Compatibility Conditions and the Convolution of Functions and Generalized Functions

Abstract

The paper is a review of certain existence theorems concerning the convolution of functions, distributions, and ultradistributions of Beurling type with supports satisfying suitable compatibility conditions. The fact that some conditions are essential for the existence of the convolution in the discussed spaces follows from earlier results and the proofs given at the end of this paper.

In memory of Professor Jan Mikusiński on the 100th anniversary of his birthday

1. Introduction

The convolution and its various generalizations play a very important role in the classical and abstract analysis as well as in other fields of mathematics, in particular in the theory of distributions (see [1–7]), ultradistributions (see [3, 4, 8–18]), hyperfunctions (see [19–21]), and other generalized functions considered for various spaces, subspaces, and approaches. The notion of convolution is a starting point in algebraic approaches to certain generalized functions: the convolution algebra of continuous functions on [0,∞) extends, due to the Titchmarsh theorem, to the field of Mikusiński operators (see [22–25]) which are generalized functions of another type than Schwartz distributions, while Boehmians stand for a common generalization of regular Mikusiński operators of Boehme (see [26]), Schwartz distributions, and other classes of generalized functions on the real line (see [27–31]). An important part of investigations connected with the convolution is the study of convolution operators and convolution semigroups for various spaces of functions and generalized functions (see, e.g., [32–37]). Therefore the problems concerning the existence of the convolution in various spaces of functions and generalized functions are crucial.

The theory developed by Colombeau (see [38]; see also [39]) and his followers (see, e.g., [40–43]) has led to constructions of algebras of new generalized functions related to the distributions and other classical generalized functions due to certain quotient procedures; consequently the algebras of new generalized functions are closed with respect to multiplication as well as other nonlinear operations. However the problem of existence of the product and the convolution of distributions and other generalized functions in the standard sense, without using Colombeau’s approach, remains important. We will analyse in this paper the existence of the convolution of distributions and tempered distributions on ℝ^𝑑 meant in the classical sense of general equivalent definitions introduced independently by several authors (see [2, 44–50]). Also the existence of the convolution of ultradistributions and tempered ultradistributions of Beurling type introduced in [18, 51] (see also [52]), corresponding to the above mentioned general definitions of the convolution of distributions and tempered distributions, will be discussed. The convolution of ultradistributions of Roumieu type will be not considered in this paper; for recently obtained results concerning the convolution of Roumieu ultradistributions we refer to the paper [53].

There exist various sufficient conditions guaranteeing existence of the convolution in various spaces of functions and generalized functions. Some of the conditions are given in the form of suitable assumptions concerning their growth at infinity, but there are conditions of another type expressed in terms of supports of given (generalized) functions. We will call them after Mikusiński (see [54], and [7, p. 124–127]) compatibility conditions. We mean the support of a given function (generalized function) in the standard way as the smallest closed set in ℝ^𝑑 outside which the function (generalized function) vanishes almost everywhere (everywhere). For modifications of the notion of support in case of distributions and tempered distributions see [7, p. 241] and [55].

We recall in this paper various versions of compatibility conditions imposed on supports of functions, distributions, and ultradistributions of Beurling type which guarantee that the convolution in the respective spaces exists (see [5, 7, 54, 56–59]; see also [60–63]). In Section 7, we prove some inverse theorems which mean that the considered compatibility conditions are optimal for the existence of the convolution in some spaces in terms of supports; that is, the conditions cannot be relaxed. We also mention new cases of compatibility of supports in which the convolution exists in the considered spaces. In particular, we show examples of the so-called spiral sets in ℝ^𝑑 with the property that the convolutions of functions, distributions, ultradistributions having supports contained in such sets always exist in the respective spaces, though the sets are unbounded in each direction of ℝ^𝑑.

2. Preliminaries

We will use the standard multidimensional notation concerning ℝ^𝑑 and ℕ^𝑑₀, using traditional symbols even in case they are formally inconsistent, because the proper meaning is easily seen from the context; for example, |𝑥| denotes the Euclidean norm of 𝑥 in ℝ^𝑑 and ⟨𝑥⟩∶=1+|𝑥|^21/2=1+𝜉²₁+⋯+𝜉²_𝑑^1/2,𝑥=𝜉₁,…,𝜉_𝑑∈ℝ^𝑑, (1) but we write also |𝑘|∶=𝜅₁+⋯+𝜅_𝑑 for 𝑘=(𝜅₁,…,𝜅_𝑑)∈ℕ^𝑑₀, though ℕ^𝑑₀⊂ℝ^𝑑.

The standard notation will be also used for the known spaces of (complex-valued) functions on ℝ^𝑑: 𝐿^𝑟(ℝ^𝑑) for 𝑟∈[1,∞] (with the norm denoted by ‖⋅‖_𝑟), 𝐿¹_loc(ℝ^𝑑), 𝒞(ℝ^𝑑), 𝒞^∞(ℝ^𝑑), ℰ(ℝ^𝑑), 𝒟(ℝ^𝑑), 𝒟_𝐾(ℝ^𝑑), 𝒮(ℝ^𝑑), and the known (sub) spaces of distributions on ℝ^𝑑: 𝒞^′(ℝ^𝑑), 𝒟^′(ℝ^𝑑), and 𝒮^′(ℝ^𝑑) (cf. [1, 7]).

For a set 𝐸⊆ℝ^𝑑 we denote 𝐸^𝑐∶=ℝ^𝑑⧵𝐸. It will be convenient to use the following notation for 𝐸⊆ℝ^𝑑 and a function 𝜙 on ℝ^𝑑: 𝐸^Δ∶=𝑥,𝑦∈ℝ^2𝑑,𝜙∶𝑥+𝑦∈𝐸_Δ𝑥,𝑦∶=𝜙𝑥+𝑦,𝑥,𝑦∈ℝ^𝑑. (2)

Clearly, for any 𝐸⊆ℝ^𝑑, the set 𝐸^Δ is bounded (compact) in ℝ^2𝑑 if and only if 𝐸=∅. Therefore if 𝜙∈𝒟(ℝ^𝑑), then 𝜙^Δ∈𝒞^∞(ℝ^2𝑑), but 𝜙^Δ∈𝒟(ℝ^2𝑑) only in case 𝜙=0. Moreover, the symbol ⊏ℝ^𝑑 will be used in the following sense: 𝐸⊏ℝ^𝑑whenever𝐸isacompactsubsetofℝ^𝑑. (3)

The words “measure” and “measurable” concerning the sets in ℝ^𝑑 are considered in the sense of Lebesgue; the Lebesgue measure of a given Lebesgue measurable set 𝐴⊆ℝ^𝑑 will be denoted by ℓ^𝑑(𝐴). If 𝐴⊆ℝ^𝑑 is a measurable set, we use the following notation: |𝐴|_∞∶=supess{|𝑥|∶𝑥∈𝐴}. (4) That is, |𝐴|_∞=infΛ(𝐴) in case Λ(𝐴)≠∅ and |𝐴|_∞=∞ in case Λ(𝐴)=∅, where Λ(𝐴)∶={𝛼>0∶ℓ^𝑑(𝑋_𝛼)=0} with 𝑋_𝛼∶={𝑥∈𝐴∶|𝑥|>𝛼}. Clearly, |𝐴|_∞=0iffℓ^𝑑(𝐴)=0. (5)

We call a measurable set 𝐴⊂ℝ^𝑑positively bounded in ℝ^𝑑 whenever 0<|𝐴|_∞<∞. An element 𝑥 of a measurable set 𝐴∈ℝ^𝑑 is called a massive point of the set 𝐴 if ℓ^𝑑(𝐴∩𝑈(𝑥))>0 for each open neighbourhood 𝑈(𝑥) of 𝑥 in ℝ^𝑑.

A measurable function 𝐹 on ℝ^𝑑 is called slowly increasing (resp., rapidly decreasing) if |𝐹(𝑥)|≤𝑃(𝑥) (resp., |𝑃(𝑥)𝐹(𝑥)|≤𝑐(𝑃)) for a certain polynomial 𝑃 on ℝ^𝑑 (resp., for every polynomial 𝑃 and some positive constant 𝑐(𝑃)) and for almost all 𝑥∈ℝ^𝑑. The sets of all measurable slowly increasing and rapidly decreasing functions on ℝ^𝑑 will be denoted by 𝒫(ℝ^𝑑) and ℛ(ℝ^𝑑), respectively.

In our considerations concerning the convolution in the spaces 𝒟^{′(𝑀𝑝)}(ℝ^𝑑) of ultradistributions and 𝒮^{′(𝑀𝑝)}(ℝ^𝑑) of tempered ultradistributions of Beurling type (the respective definitions and notation are introduced in Section 4) we always assume that a given sequence (𝑀_𝑝)=(𝑀_𝑝)_𝑝∈ℕ0 of positive numbers satisfies the following three conditions (see [10–12, 18, 51, 52, 59]):(M.1)𝑀²_𝑝≤𝑀_𝑝−1𝑀_𝑝+1,𝑝∈ℕ; (M.2)𝑀_𝑝≤𝑎𝑏^𝑝𝑀_𝑞𝑀_𝑝−𝑞,𝑝,𝑞∈ℕ,0≤𝑞≤𝑝; (M.3)∑^∞_𝑝=𝑞+1𝑀_𝑝−1𝑀_𝑝⁻¹≤𝑎𝑞𝑀_𝑞𝑀_𝑝⁻¹,𝑞∈ℕ, where 𝑎>0 and 𝑏>0 are some constants. We extend the sequence (𝑀_𝑝)_𝑝∈ℕ0 to its multidimensional version (𝑀_𝑘)_𝑘∈ℕ^𝑑₀ as follows: 𝑀_𝑘∶=𝑀_{𝜅1+⋯+𝜅𝑑},𝑘=𝜅₁,…,𝜅_𝑑∈ℕ^𝑑₀. (6)

By the associated function for the sequence (𝑀_𝑝) we mean the function on [0,∞) given by 𝑀(𝑡)∶=sup_𝑝∈ℕ0log₊𝑡^𝑝𝑀_𝑝,𝑡>0, (7) where log₊∶=max{log𝑡,0} and 𝑀(0)∶=0.

3. Convolution of Functions and Distributions

Definition 1. Let 𝐹 and 𝐺 be Lebesgue measurable functions on ℝ^𝑑. For a given 𝑥∈ℝ^𝑑 one defines (𝐹∗𝐺)(𝑥)∶=_ℝ𝑑𝐹(𝑥−𝑡)𝐺(𝑡)𝑑𝑡, (8)

saying that (𝐹∗𝐺)(𝑥) exists, whenever the function under the integral sign in (8) is Lebesgue integrable as a function of 𝑡 for the fixed 𝑥. If (𝐹∗𝐺)(𝑥) exists or, equivalently, (|𝐹|∗|𝐺|)(𝑥) exists, for all (almost all) 𝑥∈ℝ^𝑑, then one says that the convolution 𝐹∗𝐺exists everywhere (almost everywhere) on ℝ^𝑑.

Definition 2. Let 𝐹,𝐺∈𝐿¹_loc(ℝ^𝑑). One says that the convolution 𝐹∗𝐺 exists in 𝐿¹_loc if 𝐹∗𝐺 exists almost everywhere on ℝ^𝑑 and |𝐹|∗|𝐺|∈𝐿¹_loc(ℝ^𝑑).

Let us list some classical results concerning the existence in 𝐿¹_loc of the convolution of functions of certain classes:(a)if 𝐹,𝐺∈𝐿¹(ℝ^𝑑), then 𝐹∗𝐺 exists in 𝐿¹_loc and 𝐹∗𝐺∈𝐿¹(ℝ^𝑑);(b)if 𝐹∈𝐿^𝑝(ℝ^𝑑) and 𝐺∈𝐿^𝑞(ℝ^𝑑), where 𝑝,𝑞∈[1,∞) such that 1/𝑝+1/𝑞>1, then 𝐹∗𝐺 exists in 𝐿¹_loc and 𝐹∗𝐺∈𝐿^𝑟(ℝ^𝑑), where 𝑟=(1/𝑝+1/𝑞−1)⁻¹ (Young);(c)if 𝐹∈𝒫(ℝ^𝑑) and 𝐺∈ℛ(ℝ^𝑑), then 𝐹∗𝐺 exists in 𝐿¹_loc and 𝐹∗𝐺∈𝒫(ℝ^𝑑).

However, in general, the existence of the convolution of functions in 𝐿¹_loc does not imply any restriction of growth of the convolution at infinity. For instance, the convolution of two measurable slowly increasing functions may exist in 𝐿¹_loc, but their convolution may be a function of arbitrarily fast increase at infinity.

The following result, much stronger than one may expect, was proved in [56, 57] (see also [58, 63]).

Theorem 3. Let Φ be an arbitrary continuous (complex-valued) function on ℝ^𝑑. There exists a nonnegative 𝐶^∞ function 𝜑 on ℝ^𝑑 such that lim_|𝑥|→∞𝜑(𝑥)=0, (9)
the convolution 𝜑∗𝜑 exists everywhere in ℝ^𝑑, and the following inequality is satisfied: 𝜑∗𝜑(𝑥)>|Φ(𝑥)|∀𝑥∈ℝ^𝑑. (10)
Moreover, the function 𝜑 can be constructed in such a way that its support is compatible with itself (see Definition 16).

Let us underline that inequality (10) in the assertion of Theorem 3 is satisfied for each 𝑥∈ℝ^𝑑; the convergence in (9) can be slow: the faster the increase of the function Φ (i.e., the required increase of the convolution 𝜑∗𝜑) is at infinity the slower the approach of 𝜑 to 0 is as |𝑥| tends to ∞ (for details of the construction see [56, 57, 63]).

Theorem 3 has a general value. From the theorem one may easily see, for various spaces of functions or generalized functions and their various subspaces, that the convolution in the sense of the considered space leads out of the examined subspace. This concerns, in particular, the convolution in 𝒟^′ and in 𝒟^{′(𝑀𝑝)} (see Definitions 5 and 12) of elements of the subspaces 𝒮^′(ℝ^𝑑) and 𝒮^{′(𝑀𝑝)}(ℝ^𝑑) of the spaces 𝒟^′(ℝ^𝑑) and 𝒟^{′(𝑀𝑝)}(ℝ^𝑑) of Schwartz distributions and Beurling ultradistributions, respectively (see the definitions of the spaces, given in this and in the next section, and the comments at the end of this section).

Let us shortly recall (see [1]; see also, e.g., [5–7]) that distributions are elements of the strong dual 𝒟^′(ℝ^𝑑) of the basic space 𝒟(ℝ^𝑑) of test functions, that is, linear continuous functionals on the space 𝒟(ℝ^𝑑) defined as the inductive limit: 𝒟ℝ^𝑑∶=indlim_{𝐾⊏⊏ℝ𝑑}𝒟_𝐾ℝ^𝑑, (11) where the symbol 𝐾⊏⊏ℝ^𝑑 means here that 𝐾 are compact sets growing up to ℝ^𝑑. Recall that 𝒟_𝐾(ℝ^𝑑), for a fixed 𝐾⊏ℝ^𝑑, means the space of all 𝐶^∞ functions on ℝ^𝑑 whose support is contained in 𝐾, with the topology given by the family {‖⋅‖_𝐾,𝑚∶𝑚∈ℕ₀} of the norms defined as follows: 𝜑_𝐾,𝑚∶=sup𝜑^(𝑘)(𝑥)∶𝑥∈𝐾,𝑘∈ℕ^𝑑₀,|𝑘|≤𝑚,𝑚∈ℕ₀. (12) Locally integrable functions on ℝ^𝑑 may be treated as distributions (so-called regular distributions), because the space 𝐿¹_loc(ℝ^𝑑) can be naturally embedded into 𝒟^′(ℝ^𝑑). The known structural theorem describes every distribution as a derivative (in the distributional sense) locally, that is, on each relatively compact set in ℝ^𝑑, of finite order of a continuous function (see [1, 5–7]).

Recall that tempered distributions are elements of the dual 𝒮^′(ℝ^𝑑) of the complete metric space 𝒮(ℝ^𝑑) of rapidly decreasing 𝐶^∞ functions, that is, of all 𝜑∈𝒞^∞(ℝ^𝑑) such that 𝜎_𝑚𝜑∶=max⟨⋅⟩^𝑖𝜑^(𝑘)_∞∶𝑖,𝑘∈ℕ^𝑑₀,|𝑖|,|𝑘|≤𝑚<∞ (13) for all 𝑚∈ℕ₀, where ⟨⋅⟩ is defined by (1), with the metric topology induced by the family {𝜎_𝑚∶𝑚∈ℕ₀} of the norms defined in (13). Since 𝒟(ℝ^𝑑) is dense in 𝒮(ℝ^𝑑), the space 𝒮^′(ℝ^𝑑) of tempered distributions can be naturally embedded into 𝒟^′(ℝ^𝑑). The known structural theorem describes each tempered distribution as a derivative (in the distributional sense) of finite order of a continuous slowly increasing function on ℝ^𝑑 (see [1–7]).

There are several general definitions of the convolution in 𝒟^′ of Schwartz distributions given consecutively by Chevalley [2], Schwartz [44], Shiraishi [45], and Vladimirov [46] (see also [47–50, 63–72]).

Most of the mentioned definitions are equivalent (for details see, e.g., [50]). We will recall only one of them, the sequential definition of Vladimirov [46] (see also [6]), based on the notion of strong approximate unit.

Definition 4. A sequence (𝜂_𝑛) of elements of 𝒟(ℝ^𝑑) is said to be a strong approximate unit on ℝ^𝑑 if for every 𝐾⊏ℝ^𝑑 there exists an 𝑛₀∈ℕ such that 𝜂_𝑛(𝑥)=1 for 𝑥∈𝐾 and 𝑛≥𝑛₀ (hence 𝜂_𝑛→1 in ℰ(ℝ^𝑑) as 𝑛→∞) and, in addition, sup_𝑛∈ℕ𝜂_𝑛^(𝑘)_∞<∞forevery𝑘∈ℕ^𝑑₀. (14)

One denotes the set of all strong approximate units on ℝ^𝑑 by 𝕌(ℝ^𝑑).

Definition 5. For given 𝑆,𝑇∈𝒟^′(ℝ^𝑑) the convolution 𝑆∗𝑇 in 𝒟^′ is defined by 𝑆∗𝑇,𝜑∶=lim_𝑛→∞𝑆⊗𝑇,𝜂_𝑛𝜑^Δ,𝜑∈𝒟ℝ^𝑑, (15) whenever the above limit exists for every strong approximate unit (𝜂_𝑛)∈𝕌(ℝ^2𝑑) and 𝜑∈𝒟(ℝ^𝑑). One says then that the convolution 𝑆∗𝑇 exists in 𝒟^′.

By the Lebesgue theorem, if 𝐹,𝐺∈𝐿¹_loc(ℝ^𝑑) and the convolution 𝐹∗𝐺 exists in 𝐿¹_loc, then it exists in 𝒟^′ and represents a regular distribution in 𝒟^′(ℝ^𝑑) (see [6, p. 63]).

Analogously to and independently of Definition 5, one may define the convolution in 𝒮^′ of tempered distributions in various ways (see, e.g., [45, 49, 50] and other references given earlier). Again, we present below only one of several equivalent definitions of the convolution in 𝒮^′, namely, the respective counterpart of the aforementioned sequential definition of Vladimirov (cf. [6, 46]).

Definition 6. For given 𝑆,𝑇∈𝒮^′(ℝ^𝑑) one defines the convolution 𝑆∗𝑇 in 𝒮^′ by 𝑆∗𝑇,𝜓∶=lim_𝑛→∞𝑆⊗𝑇,𝜂_𝑛𝜓^Δ,𝜓∈𝒮ℝ^𝑑, (16) whenever the above limit exists for every strong approximate unit (𝜂_𝑛)∈𝕌(ℝ^2𝑑) and 𝜓∈𝒮(ℝ^𝑑). One says then that the convolution 𝑆∗𝑇 exists in 𝒮^′.

In [45], Shiraishi posed the problem of whether the assumption that the convolution 𝑆∗𝑇 of two tempered distributions 𝑆,𝑇∈𝒮^′(ℝ^𝑑) exists in 𝒟^′ implies the existence of the convolution 𝑆∗𝑇 in 𝒮^′ (in particular, whether 𝑆∗𝑇∈𝒮^′(ℝ^𝑑)).

The negative answer to Shiraishi’s problem follows directly from Theorem 3, proved by Kamiński in [56, 57] (see also [58, 63]) and presented in Professor Jan Mikusiński’s seminar in Katowice in 1971 and during the international conference on generalized functions in Rostock in 1972. This also follows from a result obtained independently by Dierolf and Voigt and published in [49]. Dierolf and Voigt constructed in [49] two tempered measures 𝑆 and 𝑇, concentrated on a countable set in ℝ¹, such that the convolution 𝑆∗𝑇 exists in 𝒟^′, but 𝑓∗𝑔∉𝒮^′(ℝ¹).

Theorem 3 is much stronger than the result of Dierolf and Voigt (it supplies counterexamples concerning the growth of the convolution in various spaces of functions and generalized functions).

To get the negative answer to Shiraishi’s problem it is enough to take in Theorem 3 for Φ the function defined by Φ(𝑥)∶=exp𝑥=exp𝜉₁+⋯+𝜉_𝑑,𝑥=𝜉₁,…,𝜉_𝑑∈ℝ^𝑑 (17) (or any continuous function of even faster increase). Clearly, Φ represents a distribution which is not tempered, but from Theorem 3 it follows that there exists a bounded (even vanishing at infinity) 𝐶^∞ function 𝜑 on ℝ^𝑑 representing a tempered distribution for which the convolution 𝜑∗𝜑 exists everywhere and exceeds everywhere on ℝ^𝑑 the function Φ. Consequently, 𝜑∗𝜑 represents a distribution but not a tempered distribution.

4. Convolution of Beurling Ultradistributions

We will recall the definitions of the space of 𝒟^{′(𝑀𝑝)}(ℝ^𝑑) of Beurling ultradistributions (see [9–12, 18, 51]) and the space 𝒮^{′(𝑀𝑝)}(ℝ^𝑑) of Beurling tempered ultradistributions (see [18, 52, 59, 73]) as well as the corresponding structural theorems characterizing elements of these spaces for a fixed numerical sequence (𝑀_𝑝) satisfying conditions (𝑀.1)--(𝑀.3) (see Section 2). Ultradistributions of Roumieu type (see [8]) are not discussed in this paper.

We start by defining Beurling spaces of ultradifferentiable functions. For a given ℎ>0 and a regular compact subset 𝐾 of ℝ^𝑑 (see [10–12]), we define the space 𝒟^(𝑀𝑝)_𝐾,ℎ(ℝ^𝑑), consisting of all functions 𝜑 from ℰ(ℝ^𝑑) with support contained in 𝐾 such that 𝜑_𝐾,ℎ∶=sup_{𝑘∈ℕ𝑑0}sup_𝑥∈𝐾𝜑^(𝑘)(𝑥)ℎ^𝑘𝑀_𝑘<∞, (18) with the topology induced by the above norm. Then the basic space 𝒟^(𝑀𝑝)(ℝ^𝑑) of test functions is defined by means of the projective and inductive limits as follows: 𝒟^(𝑀𝑝)ℝ^𝑑∶=indlim_{𝐾⊂⊂ℝ𝑑}projlim_ℎ→0𝒟^(𝑀𝑝)_𝐾,ℎℝ^𝑑, (19) where the symbol 𝐾⊂⊂ℝ^𝑑 means that 𝐾 are regular compact sets growing up to ℝ^𝑑.

In addition, for a fixed 𝑚>0, we denote by 𝒮^(𝑀𝑝₂^),𝑚(ℝ^𝑑) the space of all 𝐶^∞ functions 𝜑 such that 𝜎_𝑚,2𝜑∶=_{𝛼,𝛽∈ℕ𝑑0}𝑚^𝛼+𝛽𝑀_𝛼𝑀_𝛽⟨⋅⟩^𝛽𝜑^(𝛼)₂^21/2<∞, (20) where the symbol ⟨⋅⟩ is defined in (1), equipped with the topology induced by the above norm 𝜎_𝑚,2. Then we define 𝒮^(𝑀𝑝)ℝ^𝑑∶=projlim_𝑚→∞𝒮^(𝑀𝑝₂^),𝑚ℝ^𝑑. (21)

The strong dual of 𝒟^(𝑀𝑝)(ℝ^𝑑), denoted by 𝒟^{′(𝑀𝑝)}(ℝ^𝑑), is called the space of Beurling ultradistributions.

The following structural theorem (see [10], p. 76) says that Beurling ultradistributions are locally infinite derivatives of measures (continuous functions).

Theorem 7. Let 𝑆∈𝒟^{′(𝑀𝑝)}(ℝ^𝑑). Then, for each open, relatively compact set 𝑈 in ℝ^𝑑, there are measures 𝜇_𝛼∈𝒞^′(𝑈) for 𝛼∈ℕ^𝑑₀ and positive constants 𝐿 and 𝑐 such that 𝑆∣_𝑈=_{𝛼∈ℕ𝑑0}𝜇_𝛼^(𝛼),𝑆_{𝛼𝒞′(𝑈)}≤𝑐𝐿^𝛼𝑀_𝛼𝑓𝑜𝑟𝛼∈ℕ^𝑑₀. (22)

The space 𝒮^{′(𝑀𝑝)}(ℝ^𝑑) of all Beurling tempered ultradistributions is meant as the strong dual of the space 𝒮^(𝑀𝑝)(ℝ^𝑑) defined previously; it was introduced by Pilipović in [73] (see also [18, 52, 59]). Since 𝒟^(𝑀𝑝)(ℝ^𝑑) is dense in 𝒮^(𝑀𝑝)(ℝ^𝑑) and the inclusion mapping is continuous, we have 𝒮^{′(𝑀𝑝)}(ℝ^𝑑)⊂𝒟^{′(𝑀𝑝)}(ℝ^𝑑).

For more details concerning the definitions and properties of the aforementioned spaces of test functions and the spaces 𝒟^{′(𝑀𝑝)} of Beurling ultradistributions and 𝒮^{′(𝑀𝑝)} of Beurling tempered ultradistributions we refer to [18, 51, 52, 59, 60, 73–75].

The following result is a representation theorem for Beurling tempered ultradistributions (see [59]).

Theorem 8. Suppose that 𝑆∈𝒮^{′(𝑀𝑝)}(ℝ^𝑑). Then there are functions 𝐹_𝛼 in 𝐿²(ℝ^𝑑) for 𝛼∈ℕ^𝑑₀ and positive constants 𝜆, 𝐿, and 𝑐 such that 𝑆∣_𝑈=_{𝛼∈ℕ𝑑0}𝐸_𝜆𝐹_𝛼^(𝛼),𝐹_𝛼2≤𝑐𝐿^𝛼𝑀_𝛼for𝛼∈ℕ^𝑑₀, (23) where 𝐸_𝜆(𝑥)=𝑒^{𝑀(𝜆|𝑥|)} for 𝑥∈ℝ^𝑑.

There are various general definitions of the convolution in 𝒟^{′(𝑀𝑝)} of Beurling ultradistributions (see [51]) and of the convolution in 𝒮^{′(𝑀𝑝)} of Beurling tempered ultradistributions (see [52]). They are suitable counterparts of the known general definitions of the convolution in 𝒟^′ and the convolution in 𝒮^′ mentioned in Section 3. The fact that the mentioned definitions of the convolution in 𝒟^{′(𝑀𝑝)} of Beurling ultradistributions are equivalent and that the corresponding definitions of the convolution in 𝒮^{′(𝑀𝑝)} of Beurling tempered ultradistributions are equivalent was proved in [51, 52], respectively (see also [18]).

We will recall here only these definitions of the convolution in 𝒟^{′(𝑀𝑝)} and in 𝒮^{′(𝑀𝑝)} which correspond to Vladimirov’s definition of the convolution in 𝒟^′ and in 𝒮^′, respectively. The definitions are based on the notions of strong 𝒟^(𝑀𝑝)-approximate unit and strong 𝒮^(𝑀𝑝)-approximate unit.

Definition 9. A sequence (𝜂_𝑛) of elements of 𝒟^(𝑀𝑝)(ℝ^𝑑) is said to be a strong 𝒟^(𝑀𝑝)-approximate unit on ℝ^𝑑 if for every 𝐾⊏ℝ^𝑑 there exists an 𝑛₀∈ℕ such that 𝜂_𝑛(𝑥)=1 for 𝑥∈𝐾 and 𝑛≥𝑛₀ (hence 𝜂_𝑛→1 in ℰ^(𝑀𝑝)(ℝ^𝑑) as 𝑛→∞) and, in addition, if there exists a positive constant ℎ such that sup_𝑛∈ℕsup_{𝑘∈ℕ𝑑0}ℎ^𝑘𝑀_𝑘𝜂_𝑛^(𝑘)_∞<∞. (24)

One denotes the set of all strong 𝒟^(𝑀𝑝)-approximate units on ℝ^𝑑 by 𝕌^(𝑀𝑝)(ℝ^𝑑).

Definition 10. If in the previous definition the assumption that 𝜂_𝑛∈𝒟^(𝑀𝑝)(ℝ^𝑑) for 𝑛∈ℕ is replaced by 𝜂_𝑛∈𝒮^(𝑀𝑝)(ℝ^𝑑) for 𝑛∈ℕ and the remaining assumptions are preserved, then the sequence (𝜂_𝑛) is called a strong 𝒮^(𝑀𝑝)-approximate unit. One denotes the set of all strong 𝒮^(𝑀𝑝)-approximate units on ℝ^𝑑 by 𝕌^(𝑀𝑝)_𝑠(ℝ^𝑑).

Remark 11. According to the known Denjoy-Carleman-Mandelbrojt theorem, the previously defined class 𝕌^(𝑀𝑝)(ℝ^𝑑) of strong 𝒟^(𝑀𝑝)-approximate units as well as the class 𝕌^(𝑀𝑝)_𝑠(ℝ^𝑑) of strong 𝒮^(𝑀𝑝)-approximate units contains sufficiently many sequences, for example, all sequences of the form 𝜂_𝑛(𝑥)𝑥=𝜂𝑛,𝑥∈ℝ^𝑑,𝑛∈ℕ, (25) where 𝜂 is a function of class 𝒟^(𝑀𝑝)(ℝ^𝑑) or 𝒮^(𝑀𝑝)(ℝ^𝑑), respectively, such that 𝜂=1 in some neighbourhood of 0.

Vladimirov’s version of the definition of the convolution in 𝒟^{′(𝑀𝑝)} of Beurling ultradistributions has the following form.

Definition 12. For given Beurling ultradistributions 𝑆,𝑇∈𝒟^{′(𝑀𝑝)}(ℝ^𝑑) the convolution 𝑆∗𝑇 in 𝒟^{′(𝑀𝑝)} is defined by 𝑆∗𝑇,𝜑∶=lim_𝑛→∞𝑆⊗𝑇,𝜂_𝑛𝜑^Δ,𝜑∈𝒟^(𝑀𝑝)ℝ^𝑑, (26) whenever the limit in (26) exists for every strong approximate unit (𝜂_𝑛)∈𝕌^(𝑀𝑝)(ℝ^2𝑑) and 𝜑∈𝒟^(𝑀𝑝)(ℝ^𝑑).

Analogously, the convolution in 𝒮^{′(𝑀𝑝)} of Beurling tempered ultradistributions can be defined as follows.

Definition 13. For given two Beurling tempered ultradistributions 𝑆,𝑇∈𝒮^{′(𝑀𝑝)}(ℝ^𝑑) we define the convolution 𝑆∗𝑇 in 𝒮^{′(𝑀𝑝)} by ⟨𝑆∗𝑇,𝜓⟩∶=lim_𝑛→∞𝑆⊗𝑇,𝜂_𝑛𝜓^Δ,𝜓∈𝒮^(𝑀𝑝)ℝ^𝑑, (27) whenever the limit in (27) exists for every strong approximate unit (𝜂_𝑛)∈𝕌^(𝑀𝑝)_𝑠(ℝ^2𝑑) and 𝜓∈𝒮^(𝑀𝑝)(ℝ^𝑑).

5. Compatibility Conditions

Let us present various forms of the condition imposed on sets 𝐴,𝐵⊆ℝ^𝑑 in the context of the convolution of distributions (see [5, p. 383]; see also [6, 7]). They can be expressed in a shorter way by means of the notation introduced in (3).

Let us begin by recalling known equivalent forms of the condition in case the considered sets 𝐴 and 𝐵 are closed in ℝ^𝑑 (see, e.g., [5, p. 383]).

Proposition 14. Let 𝐴,𝐵⊆ℝ^𝑑 be arbitrary closed sets. The following conditions are equivalent:(Σ)(𝐴×𝐵)∩𝐾^▵⊏ℝ^2𝑑 for every 𝐾⊏ℝ^𝑑;(Σ^′)𝐴∩(𝐾−𝐵)⊏ℝ^𝑑 for every 𝐾⊏ℝ^𝑑;(Σ^′′)(𝐾−𝐴)∩𝐵⊏ℝ^𝑑 for every 𝐾⊏ℝ^𝑑.

The meaning of the conditions for 𝑑=1 can be seen in Figure 1 (where 𝑋 and 𝑌 mean ℝ^𝑑).

Figure 1 

Compatible sets.

As proved in [5, pp. 383–384], conditions (Σ)--(Σ^′′) are equivalent if the sets 𝐴 and 𝐵 are closed in ℝ^𝑑. In general, we have the following equivalence (see [76]).

Proposition 15. Let 𝐴,𝐵⊆ℝ^𝑑 be arbitrary sets. The following conditions are equivalent:(Σ_𝑏)(𝐴×𝐵)∩𝐾^Δ is bounded in ℝ^2𝑑 for every bounded set 𝐾 in ℝ^𝑑;(Σ^′_𝑏)𝐴∩(𝐾−𝐵) is bounded in ℝ^𝑑 for every bounded set 𝐾 in ℝ^𝑑;(Σ^′′_𝑏)(𝐾−𝐴)∩𝐵 is bounded in ℝ^𝑑 for every bounded set 𝐾 in ℝ^𝑑;(𝑀)if 𝑥_𝑛∈𝐴 and 𝑦_𝑛∈𝐵 for 𝑛∈ℕ, then lim_𝑛→∞(|𝑥_𝑛|+|𝑦_𝑛|)=∞ implies lim_𝑛→∞|𝑥_𝑛+𝑦_𝑛|=∞.
If the sets 𝐴,𝐵⊆ℝ^𝑑 are closed, then each of the above conditions is equivalent to each of conditions (Σ)--(Σ^′′).

Definition 16. Two sets 𝐴, 𝐵⊆ℝ^𝑑 are called compatible if one of equivalent conditions (Σ_𝑏)--(𝑀) is satisfied.

Remark 17. The condition in the form (𝑀) and the name “compatible sets” were introduced in [54] (see also [7, p. 124–127]).

There are two well known particular cases of compatible sets 𝐴 and 𝐵 in ℝ¹:(1^∘)at least one of the sets 𝐴 and 𝐵 is bounded;(2^∘)both 𝐴 and 𝐵 are bounded from the left or both from the right.

Case (1^∘) extends clearly to ℝ^𝑑 for 𝑑>1 and case (2^∘) can be described in ℝ^𝑑 in the following form: 𝐴,𝐵⊂ℝ^𝑑 are (or are contained in) suitable cones with vertices at 0 such that 𝐴 is an open convex cone and 𝐵⊂𝐴^∗, where 𝐴^∗ means the cone dual to 𝐴 (see [18, pp. 4–6]; [7, pp. 129-130]; [6, pp. 63-64]). For 𝑑=2, case (2^∘) is illustrated in Figure 2 (where 𝑋₁ and 𝑋₂ mean ℝ¹).

Figure 2 

Cones.

Another, less obvious, case of compatible sets in ℝ¹ is the following:(3^∘)both sets 𝐴 and 𝐵 in ℝ¹ are unbounded from both sides, that is, unbounded both from below and from above in ℝ¹.

In [63] (see also [56–58]) various pairs of compatible sets 𝐴 and 𝐵 in ℝ¹, both unbounded from both sides, are considered; for 𝐴=𝐵 in particular. They were constructed as countable sums of intervals (of length 1) situated in ℝ¹ in a specific way. A general idea lies in constructing a sequence of intervals on the positive half-line (with suitable distances between them) and in shifting in a proper way their symmetric counterparts on the negative half-line. Let (𝑝_𝑛)_𝑛∈ℕ0 be a sequence of positive integers. First denote the sets 𝑃_𝑛 of indices 𝑃_𝑛∶={0,1,…,𝑝_𝑛−1} for 𝑛∈ℕ₀. Consider the (doubly indexed) intervals 𝐼_𝑛,𝑘 and 𝐽_𝑛,𝑘 in (−∞,0) and (0,∞), respectively, defined in the following way: 𝐼_𝑛,𝑘∶=𝑎_𝑛,𝑘,𝑎_𝑛,𝑘+1,𝐽_𝑛,𝑘∶=𝑏_𝑛,𝑘,𝑏_𝑛,𝑘+1, (28) where 𝑎_𝑛,𝑘∶=−^𝑛−1_𝑗=02𝑗+1𝑝_𝑗−()(),𝑏2𝑛+1𝑘−𝑛+1_𝑛,𝑘∶=^𝑛−1_𝑗=02𝑗+1𝑝_𝑗+(2𝑛+1)𝑘 (29)(30) for any 𝑘∈𝑃_𝑛 and 𝑛∈ℕ₀ (the first sum on the right in (29) and (30) is meant to be equal to 0 for 𝑛=0). Now define the set 𝐴⊂ℝ¹ by 𝐴∶=_𝑛∈ℕ0𝐴_𝑛∪𝐵_𝑛,where𝐴_𝑛∶=_{𝑘∈𝑃𝑛}𝐼_𝑛,𝑘,𝐵_𝑛∶=_{𝑘∈𝑃𝑛}𝐽_𝑛,𝑘. (31)

It is proved in [56, 57] (see also [58]) and in [63] that the set 𝐴 defined in (31) is compatible with itself. Modifying suitably the previous formulas we can obtain examples of various kinds of compatibility of the set 𝐴 with itself, for example, polynomial compatibility or 𝑀-compatibility considered in the next section. For details see [63] (see also [56–58]).

One can extend case (3^∘) to ℝ^𝑑 in various ways, for example, constructing sets contained in certain infinite spirals in ℝ^𝑑 (see, e.g., the spiral in Figure 3 for 𝑑=2 and 𝐴=𝐵). Examples of a 𝑑-dimensional set which is compatible with itself and unbounded in each direction of ℝ^𝑑 can be obtained in this way (see [63]).

Figure 3 

Spiral.

6. Compatibility of Support and Existence of Convolution

The condition of compatibility (see Definition 16) of the supports of locally integrable functions is sufficient for existence of their convolution in 𝐿¹_loc (see, e.g., [7, p. 124]; see also [5, 6]).

Theorem 18. Let 𝐹,𝐺∈𝐿¹_loc(ℝ^𝑑). If the supports of the functions 𝐹 and 𝐺 are compatible sets, then the convolution 𝐹∗𝐺 exists in 𝐿¹_loc.

The condition of compatibility of supports is also sufficient for existence of the convolution in the space 𝒟^′(ℝ^𝑑) of distributions. The following result is very well known (see, e.g., [7, p. 156]; see also [5, 6]).

Theorem 19. Let 𝑆,𝑇∈𝒟^′(ℝ^𝑑). If the supports of the distributions 𝑆 and 𝑇 are compatible sets, then 𝑆∗𝑇 exists in 𝒟^′ and 𝑆∗𝑇∈𝒟^′(ℝ^𝑑).

However, it follows from Theorem 3 that the counterpart of Theorem 19 for the convolution in 𝒮^′ of two tempered distributions is not true under the assumption of compatibility of their supports. In particular, the compatibility of supports of two functions of the class 𝒫(ℝ^𝑑) does not guarantee that their convolution belongs to this class. This requires a suitable modification of the notion. In [56–58], the following modification of compatibility was introduced.

Definition 20. Two sets 𝐴,𝐵⊆ℝ^𝑑 are polynomially compatible, if there is a positive polynomial 𝑃 on [0,∞) such that the following implication holds: 𝑥∈𝐴,𝑦∈𝐵⟹|𝑥|+𝑦≤𝑃𝑥+𝑦. (32)

The assumption of polynomial compatibility of supports is sufficient for the convolution of two measurable slowly increasing functions to exist in 𝐿¹_loc and to represent a function of the same class (see [56–58]):

Theorem 21. Let 𝐹,𝐺∈𝒫(ℝ^𝑑). If the supports of the functions 𝐹 and 𝐺 are polynomially compatible sets, then 𝐹∗𝐺 exists in 𝐿¹_loc and 𝐹∗𝐺∈𝒫(ℝ^𝑑).

Under the assumption of polynomial compatibility of supports the result concerning the convolution in 𝒮^′ is analogous to Theorem 19. Namely, the condition is sufficient for existence of the convolution of two tempered distributions in 𝒮^′ [56, 57] (see also [50, 58]).

Theorem 22. Let 𝑆,𝑇∈𝒮^′(ℝ^𝑑). If the supports of the tempered distributions 𝑆 and 𝑇 are polynomially compatible sets, then the convolution 𝑆∗𝑇 exists in 𝒮^′ and 𝑆∗𝑇∈𝒮^′(ℝ^𝑑).

The last result was extended in different ways to the case of distributions in the Gelfand-Shilov spaces 𝒦^′(𝑀_𝑝) in [60–62] (see also [76]).

There exist results on the existence of the convolution in 𝒟^{′(𝑀𝑝)} of Beurling ultradistributions and of the convolution in 𝒮^{′(𝑀𝑝)} of Beurling tempered ultradistributions, analogous to Theorems 19 and 22. Let us recall the following existence result concerning the convolution in 𝒟^{′(𝑀𝑝)} of Beurling ultradistributions, proved in [59].

Theorem 23. Let 𝑆,𝑇∈𝒟^{′(𝑀𝑝)}(ℝ^𝑑). If the supports of the Beurling ultradistributions 𝑆 and 𝑇 are compatible sets, then the convolution 𝑆∗𝑇 exists in 𝒟^{′(𝑀𝑝)} and 𝑆∗𝑇∈𝒟^{′(𝑀𝑝)}(ℝ^𝑑).

In [59], a modification of compatibility condition corresponding to the space 𝒮^{′(𝑀𝑝)} of Beurling tempered ultradistributions was introduced via the associated function 𝑀 for the sequence (𝑀_𝑝). We present it below in a slightly relaxed form (see [77]).

Definition 24. Two sets 𝐴,𝐵⊆ℝ^𝑑 are 𝑀-compatible, if there is a constant 𝜃>0 such that 𝑀(|𝑥|)+𝑀(|𝑦|)≤𝑀(𝜃|𝑥+𝑦|)+𝜃 for all 𝑥∈𝐴 and 𝑦∈𝐵.

The following theorem is true (compare [59] and [77]).

Theorem 25. Let 𝑆,𝑇∈𝒮^{′(𝑀𝑝)}(ℝ^𝑑). If the supports the Beurling tempered ultradistributions 𝑆 and 𝑇 are 𝑀-compatible sets, then the convolution 𝑆∗𝑇 exists in 𝒮^{′(𝑀𝑝)} and 𝑆∗𝑇∈𝒮^{′(𝑀𝑝)}(ℝ^𝑑).

7. Inverse Results

It should be noted that the conditions of compatibility and polynomial compatibility of supports of distributions and tempered distributions, assumed in Theorems 19 and 22, are in a sense optimal in terms of supports for the existence of the convolutions in 𝒟^′ and in 𝒮^′, respectively, and they cannot be relaxed. The situation is precisely described by Theorems 26 and 27 formulated below which were proved in [78].

Theorem 26. Let 𝐴,𝐵⊆ℝ^𝑑. Suppose that the convolution 𝑆∗𝑇 exists in 𝒟^′ for each pair 𝑆, 𝑇∈𝒟^′(ℝ^𝑑) of distributions such that supp𝑆⊆𝐴 and supp𝑇⊆𝐵. Then 𝐴 and 𝐵 are compatible.

Theorem 27. Let 𝐴,𝐵⊆ℝ^𝑑. Suppose that the convolution 𝑆∗𝑇 exists in 𝒮^′ for each pair 𝑆,𝑇∈𝒮^′(ℝ^𝑑) of tempered distributions such that supp𝑆⊆𝐴 and supp𝑇⊆𝐵. Then 𝐴 and 𝐵 are polynomially compatible.

We prove next an analogue of Theorem 26 for the case of Beurling ultradistributions.

Theorem 28. Let 𝐴,𝐵⊆ℝ^𝑑. Assume that the convolution 𝑆∗𝑇 exists in 𝒟^{′(𝑀𝑝)} for each pair 𝑆,𝑇∈𝒟^{′(𝑀𝑝)}(ℝ^𝑑) of Beurling ultradistributions with the supports contained in 𝐴 and 𝐵, respectively. Then 𝐴 and 𝐵 are compatible.

Proof. Assume that 𝐴 and 𝐵 are not compatible. This assumption implies that there exist sequences (𝑥_𝑛) and (𝑦_𝑛) of elements of sets 𝐴 and 𝐵, respectively, such that lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑦_𝑛=∞,lim_𝑛→∞𝑥_𝑛+𝑦_𝑛=𝑧inℝ^𝑑forsome𝑧∈ℝ^𝑑. (33)
It is clear that we may assume, in addition, that 𝑥_𝑛+1−𝑥_𝑛𝑦>1,_𝑛+1−𝑦_𝑛𝑥>1,_𝑛+𝑦_𝑛−𝑧<1,for𝑛∈ℕ. (34)
Let us define 𝑆,𝑇∈𝒟^{′(𝑀𝑝)}(ℝ^𝑑) as measures of the form 𝑆∶=^∞_𝑖=1𝛿_𝑥𝑖,𝑇∶=^∞_𝑖=1𝛿_𝑦𝑖, (35) where 𝛿_𝑥 denotes the Dirac measure in 𝒟^{′(𝑀𝑝)}(ℝ^𝑑) concentrated at the point 𝑥∈ℝ^𝑑, that is, the Beurling ultradistribution defined by ⟨𝛿_𝑥,𝜑⟩∶=𝜑(𝑥) for all 𝜑∈𝒟^(𝑀𝑝)(ℝ^𝑑). Obviously, supports of 𝑆 and 𝑇 are contained in 𝐴 and 𝐵, respectively.
Choose nonnegative functions 𝜂∈𝒟^(𝑀𝑝)(ℝ^2𝑑) and 𝜔∈𝒟^(𝑀𝑝)(ℝ^𝑑), identically equal to 1 on the balls 𝐵_2𝑑(0,√2) and 𝐵_𝑑(𝑧,1), respectively. Then define the functions 𝜂_𝑘 by 𝜂_𝑘𝑥𝑥,𝑦∶=𝜂𝑘,𝑦𝑘for𝑥,𝑦∈ℝ^𝑑,𝑘∈ℕ. (36) Hence 𝜔(𝑢)||𝜂=1if𝑢−𝑧≤1,_𝑘𝑥,𝑦=1if|𝑥|≤𝑘,𝑦≤𝑘, (37) for 𝑥,𝑦, and 𝑢∈ℝ^𝑑 and 𝑘∈ℕ. Clearly, (𝜂_𝑘)∈𝕌(ℝ^𝑑).
For sufficiently large 𝑘, we have the inclusion 𝐵_𝑑(𝑧,1)⊂𝐵_𝑑(0,𝑘). For such indices 𝑘, due to (34)–(37), we get 𝑆⊗𝑇,𝜂_𝑘𝜔^Δ≥^∞_𝑖=1𝜂_𝑘𝑥_𝑖,𝑦_𝑖𝜔𝑥_𝑖+𝑦_𝑖≥𝑠_𝑘, (38) where 𝑠_𝑘 is the number of the indices 𝑖 for which 𝑥_𝑖,𝑦_𝑖∈𝐵_𝑑(0,𝑘). Since 𝑠_𝑘→∞ as 𝑘→∞, we conclude that the convolution 𝑆∗𝑇 does not exist in 𝒟^{′(𝑀𝑝)}, which contradicts the assumption. Thus the assertion of the theorem is proved.

Whether Theorem 27 has its counterpart for Beurling tempered ultradistributions remains an open problem.

Problem 29. Let 𝐴,𝐵⊆ℝ^𝑑. Assume that the convolution 𝑆∗𝑇 exists in 𝒮^{′(𝑀𝑝)} for each pair 𝑆, 𝑇∈𝒮^{′(𝑀𝑝)}(ℝ^𝑑) of Beurling ultradistributions with the supports contained in 𝐴 and 𝐵, respectively. Do the sets 𝐴 and 𝐵 have to be 𝑀-compatible?

Let us return now to the case of the convolution in 𝐿¹_loc of locally integrable functions. We will modify for this case the notion of compatible sets expressed by means of certain equivalent conditions (cf. [79]).

Proposition 30. Let 𝐴 and 𝐵 be measurable sets in ℝ^𝑑. The following conditions are equivalent.(i)For each 𝐾⊏ℝ^𝑑 there exists an 𝐿⊏ℝ^𝑑 such that ℓ^𝑑((𝑥−𝐴)∩𝐵∩𝐿^𝑐)=0 for all 𝑥∈𝐾.(ii)For each 𝐾⊏ℝ^𝑑 there is an 𝐿⊏ℝ^𝑑 such thatℓ^𝑑𝑥∈𝐾∶ℓ^𝑑(𝑥−𝐴)∩𝐵∩𝐿^𝑐>0=0. (39)(iii)For arbitrary sequences (𝐴_𝑛) and (𝐵_𝑛) of positively bounded subsets of 𝐴 and 𝐵, respectively, the following implication holds:lim_𝑛→∞𝐴_𝑛∞+𝐵_𝑛∞=∞𝑖𝑚𝑝𝑙𝑖𝑒𝑠lim_𝑛→∞𝐴_𝑛+𝐵_𝑛∞=∞. (40)(iv)For arbitrary sequences (𝑥_𝑛) and (𝑦_𝑛) of massive points of the sets 𝐴 and 𝐵, respectively, the following implication holds:lim_𝑛→∞𝑥_𝑛+𝑦_𝑛=∞𝑖𝑚𝑝𝑙𝑖𝑒𝑠lim_𝑛→∞𝑥_𝑛+𝑦_𝑛=∞. (41)

Definition 31. One calls the sets 𝐴,𝐵⊂ℝ^𝑑 quasi-compatible if one of conditions (i)–(iv) is satisfied.

It is clear that if measurable sets 𝐴,𝐵⊂ℝ^𝑑 are compatible, then they are quasi-compatible, but not conversely.

The proof of Proposition 30 is based on the two following lemmas (see [79]).

Lemma 32. If a measurable set 𝐴⊂ℝ^𝑑 is positively bounded, then for every 𝜀 with 0<𝜀<|𝐴|_∞ there exists a massive point 𝑥 of 𝐴 such that |𝑥|≥|𝐴|_∞−𝜀.

Lemma 33. For any two measurable sets 𝐴 and 𝐵 in ℝ^𝑑 such that ℓ^𝑑(𝐴)>0 and ℓ^𝑑(𝐵)>0 there exists a pair (𝑥,𝐶), where 𝑥 is a point in ℝ^𝑑 and 𝐶 is a measurable subset of 𝐵 with ℓ^𝑑(𝐶)>0, having the following property: for every 𝑦∈𝐶 there exists a measurable subset 𝐷_𝑦 of 𝐵 with ℓ^𝑑(𝐷_𝑦)>0 such that the inclusions 𝑥+𝑦−𝐷_𝑦⊆𝐴 hold for all 𝑦∈𝐶.

The following assertion is an easy corollary from Theorem 18.

Theorem 34. If 𝐹,𝐺∈𝐿¹_loc(ℝ^𝑑) and the supports of the functions 𝐹 and 𝐺 are (contained in) quasi-compatible sets, then the convolution 𝐹∗𝐺 exists and represents a locally integrable function.

The condition of quasi-compatibility of measurable sets in ℝ^𝑑 can be described in the following way in the context of the existence of the convolutions in the space 𝐿¹_loc(ℝ^𝑑) of locally integrable functions (cf. [79]).

Theorem 35. Let 𝐴 and 𝐵 be measurable sets in ℝ^𝑑. Suppose that convolution 𝐹∗𝐺 exists and represents a locally integrable function in ℝ^𝑑 for any 𝐹,𝐺∈𝐿¹_loc(ℝ^𝑑) whose supports are contained in 𝐴 and 𝐵, respectively. Then 𝐴 and 𝐵 are quasi-compatible.

Proof. Suppose that 𝐴 and 𝐵 are not quasi-compatible sets in ℝ^𝑑. Then, by Proposition 30, one can find two sequences (𝑥_𝑛) and (𝑦_𝑛) of massive points of the sets 𝐴 and 𝐵, respectively, such that lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑦_𝑛=∞,𝜌∶=sup_𝑛∈ℕ𝑥_𝑛+𝑦_𝑛<∞. (42) We may assume, in addition, that 𝑥_𝑛+1−𝑥_𝑛>2,𝑦_𝑛+1−𝑦_𝑛>2,for𝑛∈ℕ. (43)
Let 𝐵(𝑥_𝑛) and 𝐵(𝑦_𝑛) be the closed balls in ℝ^𝑑 of radius 1 with centers at 𝑥_𝑛 and 𝑦_𝑛, respectively, for 𝑛∈ℕ. Let 𝐶 be the closed ball with the center at 0 and radius 𝜌+2. Put 𝐴_𝑛∶=𝐴∩𝐵𝑥_𝑛,𝐵_𝑛∶=𝐵∩𝐵𝑦_𝑛,for𝑛∈ℕ. (44)
By (42), (43), and Proposition 30, we have 𝐴_𝑛+𝐵_𝑛⊆𝐶 for all 𝑛∈ℕ and, moreover, 𝐴_𝑛∩𝐴_𝑚=∅ and 𝐵_𝑛∩𝐵_𝑚=∅ for 𝑛≠𝑚; in addition, ℓ^𝑑(𝐴_𝑛)>0 and ℓ^𝑑(𝐵_𝑛)>0 for all 𝑛∈ℕ.
Define the functions 𝐹,𝐺, and 𝜓 in the following way: 𝐹=^∞_𝑛=1𝛼_𝑛𝜒_𝐴𝑛,𝐺=^∞_𝑛=1𝛽_𝑛𝜒_𝐵𝑛,𝜓=𝜒_𝐶, (45) where 𝛼_𝑛=[ℓ^𝑑(𝐴_𝑛)]⁻¹ and 𝛽_𝑛=[ℓ^𝑑(𝐵_𝑛)]⁻¹ for 𝑛∈ℕ. According to (44) and (43), 𝐹 and 𝐺 are well defined nonnegative locally integrable functions with the supports contained in 𝐴 and 𝐵, respectively. Moreover, 𝜓∈𝐿¹_loc and supp𝜓=𝐶⊏ℝ^𝑑. Hence _ℝ𝑑𝐹|∗|𝐺𝑥)𝜓(𝑥)≥(||)(𝑑𝑥_{∞𝑛=1𝐴𝑛𝐵𝑛}𝐹(𝑠)𝐺(𝑡)𝜒_𝐶(≥𝑠+𝑡)𝑑𝑠𝑑𝑡_∞𝑛=1𝛼_𝑛𝛽_𝑛𝐴𝑛𝑑𝑠_𝐵𝑛𝑑𝑡=∞, (46) which contradicts our assumption that 𝐹∗𝐺 exists in 𝐿¹_loc and thus the proof of our assertion is completed.

Acknowledgments

The authors thank the referees for valuable comments which helped in improving the paper. Special thanks are addressed to Mr. K. Majchrowski for preparing the figures placed in the text. This work was partly supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge.