1/pρ(x)dσ(x)<∞

for every affine hyperplane in which does not pass through the origin.

Moreover, they conjectured that for the condition (1.3) is indeed necessary and sufficient for the boundedness of the maximal operator on at least if for instance is compact and

Remark 1.7.

Notice that condition (1.3) is easily seen to be true for every affine hyperplane which is nowhere tangential to so that it is in fact a condition on affine tangent hyperplanes to only. Moreover, if Assumption 1.1 is satisfied, then there are no affine tangent hyperplanes which pass through the origin, so that in this case it is a condition on all affine tangent hyperplanes.

In Section 11, we shall prove

Proposition 1.8.

Suppose is a smooth hypersurface in and let be a fixed point. Then, for every we have

 (1.4) ∫S∩UdT,x0(x)−1/pdσ(x)=∞

for every neighborhood of Moreover, if is analytic near then (1.4) holds true also for

Notice that this result does not require Assumption 1.1.

As an immediate consequence of Theorem 1.2, Theorem 1.6 and Proposition 1.8 we obtain

Corollary 1.9.

Assume that satisfies Assumption 1.1, and let be a fixed point. Moreover, let

Then, if is analytic near there exists a neighborhood of the point such that for any with the associated maximal operator is bounded on if and only if condition (1.3) holds for every affine hyperplane in which does not pass through the origin.

If is only assumed to be smooth near then the same conclusion holds true, with the possible exception of the exponent

This confirms the conjecture by Iosevich and Sawyer in our setting for analytic , and for smooth with the possible exception of the exponent For the critical exponent if is not analytic near examples show that unlike in the analytic case it may happen that is bounded on (see, e.g., [15]), and the conjecture remains open for this value of For further details, we refer to Section 11.

As mentioned before, the estimates of the maximal operator on Lebesgue spaces are intimately connected with the decay rate of the Fourier transform

 ˆρdσ(ξ)=∫Se−iξ⋅xρ(x)dσ(x),ξ∈Rn,

of the superficial measure Estimates of such oscillatory integrals will naturally play a central role also in our proof Theorem 1.2. Indeed our proof of Theorem 1.2 will provide enough information that it will also be easy to derive from it the following uniform estimate for the Fourier transform of surface carried measures on

Theorem 1.10.

Let be a smooth hypersurface of finite type in and let be a fixed point in Then there exists a neighborhood of the point such that for every the following estimate holds true:

 (1.5) |ˆρdσ(ξ)|≤C||ρ||C3(S)log(2+|ξ|)(1+|ξ|)−1/h(x0,S)  for every ξ∈R3.

This estimate generalizes Karpushkin’s estimates in [17] from the analytic to the finite type setting, at least for linear perturbations.

The next result establishes a direct link between the decay rate of and Iosevich-Sawyer’s condition (1.3). In combination with Proposition 1.8 it shows in particular that the exponent in estimate (1.5) is sharp (for the case of analytic hypersurfaces, the latter follows also from Varchenko’s asymptotic expansions of oscillatory integrals in [30]).

Theorem 1.11.

Let be a smooth hypersurface in and let be a smooth cut-off function and assume that

 (1.6) |ˆρdσ(ξ)|≤Cβ(1+|ξ|)−β for % every ξ∈Rn,

for some Then for every such that

 (1.7) ∫SdH(x)−1/pρ(x)dσ(x)<∞,

for every affine hyperplane in

In combination with Proposition 1.8 this result easily implies (see Section 11)

Corollary 1.12.

Suppose is a smooth hypersurface in let be a fixed point and assume that the estimate (1.6) holds true for some If and if is supported in a sufficiently small neighborhood of then necessarily

Indeed, more is true. Let us introduce the following quantities. In analogy with V. I. Arnol’d’s notion of the ”singularity index” [2], we define the uniform oscillation index of the hypersurface at the point as follows:

Let denote the set of all for which there exists an open neighborhood of in such that estimate (1.6) holds true for every function Then

 βu(x0,S):=sup{β:β∈Bu(x0,S)}.

If we restrict our attention to the normal direction to at only, then we can define analogously the notion of oscillation index of the hypersurface at the point More precisely, if is a unit normal to at then we let denote the set of all for which there exists an open neighborhood of in such that estimate (1.6) holds true along the line for every function i.e.,

 (1.8) |ˆρdσ(λn(x0))|≤Cβ(1+|λ|)−γ for every λ∈R.

Then

 β(x0,S):=sup{β:β∈B(x0,S)}.

If we regard locally as the graph of a function then we can introduce related notions and for regarded as the phase function of an oscillatory integral (cf. [11], and also Section 11).

We also define the uniform contact index of the hypersurface at the point as follows:

Let denote the set of all for which there exists an open neighborhood of in such that the estimate

 (1.9) ∫UγdH(x)−γdσ(x)<∞

holds true for every affine hyperplane in Then we put

 γu(x0,S):=sup{γ:γ∈Cu(x0,S)}.

Similarly, we let denote the set of all for which there exists an open neighborhood of in such

 (1.10) ∫UγdT,x0(x)−γdσ(x)<∞,

and call

 γ(x0,S):=sup{γ:γ∈C(x0,S)}

the contact index of the hypersurface at the point Then clearly

 (1.11) βu(x0,S)≤β(x0,S),γu(x0,S)≤γ(x0,S).

At least for hypersurfaces in a lot more is true.

Theorem 1.13.

Let by a smooth, finite type hypersurface in and let be a fixed point. Then

 βu(x0,S)=β(x0,S)=γu(x0,S)=γ(x0,S)=1/h(x0,S).

Let us recall at this point a result by A. Greenleaf. In [9] he proved that if and if then the maximal operator is bounded on whenever The case remained open.

For E. M. Stein and later for the full range A. Iosevich and E. Sawyer [15] conjectured that if is a smooth, compact hypersurface in such that

 |ˆρdσ(ξ)|=O(|ξ|−β)for some 0<β≤1/2,

then the maximal operator is bounded on for every at least if we assume

A partial confirmation of Stein’s conjecture has been given by C. D. Sogge [26] who proved that if the surface has at least one non-vanishing principal curvature everywhere, then the maximal operator is -bounded for every Certainly, if the surface has at least one non-vanishing principal curvature then the estimate above holds for .

Now, if and if then for every point so that our Theorem 1.13 implies that Then, if we have Therefore, by means of a partition of unity argument, we obtain from Theorem 1.2 the following confirmation of the Stein-Iosevich-Sawyer conjecture in this case.

Corollary 1.14.

Let be a smooth compact hypersurface in satisfying Assumption 1.1, and let be a smooth density on We assume that there is some such that

 |ˆρdσ(ξ)|=O(|ξ|−β).

Then the associated maximal operator is bounded on for every

We finally remark that the case behaves quite differently, and examples show that neither condition (1.3) nor the notation of height will be suitable to determine the range of exponents for which the maximal operator is -bounded (see, e.g., [16]). The study of this range for is work in progress.

1.1. Outline of the proof of Theorem 1.2 and organization of the article

The proof of our main result, Theorem 1.2, will strongly make use of the results in [11] on the existence of a so-called ”adapted” coordinate system for a smooth, finite type function defined near the origin in (see Section 2 for some basic notation). These results generalize the corresponding results for analytic by A. N. Varchenko [30], by means of a simplified approach inspired by the work of Phong and Stein [21]. According to these results, one can always find a change of coordinates of the form

 y1:=x1, y2:=x2−ψ(x1)

which leads to adapted coordinates The function can be constructed from the Pusieux series expansion of roots of (at least if is analytic) as the so-called principal root jet (cf.[11]). Somewhat simplifying, it agrees with a real-valued leading part of the (complex) root of near which is ”small of highest order” in an averaged sense. One would preferably like to work in these adapted coordinates since the height of when expressed in these adapted coordinates, can be read off directly from the Newton polyhedron of as the so-called ”distance.” However, this change of coordinates leads to substantial problems, since it is in general non-linear.

Now, away from the curve it turns out that one can find some with such that This suggests that one may apply the results on maximal functions on curves in [13]. Indeed this is possible, but we need estimates for such maximal operators along curves which are stable under small perturbations of the given curve. Such results, which will be based on the local smoothing estimates by G. Mockenhaupt, A. Seeger and C. Sogge in [18], and related estimates for maximal operators along surfaces, are derived in Section 3. The necessary control on partial derivatives will be obtained from the study of mixed homogeneous polynomials in Section 4. Indeed, in a similar way as the Schulz polynomial is used in the convex case to approximate the given function we shall approximate the function in domains close to a given root of by a suitable mixed homogeneous polynomial, following here some ideas in [21].

The case where our original coordinates are adapted or where the height is strictly less than is the simplest one, since we can here avoid non-linear changes of coordinates. This case is dealt with in Section 5.

We then concentrate on the situation where and where the coordinates are not adapted. The contributions to the maximal operator by a suitable homogeneous domain away from the curve require a lot more effort and are estimated in Section 6 by means of the results in Sections 3 and 4.

There remains the domain near the curve For this domain, it is in general no longer possible to reduce its contribution to the maximal operator to maximal operators along curves, and we have to apply two-dimensional oscillatory integral technics. Indeed, we shall need estimates for certain classes of oscillatory integrals with small parameters, which will be given in Section 9. These results will be applied in Sections 7 and 8 in order to complete the proof of Theorem 1.2.

We remark that our proof does not make use of any damping technics, which had been crucial to many other approaches.

The proof of Theorem 1.10, which will be given in Section 10, can easily be obtained from the results established in the course of the proof of Theorem 1.2, except for the case which, however, has been studied in a complete way by Duistermaat [8]. The main difference is that we have to replace the estimates for maximal operators in Section 3 by van der Corput type estimates due to J. E. Björk and G. I. Arhipov.

In the last Section 11, we shall give proofs of all the other results stated above.

2. Newton diagrams and adapted coordinates

We recall here some basic notation (compare, e.g., [11] for further information). Let be a smooth real-valued function defined on a neighborhood of the origin in with and consider the associated Taylor series

 ϕ(x1,x2)∼∞∑j,k=0cjkxj1xk2

of centered at the origin. The set

 T(ϕ):={(j,k)∈N2:cjk=1j!k!∂jx1∂kx2ϕ(0,0)≠0}

will be called the Taylor support of at We shall always assume that

 T(ϕ)≠∅,

i.e., that the function is of finite type at the origin. If is real analytic, so that the Taylor series converges to near the origin, this just means that The Newton polyhedron of at the origin is defined to be the convex hull of the union of all the quadrants in with The associated Newton diagram in the sense of Varchenko [30] 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 for points in the plane containing the Newton polyhedron, in order to distinguish this plane from the - plane.

The distance between the Newton polyhedron and the origin in the sense of Varchenko is given by the coordinate of the point at which the bisectrix intersects the boundary of the Newton polyhedron.

The principal face of the Newton polyhedron of is the face of minimal dimension containing the point . Deviating from the notation in [30], we shall call the series

 ϕp(x1,x2):=∑(j,k)∈π(ϕ)cjkxj1xk2

the principal part of In case that is compact, is a mixed homogeneous polynomial; otherwise, we shall consider as a formal power series.

Note that the distance between the Newton polyhedron and the origin depends on the chosen local coordinate system in which is expressed. By a local analytic (respectively smooth) coordinate system at the origin we shall mean an analytic (respectively smooth) coordinate system defined near the origin which preserves If we work in the category of smooth functions we shall always consider smooth coordinate systems, and if is analytic, then one usually restricts oneself to analytic coordinate systems (even though this will not really be necessary for the questions we are going to study, as we will see). The height of the analytic (respectively smooth) function is defined by

 h(ϕ):=sup{dx},

where the supremum is taken over all local analytic (respectively smooth) coordinate systems at the origin, and where is the distance between the Newton polyhedron and the origin in the coordinates .

A given coordinate system is said to be adapted to if

2.1. The principal part of ϕ associated to a supporting line of the Newton polyhedron as a mixed homogeneous polynomial

Let with be a given weight, with associated one-parameter family of dilations A function on is said to be -homogeneous of degree if for every Such functions will also be called mixed homogeneous. The exponent will be denoted as the -degree of For instance, the monomial has -degree

If is an arbitrary smooth function near the origin, consider its Taylor series around the origin. We choose so that the line is the supporting line to the Newton polyhedron of Then the non-trivial polynomial

 ϕκ(x1,x2):=∑(j,k)∈Lκcjkxj1xk2

is -homogeneous of degree it will be called the -principal part of By definition, we then have

 (2.1) ϕ(x1,x2)=ϕκ(x1,x2)+ terms of higher κ-degree.

More precisely, we mean by this that every point in the Taylor support of the remainder term lies on a line with parallel to, but above the line i.e., we have Moreover, clearly

 Nd(ϕκ)⊂Nd(ϕ).

3. Uniform estimates for maximal operators associated to families of finite type curves and related surfaces

3.1. Finite type curves

In this subsection, we shall prove an extension of some results by Iosevich [Ios], which allows for uniforms estimates for maximal operators associated to families of curves which arise as small perturbations of a given curve.

We begin with a result whose proof is based on Iosevich’s approach in [Ios].

Proposition 3.1.

Consider averaging operators along curves in the plane of the form

 Atf(x)=A(ρ,η,τ)tf(x):=∫Rf(x1−t(ρ1s+η1),x2−t(η2+τs+ρ2g(s)))ψ(s)ds,

where , is supported in a bounded interval I containing the origin, and where

 (3.1) g(s)=sm(b(s)+R(s)),s∈I,m∈N,m≥2,

with satisfying . Moreover, is a smooth perturbation term.

By we denote the associated maximal operator

 M(ρ,η,τ)f(x):=supt>0|A(ρ,η,τ)tf(x)|.

Then there exist a neighborhood of the origin in and , , such that for ,

 (3.2)

for every supported in and every with , with a constant depending only on and the -norm of (such constants will be called “admissible”).

Proof. Consider the linear operator

 Tf(x1,x2)=(ρ1ρ2)−1/pf(ρ−11x1,ρ−12(x2−τρ1x1)).

Then is isometric on , and one computes that is given by

 ~Atf(x)=~Aσtf(x)=∫f(x1−t(s+σ1),x2−t(σ2+g(s)))ψ(s)ds,

where is given by

 σ1=η1ρ1,σ2=η2ρ2−τρ2η1ρ1.

Put

 ~Mf(x)=supt>0|~Atf(x)|.

Then (3.2) is equivalent to the following estimate for :

 (3.3) ||~Mf||p≤Cp(|σ|+1)1/p||f||p,f∈S(R2),

for every , where is an admissible constant.

a) We first consider the case .

By means of the Fourier inversion formula, we can write

 ~Atf(x)=1(2π)2∫R2ei(x−tσ)⋅ξH(tξ)^f(ξ)dξ,

where

 H(ξ1,ξ2):=∫Re−i(ξ1s+ξ2g(s))ψ(s)ds.

If is a critical point of the phase of , then , where by our assumptions on we have .

This shows that we can choose a neighborhood of in and some such that for any with the phase function has a unique non-degenerate critical point

 s0(ξ1/ξ2)=−ξ1ξ2ω(ξ1ξ2,R)∈U,

where depends smoothly on and the error term , and . Moreover, if , we may assume that no critical point belongs to . In the last case, we may integrate by parts to see that

 |DαξH(ξ)|≤Cα,N(1+|ξ|)−N,|ξ1/ξ2|≥ε1,

for every with and , where the constants are admissible.

A similar estimate holds obviously true for . Applying then the stationary phase method to the remaining frequency region, and combining these estimates, we get:

 H(ξ)=eiq(ξ)χ(ξ1ξ2)A(ξ)(1+|ξ|)1/2+B(ξ),

where is a smooth function supported on a small neighborhood of the origin,

 q(ξ)=q(ξ,R)

is a smooth function of and which is homogenous of degree 1, and which can be considered as a small perturbation of , if is contained in a sufficiently small neighborhood of 0 in . It is also important to notice that the Hessian has rank 1, so that the same applies to for small perturbations . Moreover, is a symbol of order zero such that

 A(ξ)=0,if|ξ|≤C,

and

 (3.4) |ξαDαξA(ξ)|≤Cα,α∈N2,|α|≤3,

where the are admissible constants. Finally, is a remainder term satisfying

 (3.5) |DαξB(ξ)|≤Cα,N(1+|ξ|)−N,|α|≤3,0≤N≤3,

If we put

 ~A0tf(x):=1(2π)2∫R2ei(x−tσ)⋅ξB(tξ)^f(ξ)dξ,

then by (3.5),

 ~A0tf(x)=f∗kσt(x),

where

 kσt(x)=t−2k(xt)

and where is the translate

 (3.6) kσ(x):=k(x−σ)

of by the vector of a fixed function satisfying an estimate of the form

 (3.7) |k(x)|≤C(1+|x|)−3.

Let denote the corresponding maximal operator. (3.6) and (3.7) show that

 ||~M0||L∞→L∞≤C,

with a constant which does not depend on .

Moreover, scaling by the factor in direction of the vector we see that

 ||~M0||L1→L1,∞≤C(|σ|+1),

since we then can compare with , where is the Hardy-Littlewood maximal operator. By interpolation, these estimates imply that

 ||~M0||Lp→Lp≤Cp(|σ|+1)1/p,

if .

There remains the maximal operator corresponding to the family of averaging operators

 ~A1t(x):=1(2π)2∫R2ei[ξ⋅x−t(σ⋅ξ+q(ξ))]χ(ξ1/ξ2)A(tξ)(1+|tξ|)1/2^f(ξ)dξ

(notice that ).

As usually we choose a non-negative function such that

 suppβ⊂[1/2,2],∞∑j=−∞β(2−jr)=1for r>0,

and put

 Aj,tf(x):=∫R2ei[ξ⋅x−t(σ⋅ξ+q(ξ)