On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions

Abstract

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 𝐿_𝑝 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 𝐴(𝐷)=_|𝛼|=𝑚𝑐_𝛼𝐷^𝛼 (1.1) be a homogeneous elliptic differential operator of order 𝑚. Let us consider its symbol defined as polynomial: 𝐴(𝑥)=_|𝛼|=𝑚𝑐_𝛼𝑥^𝛼, (1.2) and assume that the Gaussian curvature of surface 𝑆={𝑥∈𝑅^𝑛∶𝐴(𝑥)=1} is always strictly positive.

We recall that for 𝑓∈𝐿₂(𝑅^𝑛) its Fourier transform is defined as 𝑓(𝜉)=𝑓𝑦𝑒^{−𝑖𝑦𝜉}𝑑𝑦 (1.3) and partial Fourier integral associated with elliptic operator (1.1): 𝐸_𝜆𝑓(𝑥)=(2𝜋)^−𝑛_{𝐴(𝜉)<𝜆}𝑓(𝜉)𝑑𝜉 (1.4) (note that throughout the paper we consider only Lebesque measure on 𝑅^𝑛 and ∫=∫_𝑅𝑛). 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 𝑠 are defined as 𝐸^𝑠_𝜆𝑓(𝑥)=(2𝜋)^−𝑛_{𝐴(𝜉)≤𝜆}𝜉1−²𝜆^2𝑠𝑓(𝜉)𝑒^𝑖𝜉𝑥𝑑𝜉. (1.5)

As an example, one can consider Laplacian 𝐴(𝐷)=∑^𝑛_𝑖=1(𝜕²/𝜕𝑥²_𝑖), and note that the level surfaces of its symbol are Euclidean spheres. Thus, Fourier inversion associated with Laplace operator has the form: 𝐸_𝜆𝑓(𝑥)=(2𝜋)^−𝑛_|𝜉|2<𝜆𝑓(𝜉)𝑒^𝑖𝜉𝑥𝑑𝜉 (1.6) and known as spherical partial Fourier integrals. The question of 𝐸_𝜆𝑓(𝑥) convergence to 𝑓(𝑥) almost everywhere is not solved in 𝑅^𝑛,𝑛≥2 even for classical 𝐿₂ 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 𝐸^𝑠_𝜆.

Definition 1.1. We say that, for the Riesz means of order 𝑠, the generalized localization principle in function class 𝔉 is satisfied, if for any function 𝑓∈𝔉, the equality lim_𝜆→∞𝐸^𝑠_𝜆𝑓(𝑥)=0 (1.7) is true for a.e. 𝑥∈𝑅^𝑛⧵supp𝑓.

This localization principle generalizes the classical Riemann localization principle and for 𝐿_𝑝 functions was intensively investigated by Sjölin [2], Carbery and Soria [3, 4], Bastis [5–7], and Ashurov et al. [8]. It was established that 𝑅^𝑛 localization holds true in 𝐿_𝑝, where 𝑝∈[2,2𝑛/(𝑛−1)] 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., [9–12]). 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 𝑆(𝑅^𝑛) 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 𝑆(𝑅^𝑛), being equipped with a family of seminorms 𝑑_𝛼,𝛽𝜙=sup_{𝑥∈𝑅𝑛}𝑥^𝛼𝐷^𝛽𝜙(𝑥), (2.1) is a Frechet space (here 𝛼,𝛽 are multi-indices and 𝐷 is a partial derivative). As usual, we also consider class of tempered distributions 𝑆^′ defined as dual to 𝑆.

Let ℰ be the space of infinitely differentiable functions with topology 𝜏_𝐸 such that 𝜙_𝑛→0 in 𝜏_𝐸 if and only if for each multiindex 𝛼 and compact 𝐾sup_𝑥∈𝐾𝐷^𝛼𝜙_𝑛(𝑥)⟶0. (2.2) As usual we denote its conjugate space by ℰ^′.

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

We also note that for 𝑓∈𝐿₂ the Riesz mean 𝐸^𝑠_𝜆𝑓 can be considered as an integral operator: 𝐸^𝑠_𝜆𝑓(𝑥)=(2𝜋)^−𝑛𝑓𝑦𝜃^𝑠_𝜆𝑥−𝑦𝑑𝑦, (2.3) with kernel 𝜃^𝑠_𝜆(𝑦)=𝑚^𝑠_𝜆(𝑦) where 𝑚^𝑠_𝜆𝑦=𝐴𝑦1−𝜆^𝑠₊, (2.4) where (1−𝐴(𝑦)/𝜆)^𝑠₊=(1−𝐴(𝑦)/𝜆)^𝑠⋅𝜒_{𝐴(𝑦)<𝜆}(𝑦).

Representation (2.3) has its natural analogue for 𝑓∈ℰ^′. Let 𝜓_𝑛 be a sequence of Schwartz functions such that 𝜓_𝑛(𝑦)=0 as |𝑦|>𝜆 and 𝜓_𝑛(𝑦)→𝑚^𝑠_𝜆(𝑦) in 𝐿₁ norm. Then: 𝐸^𝑠_𝜆𝑓(𝑥)=lim_𝑛→∞(2𝜋)^−𝑛𝑓(𝜉)𝜓_𝑛(𝜉)𝑒^𝑖𝑥𝜉=()𝑑𝜉2𝜋^−𝑛lim_𝑛→∞𝑓(𝜉),𝜓_𝑛(𝜉)𝑒^𝑖𝑥𝜉=(2𝜋)^−𝑛lim_𝑛→∞𝑓𝑦,𝜓_𝑛𝑥−𝑦. (2.5) Note that inequality ‖𝑔‖_∞≤‖𝑔‖₁ implies that 𝜓_𝑛→𝑚^𝑠_𝜆 in ℰ and since 𝑓 is continuous on ℰ𝐸^𝑠_𝜆𝑓(𝑥)=(2𝜋)^−𝑛𝑓(⋅),𝜃^𝑠_𝜆(𝑥−⋅). (2.6)

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

Definition 2.1. We say that tempered distribution 𝑓 belongs to Sobolev class 𝐻^𝑙 if 𝑓 is a regular distribution such that 𝑓²_𝐻𝑙=𝑓(𝜉)²1+𝜉^2𝑙𝑑𝜉<∞. (2.7)
One can see that, in particular, 𝐻⁰=𝐿₂. We also remark that for every 𝑓∈ℰ′ there is 𝑙∈ℝ such that 𝑓∈𝐻^𝑙 (for proof see, e.g., [16]).

In other respects, we make the following conventions:(i)symbol 𝐽_𝜈 is used to denote Bessel function of the first kind and order 𝜈≥0, (ii)𝜒_𝐸 is preserved for an indicator function of 𝐸⊂𝑅^𝑛, (iii)unless otherwise indicated, all functions are assumed to be defined on 𝑅^𝑛 and by definition 𝐿_𝑝(Ω)≡{𝑓∈𝐿_𝑝(𝑅^𝑛)∶supp𝑓⊂Ω⊂𝑅^𝑛}.

3. Main Result

As has been mentioned above, every 𝑓∈ℰ^′ belongs to some Sobolev classes 𝐻^𝑙, 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 𝑓∈ℰ^′∩𝐻^−𝑙, 𝑙≥0. Then, for integer 𝑠≥𝑙, equality lim_𝜆→∞𝐸^𝑠_𝜆𝑓(𝑥)=0 (3.1) holds true a.e. on 𝑅^𝑛⧵supp𝑓.

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 𝑎(𝑥)=[𝐴(𝑥)]^1/𝑚 and 𝐾={𝑥∈𝑅^𝑛∶𝑎(𝑥)≤1}. Then, 𝐾 is a symmetric body that is convex compact symmetric set. We recall that set 𝐾^∗={𝑦∶|𝑥⋅𝑦|≤1,∀𝑥∈𝐾} is called polar set with respect to 𝐾.

As it is done in [17], we will introduce the norm ‖⋅‖_𝑎 generated by 𝑎(𝑥) as ‖𝑥‖_𝑎=𝑎(𝑥) (4.1) and dual norm ‖⋅‖^∗_𝑎 as 𝑦^∗_𝑎=sup_{‖𝑥‖𝑎≤1}𝑥⋅𝑦=sup_{‖𝑥‖𝑎=1}𝑥⋅𝑦. (4.2) Next, let 𝑆 and 𝑆^∗ be the boundaries of 𝐾 and 𝐾^∗, respectively.

It is not difficult to show that 𝑆^∗={∇𝑎(𝑥),𝑥∈𝑆}. Indeed on the one hand 𝑎(𝜆𝑥)=𝜆𝑎(𝑥) and, therefore, for 𝑥∈𝑆𝑥⋅∇𝑎(𝑥)=𝑑𝑎(𝑥)(𝑥)()(𝑥)𝑑𝑟=𝑎=1,−𝑥⋅∇𝑎=−𝑑𝑎(𝑥)(𝑥)𝑑𝑟=𝑎=−1, (4.3) which means that ‖∇𝑎(𝑥)‖^∗_𝑎≥1. On the other hand, for any 𝑦∈𝑆, one can consider 𝐹(𝑦)=𝑦⋅∇𝑎(𝑥) and examine its local extremums on the surface 𝑆. Since 𝑆 is compact, 𝐹(𝑦) reaches its extremum values and it is known that, at extremum points, ∇𝐹(𝑦) must be parallel to the normal to 𝑆 at point 𝑦, which is parallel to ∇𝑎(𝑦). Since ∇𝐹(𝑦)=∇𝑎(𝑥), we can conclude that ∇𝑎(𝑥)‖∇𝑎(𝑦) at the extremum points. Since 𝑆 is strictly convex, it is possible only for 𝑦=±𝑥, that implies ‖∇𝑎(𝑥)‖^∗_𝑎≤1.

It is convenient for given 𝑥∈𝑅^𝑛 to use the notation 𝜃(𝑥) to denote the point on 𝑆 such that the outer normal to 𝑆 at 𝜃(𝑥) is parallel to 𝑥. Similarly, we denote 𝜂(𝑥) the point on 𝑆^∗ such that the outer normal to 𝑆^∗ at 𝜂(𝑥) is parallel to 𝑥. One can remark that we have just seen that for 𝑦∈𝑆^∗𝑦⋅𝜃𝑦=1. (4.4)

5. Technical Lemmas for Theorem 3.1

We will need the asymptotic representation of 𝜃^𝑠_𝜆(𝑦), which can be derived by stationary-phase method (see, e.g., [18]): 𝜃^𝑠_𝜆𝑦=𝜆^(𝑛−1)/2𝑦^{−(𝑛+1)/2}⋅𝑅^𝑠₊𝑦,𝜆𝑒^{𝑖𝜆𝑦⋅𝜃(𝑦)}+𝑅^𝑠₋𝑦,𝜆𝑒^{−𝑖𝜆𝑦⋅𝜃(𝑦)}, (5.1) where functions 𝑅^𝑠_±(𝑦,𝜆)∈𝐶^∞({𝑦∶|𝑦|>𝜖}×[1,∞)) and 𝐷^𝛼_𝑦𝐷^𝛽_𝜆𝑅^𝑠_±𝑦,𝜆=𝑂𝜆^{−𝑠−𝛽}, (5.2) uniformly on |𝑦|>𝛿 and 𝜆>𝛿.

Now, let us consider positive numbers 𝜀 and 𝑅, 𝜀<𝑅 and function 𝜙(𝑥)=𝜙(‖𝑥‖^∗_𝑎)∈𝐶^∞₀ vanishing on {𝑥∶(‖𝑥‖^∗_𝑎<𝜀)∨(‖𝑥‖^∗_𝑎>𝑅)}. Then, for 𝑠≥0, we set by definition Θ^𝑠_𝜆(𝑥)=𝜙(𝑥)𝜃^𝑠_𝜆(𝑥), (5.3) where 𝜃^𝑠_𝜆 as in (2.3).

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

Lemma 5.1. Let 𝑡≥𝛿>0 and |𝜉|<1. Then, for any 𝛼>0Θ^𝑠_𝑡(𝜉)≤𝑂𝑡^−𝛼. (5.4)

Proof. This estimate easily follows from the definition of Θ^𝑠_𝑡. Indeed, Θ^𝑠_𝑡(𝜉)=𝜃^𝑠_𝑡(𝑥)𝜙(𝑥)𝑒^{−𝑖𝜉𝑥}𝐴𝑦𝑑𝑥=1−𝑡^𝑠₊(𝑥)𝜙(𝑥)𝑒^{−𝑖𝜉𝑥}=𝐴(𝑥)𝑑𝑥1−𝑡^𝑠₊𝜙(𝑥+𝜉)𝑑𝑥=_{𝐴(𝑥)<𝑡}𝜙()+𝑥+𝜉𝑑𝑥_𝑠𝑘=1𝐶_𝑘𝑡^𝑘_{𝐴(𝑥)<𝑡}𝜙_𝑘(⋅,𝜉)(𝑥)𝑑𝑥, (5.5) where 𝜙_𝑘(𝑦,𝜉)=𝐵^𝑘(𝐷_𝑦)[𝜙(𝑦)𝑒^{−𝑖𝑦𝜉}] and 𝐵(𝐷) is formally conjugate to operator 𝐴(𝐷). Since 𝜙(0)=𝜙_𝑘(0,𝜉)=0, Θ^𝑠_𝑡(𝜉)=_{𝐴(𝑥)>𝑡}𝜙(𝑥+𝜉)𝑑𝑥+^𝑠_𝑘=1𝐶_𝑘𝑡^𝑘_{𝐴(𝑥)>𝑡}𝜙_𝑘(⋅,𝜉)(𝑥)𝑑𝑥. (5.6) Further, we notice that since 𝜙_𝑘(𝑦,𝜉)∈𝐶^∞₀ then for any 𝛼>0 there is 𝐶_𝛼 such that functions 𝜙_𝑘(𝑥,𝜉)=𝐶_𝛼/(1+𝐴(𝑥))^𝛼, uniformly for 𝑘=1,…,𝑠 and |𝜉|<1. For the same reason, for any 𝛼>0, one has 𝜙(𝑥)≤𝑂((1+𝑥)^−𝛼). Now substituting these estimates into (5.6), we complete the proof.

Lemma 5.2. Let 𝑡≥𝛿>0 and |𝜉|≥1. Then, for any 𝛼>0, Θ^𝑠_𝑡(𝜉=𝑂(1)𝑡)^−𝑠1+𝜉_𝑎−𝑡^𝛼. (5.7)

Proof. By definition, Θ^𝑠_𝑡(𝜉)=_{𝜖<‖𝑦‖∗𝑎<𝑅}𝜙𝑦𝜃^𝑠_𝑡𝑦𝑒^{−𝑖𝜉⋅𝑦}𝑑𝑦. (5.8) Let us pass to a new coordinate system 𝑦→(𝑟=‖𝑦‖^∗_𝑎,𝜂=𝜂(𝑦)). Then, Θ^𝑠_𝑡(𝜉)=^𝑅_𝜖𝜙(𝑟)𝑟^𝑛−1_{𝜂∈𝑆∗}𝜃^𝑠_𝑡𝑟𝜂𝑒^{−𝑖𝑟𝜉⋅𝜂}𝑑𝜎𝜂𝑑𝑟, (5.9) where 𝑑𝜎(𝜂) is a Lebesgue surface measure of 𝑆^∗.
Using (5.1), we have Θ^𝑠_𝑡(𝜉)=𝑡^{(𝑛−1)/2𝑅}_𝜖𝜙(𝑟)𝑟^(𝑛−3)/2_{𝜂∈𝑆∗}𝜂^{−(𝑛+1)/2}𝑒^{𝑖𝑡𝑟[𝜂⋅𝜃(𝜂)]}𝑅^𝑠₊𝑟𝜂,𝑡𝑒^{−𝑖𝑟𝜉⋅𝜂}𝑑𝜎𝜂𝑑𝑟+𝑡^{(𝑛−1)/2𝑅}_𝜖𝜙(𝑟)𝑟^(𝑛−3)/2_{𝜂∈𝑆∗}𝜂^{−(𝑛+1)/2}𝑒^{−𝑖𝑡𝑟[𝜂⋅𝜃(𝜂)]}𝑅^𝑠₋𝑟𝜂,𝑡𝑒^{−𝑖𝑟𝜉⋅𝜂}𝑑𝜎𝜂𝑑𝑟. (5.10) We will focus on the first term since the second one can be handled alike 𝐼^𝑠_𝑡(𝜉)=𝑡^{(𝑛−1)/2𝑅}_𝜖𝜙(𝑟)𝑟^(𝑛−3)/2_{𝜂∈𝑆∗}𝜂^{−(𝑛+1)/2}𝑒^{𝑖𝑡𝑟[𝜂⋅𝜃(𝜂)]}𝑅^𝑠₊𝑟𝜂,𝑡𝑒^{−𝑖𝑟𝜉⋅𝜂}𝑑𝜎𝜂𝑑𝑟 (5.11) and note that due to (4.4) 𝜂⋅𝜃(𝜂)=1, and thus 𝐼^𝑠_𝑡(𝜉)=𝑡^{(𝑛−1)/2𝑅}_𝜖𝜙(𝑟)𝑟^(𝑛−3)/2𝑒^𝑖𝑡𝑟_{𝜂∈𝑆∗}𝜂^{−(𝑛+1)/2}𝑅^𝑠₊𝑟𝜂,𝑡𝑒^{−𝑖𝑟𝜉⋅𝜂}𝑑𝜎𝜂𝑑𝑟. (5.12) One can use the expression 𝜉=‖𝜉‖_𝑎𝜃(𝜉) and employ stationary phase method to obtain _{𝜂∈𝑆∗}𝜂^{−(𝑛+1)/2}𝑅^𝑠₊𝑟𝜂,𝑡𝑒^{−𝑖𝑟𝜉⋅𝜂}𝑑𝜎𝜂=𝜉_𝑎^{−(𝑛−1)/2}𝑒^{𝑖𝑟‖𝜉‖𝑎[𝜃(𝜉)⋅𝜂(𝜃(𝜉))]}𝑃^𝑠₊()+𝜉𝑟𝜉,𝑡_{𝑎−(𝑛−1)/2}𝑒^{−𝑖𝑟‖𝜉‖𝑎[𝜃(𝜉)⋅𝜂(𝜃(𝜉))]}𝑃^𝑠₋(𝑟𝜉,𝑡), (5.13) where 𝑃^𝑠_± are smooth functions such that 𝐷^𝛼_𝑡𝐷^𝛽_𝑧𝑃^𝑠_±(𝑧,𝑡)=𝑂(𝑡^{−𝑠−𝛼}). Using this expression, we have 𝐼^𝑠_𝑡(𝜉)=𝑡𝜉_𝑎^{(𝑛−1)/2𝑅}_𝜖𝜙(𝑟)𝑟^(𝑛−3)/2𝑒^{𝑖𝑟(𝑡−‖𝜉‖𝑎)}𝑃^𝑠₊()+𝑡𝑟𝜉,𝑡𝑑𝑟𝜉_𝑎^{(𝑛−1)/2𝑅}_𝜖𝜙(𝑟)𝑟^(𝑛−3)/2𝑒^{−𝑖𝑟(𝑡−‖𝜉‖𝑎)}𝑃^𝑠₋(𝑟𝜉,𝑡)𝑑𝑟. (5.14)
Further integrating by parts the integrals, one can see that for any 𝑁>0 both integrals are controlled by (𝐶_𝑁𝑡^−𝑠)/(1+|𝑡−‖𝜉‖_𝑎|)^𝑁. As a result, we have 𝐼^𝑠_𝑡(𝜉≤𝑡)𝜉_𝑎^(𝑛−1)/2𝐶_𝑁𝑡^−𝑠1+𝑡−𝜉_𝑎^𝑁≤𝐷_𝑁𝑡^−𝑠1+𝑡−𝜉_𝑎^{𝑁−((𝑛−1)/2)}, (5.15) uniformly for |𝜉|>1 and 𝑡>𝛿. Finally, substituting into (5.10), we obtain (5.7).

Now, combining Lemmas 5.1 and 5.2, we can claim that, in fact, for 𝑡>𝛿 and any 𝜉∈𝑅^𝑛, Θ^𝑠_𝑡(𝜉≤𝑂(1)𝑡)^−𝑠1+𝜉_𝑎−𝑡^𝛼. (5.16)

Lemma 5.3. Let Θ^𝑠_𝜆(𝑥) be defined by (5.3). Then, for any 𝛿>0 there is 𝐶_𝛿>0 such that ^∞_𝛿Θ^𝑠_𝑡(𝜉)²𝐶𝑑𝑡≤^𝛿1+𝜉^2𝑠. (5.17)

Proof. As it follows from (5.16), ^∞_𝛿Θ^𝑠_𝑡(𝜉)²𝑑𝑡≤𝑂(1)^∞_𝛿𝑡^−2𝑠𝑑𝑡1+𝜉_𝑎−𝑡^2𝛼, (5.18) where 𝛼>0 can be chosen arbitrary large. Changing the variables 𝑢=𝜉−𝑡, one has ^∞_𝛿|𝑡|^−2𝑠𝑑𝑡1+𝜉_𝑎−𝑡^2𝛼=_{𝛿<|‖𝜉‖𝑎−𝑢|<‖𝜉‖𝑎/2}𝜉_𝑎−𝑢^−2𝑠𝑑𝑡(1+|𝑢|)^2𝛼+_{max(𝛿,‖𝜉‖𝑎/2)<|‖𝜉‖𝑎−𝑢|}𝜉_𝑎−𝑢^−2𝑠𝑑𝑡(1+|𝑢|)^2𝛼. (5.19) It is not difficult to see that for the values 𝑢 in the first integral |𝑢|>|‖𝜉‖_𝑎−∣𝑢−‖𝜉‖_𝑎|>‖𝜉‖_𝑎/2, and thus choosing 𝛼>max(2𝑠,1)_{𝛿<|‖𝜉‖𝑎−𝑢|<‖𝜉‖𝑎/2}𝜉_𝑎−𝑢^−2𝑠𝑑𝑡(1+|𝑢|)^2𝛼≤𝑂(1)1+𝜉_𝑎^𝛼≤𝑂(1)1+𝜉_𝑎^2𝑠. (5.20) Moreover, it is clear that for such 𝛼_{𝛿<|‖𝜉‖𝑎−𝑢|<‖𝜉‖𝑎/2}𝜉_𝑎−𝑢^−2𝑠𝑑𝑡(1+|𝑢|)^2𝛼≤𝑂(1)1+𝜉_𝑎^2𝑠. (5.21)
Therefore, ^∞_𝛿Θ^𝑠_𝑡(𝜉)²𝑂(1)𝑑𝑡≤1+𝜉_𝑎^2𝑠. (5.22)
Since all norms in 𝑅^𝑛 are equivalent, the lemma is proved.

Lemma 5.4. Let Θ^𝑠_𝜆 be defined by (5.3). Then, for any 𝛿>0, there is 𝐶_𝛿 such that ^∞_𝛿𝑑Θ𝑑𝑡^𝑠𝑡(𝜉)²𝐶𝑑𝑡≤^𝛿1+𝜉^2𝑠. (5.23)

Proof. For any 𝑡>1, using the Fubini theorem, one has ^𝑡₁𝑑Θ𝑑𝑢^𝑠𝑢𝑦𝑒^{−𝑖𝜉𝑦}𝑑𝑦𝑑𝑢=𝑒^{−𝑖𝜉𝑦𝑡}₁𝑑Θ𝑑𝑢^𝑠𝑢𝑦𝑑𝑢𝑑𝑦=Θ^𝑠_𝑡(𝜉)−Θ^𝑠₁(𝜉), (5.24) which implies (𝑑/𝑑𝑡)Θ^𝑠_𝑡(𝜉)=(𝑑/𝑑𝑡)Θ^𝑠_𝑡(𝜉).
If 𝑠>0𝑑Θ𝑑𝑡^𝑠𝑡(𝑥)(𝑥)𝑑=𝜙𝜃𝑑𝑡^𝑠𝑡(𝑥)(𝑥)𝑑=𝜙𝑑𝑡^𝑡₀𝑢1−²𝑡^2𝑠𝑑𝜃_𝑢(𝑥)=2𝑠𝜙(𝑥)𝑡^𝑡₀𝑢1−²𝑡^2𝑠−1𝑢²𝑡²𝑑𝜃_𝑢(𝑥)=2𝑠𝑡Θ_𝑡^𝑠−1(𝑥)−Θ^𝑠_𝑡(𝑥). (5.25) Thus, using inequality (𝑎+𝑏)²≤2𝑎²+2𝑏², one has ^∞_𝛿𝑑Θ𝑑𝑡^𝑠𝑡(𝜉)²𝑑𝑡≤𝐶^∞_𝛿𝑡⁻²Θ_𝑡^𝑠−1(𝜉)²𝑑𝑡+𝐶^∞_𝛿𝑡⁻²Θ^𝑠_𝑡(𝜉)²𝑑𝑡. (5.26) Now, one can use estimate (5.16) to each integral on the right side and complete the proof.
If 𝑠=0, then for any 𝜉∈𝑅^𝑛, 𝑑Θ𝑑𝑡^𝑡(𝜉=𝑑)𝑑𝑡_{𝐴(𝑦)≤𝑡}𝜙=𝜉+𝑦𝑑𝑦_{𝐴(𝑦)=𝑡}𝜙𝑛𝑦𝑦≤𝑂(1)𝜉+𝑦𝑑𝜎1+𝜉_𝑎−𝑡^𝛼,∀𝛼>0. (5.27) Using this estimate and the reasoning presented in the previous lemma, we obtain the required estimate.

6. Proof of Theorem 3.1

Let 𝑓∈𝐻^−𝑙∩ℰ′ be such that supp𝑓⊂Ω. For 0<𝜖<1/2, we set 𝐸_𝜖=𝑥∶2𝜖<dist(𝑥,Ω)<(2𝜖)⁻¹ (6.1) and consider an arbitrary radial function 𝜙_𝜖∈𝐶^∞₀ such that 𝜙_𝜖(𝑥)=1,3𝜖2≤|𝑥|≤1𝜖|𝑥|+diamΩ;0,≤𝜖. (6.2)

It is clear that to prove the theorem it is sufficient to show that for any 𝜖>0, lim_𝜆→∞𝐸^𝑠_𝜆𝑓(𝑥)=0, a.e. 𝑥∈𝐸_𝜖

In this case, as 𝑥∈𝐸_𝜖 due to (2.6) 𝐸^𝑠_𝜆𝑓(𝑥)=𝑓(𝜉)𝜒_Ω(⋅)𝜃^𝑠_𝜆((̂)𝑓(𝜉𝜙𝑥−⋅)−𝜉𝑑𝜉=)_𝜖(𝑥−⋅)𝜃^𝑠_𝜆((̂)=𝑓(𝜉𝜙𝑥−⋅)−𝜉𝑑𝜉)_𝜖𝜃^𝑠_𝜆(̂𝜉)𝑒^𝑖𝜉𝑥𝑑𝜉, (6.3) or using notation (5.3) 𝐸^𝑠_𝜆𝑓(𝑥)=𝑓(𝜉)Θ^𝑠_𝜆(𝜉)𝑒^𝑖𝜉𝑥𝑑𝜉=𝑓(𝜉)Θ^𝑠_𝜆(𝜉)(−𝑥). (6.4)

Further, we consider maximal operator: 𝐸^𝑠_∗𝑓(𝑥)=sup_𝜆>1𝐸^𝑠_𝜆𝑓(𝑥). (6.5) We recall that to prove a.e. convergence on 𝐸_𝜖 one can use the standard technique of Banach principle (see, e.g., [19]) according to which it is sufficient to estimate maximal operator on 𝐸_𝜖⊂𝑅^𝑛⧵supp𝑓 as 𝐸^𝑠_∗𝑓(𝑥)_{𝐿2(𝐸𝜖)}≤𝐶𝑓_𝐻−𝑙. (6.6)

Let 𝛾(𝑡)∶ℝ→ℝ₊ be a 𝐶^∞ function such that 𝛾(𝑡)=10,𝑡≤3;21,𝑡≥3. (6.7) If we set 𝐸^𝑠_𝜆𝑓(𝑥)=𝛾(𝜆)𝐸^𝑠_𝜆𝑓(𝑥), then by (6.4), 𝐸^𝑠_𝜆𝑓(𝑥)=𝛾(𝜆)<𝑓(⋅),Θ^𝑠_𝜆(𝑥−⋅)>=𝛾(𝜆)𝑓(𝜉)Θ^𝑠_𝜆(𝜉)(−𝑥). (6.8) According to Sobolev's embedding theorem (see, e.g., [20]) for any 𝑓∈𝐻¹(𝑅¹), 𝑓_𝐿∞≤𝐶𝑓_𝐻1. (6.9) Using this fact, we have 𝐸^𝑠_∗𝑓(𝑥)≤𝐸^𝑠_𝜆𝑓(𝑥)_{𝐿∞(𝑅)}≤𝐸^𝑠_𝜆𝑓(𝑥)_𝐻1(𝑅). (6.10) And, therefore, in order to obtain (6.6), it is sufficient to show that there are constants 𝐶₁,𝐶₂ such that the following estimates are true: 𝐸^𝑠_𝜆𝑓(𝑥)²_𝐿2(𝑅)𝑑𝑥≤𝐶₁𝑓_𝐻−𝑙,𝑑𝐸𝑑𝜆^𝑠𝜆𝑓(𝑥)²_𝐿2(𝑅)𝑑𝑥≤𝐶₂𝑓_𝐻−𝑙. (6.11)

First, we note that estimate (5.16) and 𝑓∈𝐻^−𝑙 imply that 𝑓Θ^𝑠_𝜆∈𝐿₂ which in turn with (6.8) implies the fact 𝐸^𝑠_𝜆𝑓∈𝐿₂.

Further, using the Plancherel theorem, we have 𝐸^𝑠_𝜆𝑓(𝑥)²𝑑𝜆𝑑𝑥≤𝛾²(𝜆)𝑓(𝜉)²Θ^𝑠_𝜆(𝜉)²𝑑𝜉𝑑𝜆≤^∞_1/3𝛾²(𝑡)1+𝜉^2𝑙Θ^𝑠_𝑡(𝜉)²1+𝜉^2−𝑙𝑓(𝜉)²𝑑𝜉𝑑𝑡≤sup_{𝜉∈𝑅𝑛}1+𝜉^2𝑙∞_1/3Θ^𝑠_𝑡(𝜉)²𝑑𝑡×𝑓²_𝐻−𝑙≤𝐶𝑓²_𝐻−𝑙 (6.12) (the last inequality follows from Lemma 5.3).

For the same reason, (6.11) can be proved using Lemmas 5.3 and 5.4: ^∞_1/3𝑑𝛾(𝑡)Θ𝑑𝑡^𝑠𝑡∗𝑓(𝑥)²𝑑𝑡𝑑𝑥≤𝛾^′2(𝜆)_∞^∞_1/3Θ^𝑠_𝑡∗𝑓(𝑥)²+𝑑𝑡𝑑𝑥_∞1/3𝑑Θ𝑑𝜆^𝑠𝜆∗𝑓(𝑥)²𝑑𝜆𝑑𝑥≤𝐶𝑓²_𝐻−𝑙. (6.13)

Acknowledgments

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.