We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classical spaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establish sharp conditions for generalized localization in the class of finitely supported distributions.

1. Introduction

In this paper, we study the behavior of spherical Fourier integrals and pointwise convergence and summability of Fourier inversion.

Let be a homogeneous elliptic differential operator of order . Let us consider its symbol defined as polynomial: and assume that the Gaussian curvature of surface is always strictly positive.

We recall that for its Fourier transform is defined as and partial Fourier integral associated with elliptic operator (1.1): (note that throughout the paper we consider only Lebesque measure on and ). For some functions, Fourier integrals do not converge pointwisely and various summation techniques are applied to recover convergence property. In this paper, we consider the method of the Riesz means. The Riesz means of order are defined as

As an example, one can consider Laplacian , and note that the level surfaces of its symbol are Euclidean spheres. Thus, Fourier inversion associated with Laplace operator has the form: and known as spherical partial Fourier integrals. The question of convergence to almost everywhere is not solved in even for classical functions and presents one of the most challenging open problems of classical harmonic analysis, and even special cases of this problem are of particular interest. One of such special cases is the problem of generalized localization, which for the first time was formulated by V. Ii'in in [1]. For convenience, we give its definition for the Riesz means .

Definition 1.1. We say that, for the Riesz means of order , the generalized localization principle in function class is satisfied, if for any function , the equality is true for a.e. .

This localization principle generalizes the classical Riemann localization principle and for functions was intensively investigated by Sjölin [2], Carbery and Soria [3, 4], Bastis [57], and Ashurov et al. [8]. It was established that localization holds true in , where and fails otherwise.

Over the last several years, a number of Fourier inversion studies considered distributions and investigated the behavior of their Fourier integrals (see, e.g., [912]). In particular, Alimov in [13] considered the classical Riemann localization principle for compactly supported distributions and established criteria for its validity (see also [14, 15]).

In this paper, we study generalized localization principle for compactly supported distributions and present conditions for its fulfillment.

2. Notation and Definitions

We define Schwartz space as the function class of all infinitely differentiable functions that are rapidly decreasing at infinity along with all partial derivatives. It is well known that , being equipped with a family of seminorms is a Frechet space (here are multi-indices and is a partial derivative). As usual, we also consider class of tempered distributions defined as dual to .

Let be the space of infinitely differentiable functions with topology such that in if and only if for each multiindex and compact As usual we denote its conjugate space by .

It is known (see, e.g., [16]) that each has finite support and equivalent to the class of finitely supported tempered distributions. Thus, it follows from the Paley-Wienner theorem that, for each , its Fourier transform . Since is locally integrable, it is natural to define Fourier integral of and its Riesz means by (1.4) and (1.5), respectively.

We also note that for the Riesz mean can be considered as an integral operator: with kernel where where .

Representation (2.3) has its natural analogue for . Let be a sequence of Schwartz functions such that as and in norm. Then: Note that inequality implies that in and since is continuous on

We will need Sobolev's classes which can be defined for in the following way.

Definition 2.1. We say that tempered distribution belongs to Sobolev class if is a regular distribution such that
One can see that, in particular, . We also remark that for every there is such that (for proof see, e.g., [16]).

In other respects, we make the following conventions:(i)symbol is used to denote Bessel function of the first kind and order , (ii) is preserved for an indicator function of , (iii)unless otherwise indicated, all functions are assumed to be defined on and by definition .

3. Main Result

As has been mentioned above, every belongs to some Sobolev classes , in this paper, we use this fact to establish criterion of generalized localization for finitely supported distributions. The following theorems present major results of current study.

Theorem 3.1. Let , . Then, for integer , equality holds true a.e. on .

Our approach is based on the methods by Carbery and Soria [3] and in order to prove Theorem 3.1, we will follow his idea first proving some auxiliary facts in the following section.

4. Dual Sets

Let and . Then, is a symmetric body that is convex compact symmetric set. We recall that set is called polar set with respect to .

As it is done in [17], we will introduce the norm generated by as and dual norm as Next, let and be the boundaries of and , respectively.

It is not difficult to show that . Indeed on the one hand and, therefore, for which means that . On the other hand, for any , one can consider and examine its local extremums on the surface . Since is compact, reaches its extremum values and it is known that, at extremum points, must be parallel to the normal to at point , which is parallel to . Since , we can conclude that at the extremum points. Since is strictly convex, it is possible only for , that implies .

It is convenient for given to use the notation to denote the point on such that the outer normal to at is parallel to . Similarly, we denote the point on such that the outer normal to at is parallel to . One can remark that we have just seen that for

5. Technical Lemmas for Theorem 3.1

We will need the asymptotic representation of , which can be derived by stationary-phase method (see, e.g., [18]): where functions and uniformly on and .

Now, let us consider positive numbers and , and function vanishing on . Then, for , we set by definition where as in (2.3).

We will need some estimates for the Fourier transform of . With this aim, we will need the following lemmas.

Lemma 5.1. Let and . Then, for any

Proof. This estimate easily follows from the definition of . Indeed, where and is formally conjugate to operator . Since , Further, we notice that since then for any there is such that functions , uniformly for and . For the same reason, for any , one has . Now substituting these estimates into (5.6), we complete the proof.

Lemma 5.2. Let and . Then, for any ,

Proof. By definition, Let us pass to a new coordinate system . Then, where is a Lebesgue surface measure of .
Using (5.1), we have We will focus on the first term since the second one can be handled alike and note that due to (4.4) , and thus One can use the expression and employ stationary phase method to obtain where are smooth functions such that . Using this expression, we have
Further integrating by parts the integrals, one can see that for any both integrals are controlled by . As a result, we have uniformly for and . Finally, substituting into (5.10), we obtain (5.7).

Now, combining Lemmas 5.1 and 5.2, we can claim that, in fact, for and any ,

Lemma 5.3. Let be defined by (5.3). Then, for any there is such that

Proof. As it follows from (5.16), where can be chosen arbitrary large. Changing the variables , one has It is not difficult to see that for the values in the first integral , and thus choosing Moreover, it is clear that for such
Since all norms in are equivalent, the lemma is proved.

Lemma 5.4. Let be defined by (5.3). Then, for any , there is such that

Proof. For any , using the Fubini theorem, one has which implies .
If Thus, using inequality , one has Now, one can use estimate (5.16) to each integral on the right side and complete the proof.
If , then for any , Using this estimate and the reasoning presented in the previous lemma, we obtain the required estimate.

6. Proof of Theorem 3.1

Let be such that . For , we set and consider an arbitrary radial function such that

It is clear that to prove the theorem it is sufficient to show that for any , , a.e.

In this case, as due to (2.6) or using notation (5.3)

Further, we consider maximal operator: We recall that to prove a.e. convergence on one can use the standard technique of Banach principle (see, e.g., [19]) according to which it is sufficient to estimate maximal operator on as

Let be a function such that If we set , then by (6.4), According to Sobolev's embedding theorem (see, e.g., [20]) for any , Using this fact, we have And, therefore, in order to obtain (6.6), it is sufficient to show that there are constants such that the following estimates are true:

First, we note that estimate (5.16) and imply that which in turn with (6.8) implies the fact .

Further, using the Plancherel theorem, we have (the last inequality follows from Lemma 5.3).

For the same reason, (6.11) can be proved using Lemmas 5.3 and 5.4:


The authors are thankful to the University Putra Malaysia for the support under RUGS (Grant no. 05-03-11-1450RU). A. Butaev would also like to expresses his gratitude to for the support under IGRF scheme.