# Compatibility Conditions and the Convolution of Functions and Generalized Functions

Authors: Andrzej Kamiński; Svetlana Mincheva-Kaminska
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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.


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## Body

## 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 ℝd 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 ℝd 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 ℝd 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 ℝd.


## 2. Preliminaries

We will use the standard multidimensional notation concerningℝd and ℕ0d, 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 ℝd and
(1)〈x〉:=(1+|x|2)1/2=(1+ξ12+⋯+ξd2)1/2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiix=(ξ1,…,ξd)∈ℝd,
but we write also |k|:=κ1+⋯+κd for k=(κ1,…,κd)∈ℕ0d, though ℕ0d⊂ℝd.The standard notation will be also used for the known spaces of (complex-valued) functions onℝd: Lr(ℝd) for r∈[1,∞] (with the norm denoted by ∥·∥r), Lloc1(ℝd), 𝒞(ℝd), 𝒞∞(ℝd), ℰ(ℝd), 𝒟(ℝd), 𝒟K(ℝd), 𝒮(ℝd), and the known (sub) spaces of distributions on ℝd: 𝒞′(ℝd), 𝒟′(ℝd), and 𝒮′(ℝd) (cf. [1, 7]).For a setE⊆ℝd we denote Ec:=ℝd∖E. It will be convenient to use the following notation for E⊆ℝd and a function ϕ on ℝd:
(2)EΔ:={(x,y)∈ℝ2d:x+y∈E},ϕΔ(x,y):=ϕ(x+y),x,y∈ℝd.Clearly, for anyE⊆ℝd, the set EΔ is bounded (compact) in ℝ2d if and only if E=∅. Therefore if ϕ∈𝒟(ℝd), then ϕΔ∈𝒞∞(ℝ2d), but ϕΔ∈𝒟(ℝ2d) only in case ϕ=0. Moreover, the symbol ⊏ℝd will be used in the following sense:
(3)E⊏ℝdwheneverEisacompactsubsetofℝd.The words “measure” and “measurable” concerning the sets inℝd are considered in the sense of Lebesgue; the Lebesgue measure of a given Lebesgue measurable set A⊆ℝd will be denoted by ℓd(A). If A⊆ℝd is a measurable set, we use the following notation:
(4)|A|∞:=supess{|x|:x∈A}.
That is, |A|∞=infΛ(A) in case Λ(A)≠∅ and |A|∞=∞ in case Λ(A)=∅, where Λ(A):={α>0:ℓd(Xα)=0} with Xα:={x∈A:|x|>α}. Clearly,
(5)|A|∞=0iffℓd(A)=0.We call a measurable setA⊂ℝdpositively bounded in ℝd whenever 0<|A|∞<∞. An element x of a measurable set A∈ℝd is called a massive point of the set A if ℓd(A∩U(x))>0 for each open neighbourhood U(x) of x in ℝd.A measurable functionF on ℝd 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 ℝd (resp., for every polynomial P and some positive constant c(P)) and for almost all x∈ℝd. The sets of all measurable slowly increasing and rapidly decreasing functions on ℝd will be denoted by 𝒫(ℝd) and ℛ(ℝd), respectively.In our considerations concerning the convolution in the spaces𝒟′(Mp)(ℝd) of ultradistributions and 𝒮′(Mp)(ℝd) of tempered ultradistributions of Beurling type (the respective definitions and notation are introduced in Section 4) we always assume that a given sequence (Mp)=(Mp)p∈ℕ0 of positive numbers satisfies the following three conditions (see [10–12, 18, 51, 52, 59]):(M.1)
Mp2≤Mp-1Mp+1,p∈ℕ;(M.2)
Mp≤abpMqMp-q,p,q∈ℕ,0≤q≤p;(M.3)
∑p=q+1∞‍Mp-1Mp-1≤aqMqMp-1,q∈ℕ, where a>0 and b>0 are some constants. We extend the sequence (Mp)p∈ℕ0 to its multidimensional version (Mk)k∈ℕ0d as follows:
(6)Mk:=Mκ1+⋯+κd,k=(κ1,…,κd)∈ℕ0d.By theassociated function for the sequence (Mp) we mean the function on [0,∞) given by
(7)M(t):=supp∈ℕ0log+(tpMp),t>0,
where log+:=max{logt,0} and M(0):=0.


## 3. Convolution of Functions and Distributions

Definition 1.
LetF and G be Lebesgue measurable functions on ℝd. For a given x∈ℝd one defines
(8)(F*G)(x):=∫ℝd‍F(x-t)G(t)dt,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∈ℝd, then one says that the convolution F*Gexists everywhere (almost everywhere) on ℝd.Definition 2.
LetF,G∈Lloc1(ℝd). One says that the convolution F*G exists in Lloc1 if F*G exists almost everywhere on ℝd and |F|*|G|∈Lloc1(ℝd).Let us list some classical results concerning the existence inLloc1 of the convolution of functions of certain classes:(a)
ifF,G∈L1(ℝd), then F*G exists in Lloc1 and F*G∈L1(ℝd);(b)
ifF∈Lp(ℝd) and G∈Lq(ℝd), where p,q∈[1,∞) such that 1/p+1/q>1, then F*G exists in Lloc1 and F*G∈Lr(ℝd), where r=(1/p+1/q-1)-1 (Young);(c)
ifF∈𝒫(ℝd) and G∈ℛ(ℝd), then F*G exists in Lloc1 and F*G∈𝒫(ℝd).However, in general, the existence of the convolution of functions inLloc1 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 Lloc1, 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.
LetΦ be an arbitrary continuous (complex-valued) function on ℝd. There exists a nonnegative C∞ function φ on ℝd such that
(9)lim|x|→∞φ(x)=0,
the convolutionφ*φ exists everywhere in ℝd, and the following inequality is satisfied:
(10)(φ*φ)(x)>|Φ(x)|∀x∈ℝd.
Moreover, the functionφ 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∈ℝd; 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]).Theorem3 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′(ℝd) and 𝒮′(Mp)(ℝd) of the spaces 𝒟′(ℝd) and 𝒟′(Mp)(ℝd) 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′(ℝd) of the basic space 𝒟(ℝd) of test functions, that is, linear continuous functionals on the space 𝒟(ℝd) defined as the inductive limit:
(11)𝒟(ℝd):=indlimK⊏⊏ℝd𝒟K(ℝd),
where the symbol K⊏⊏ℝd means here that K are compact sets growing up to ℝd. Recall that 𝒟K(ℝd), for a fixed K⊏ℝd, means the space of all C∞ functions on ℝd whose support is contained in K, with the topology given by the family {∥·∥K,m:m∈ℕ0} of the norms defined as follows:
(12)∥φ∥K,m:=sup{|φ(k)(x)|:x∈K,k∈ℕ0d,|k|≤m},hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhiim∈ℕ0.
Locally integrable functions on ℝd may be treated as distributions (so-called regular distributions), because the space Lloc1(ℝd) can be naturally embedded into 𝒟′(ℝd). The known structural theorem describes every distribution as a derivative (in the distributional sense) locally, that is, on each relatively compact set in ℝd, of finite order of a continuous function (see [1, 5–7]).Recall that tempered distributions are elements of the dual𝒮′(ℝd) of the complete metric space 𝒮(ℝd) of rapidly decreasing C∞ functions, that is, of all φ∈𝒞∞(ℝd) such that
(13)σm(φ):=max{∥〈·〉iφ(k)∥∞:i,k∈ℕ0d,|i|,|k|≤m}<∞
for all m∈ℕ0, where 〈·〉 is defined by (1), with the metric topology induced by the family {σm:m∈ℕ0} of the norms defined in (13). Since 𝒟(ℝd) is dense in 𝒮(ℝd), the space 𝒮′(ℝd) of tempered distributions can be naturally embedded into 𝒟′(ℝd). 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 ℝd (see [1–7]).There are several general definitions of the convolution in𝒟′ 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(ηn) of elements of 𝒟(ℝd) is said to be a strong approximate unit on ℝd if for every K⊏ℝd there exists an n0∈ℕ such that ηn(x)=1 for x∈K and n≥n0 (hence ηn→1 in ℰ(ℝd) as n→∞) and, in addition,
(14)supn∈ℕ∥ηn(k)∥∞<∞foreveryk∈ℕ0d.One denotes the set of all strong approximate units onℝd by 𝕌¯(ℝd).Definition 5.
For givenS,T∈𝒟′(ℝd) the convolution S*T in 𝒟′ is defined by
(15)〈S*T,φ〉:=limn→∞〈S⊗T,ηnφΔ〉,φ∈𝒟(ℝd),
whenever the above limit exists for every strong approximate unit (ηn)∈𝕌¯(ℝ2d) and φ∈𝒟(ℝd). One says then that the convolution S*T exists in 𝒟′.By the Lebesgue theorem, ifF,G∈Lloc1(ℝd) and the convolution F*G exists in Lloc1, then it exists in 𝒟′ and represents a regular distribution in 𝒟′(ℝd) (see [6, p. 63]).Analogously to and independently of Definition5, 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 givenS,T∈𝒮′(ℝd) one defines the convolution S*T in 𝒮′ by
(16)〈S*T,ψ〉:=limn→∞〈S⊗T,ηnψΔ〉,ψ∈𝒮(ℝd),
whenever the above limit exists for every strong approximate unit (ηn)∈𝕌¯(ℝ2d) and ψ∈𝒮(ℝd). 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′(ℝd) exists in 𝒟′ implies the existence of the convolution S*T in 𝒮′ (in particular, whether S*T∈𝒮′(ℝd)).The negative answer to Shiraishi’s problem follows directly from Theorem3, 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).Theorem3 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 Theorem3 for Φ the function defined by
(17)Φ(x):=expx=exp(ξ1+⋯+ξd),x=(ξ1,…,ξd)∈ℝd
(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 ℝd representing a tempered distribution for which the convolution φ*φ exists everywhere and exceeds everywhere on ℝd 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 of𝒟′(Mp)(ℝd) of Beurling ultradistributions (see [9–12, 18, 51]) and the space 𝒮′(Mp)(ℝd) 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 (Mp) 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 ofultradifferentiable functions. For a given h>0 and a regular compact subset K of ℝd (see [10–12]), we define the space 𝒟K,h(Mp)(ℝd), consisting of all functions φ from ℰ(ℝd) with support contained in K such that
(18)∥φ∥K,h:=supk∈ℕ0dsupx∈K|φ(k)(x)|hkMk<∞,
with the topology induced by the above norm. Then the basic space 𝒟(Mp)(ℝd) of test functions is defined by means of the projective and inductive limits as follows:
(19)𝒟(Mp)(ℝd):=indlimK⊂⊂ℝdprojlimh→0𝒟K,h(Mp)(ℝd),
where the symbol K⊂⊂ℝd means that K are regular compact sets growing up to ℝd.In addition, for a fixedm>0, we denote by 𝒮2(Mp),m(ℝd) the space of all C∞ functions φ such that
(20)σm,2(φ):=[∑α,β∈ℕ0d‍(mα+βMαMβ∥〈·〉βφ(α)∥2)2]1/2<∞,
where the symbol 〈·〉 is defined in (1), equipped with the topology induced by the above norm σm,2. Then we define
(21)𝒮(Mp)(ℝd):=projlimm→∞𝒮2(Mp),m(ℝd).The strong dual of𝒟(Mp)(ℝd), denoted by 𝒟′(Mp)(ℝd), 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.
LetS∈𝒟′(Mp)(ℝd). Then, for each open, relatively compact set U in ℝd, there are measures μα∈𝒞′(U¯) for α∈ℕ0d and positive constants L and c such that
(22)S∣U=∑α∈ℕ0d‍μα(α),∥Sα∥𝒞′(U¯)≤cLαMαforα∈ℕ0d.The space𝒮′(Mp)(ℝd) of all Beurling tempered ultradistributions is meant as the strong dual of the space 𝒮(Mp)(ℝd) defined previously; it was introduced by Pilipovic´ in [73] (see also [18, 52, 59]). Since 𝒟(Mp)(ℝd) is dense in 𝒮(Mp)(ℝd) and the inclusion mapping is continuous, we have 𝒮′(Mp)(ℝd)⊂𝒟′(Mp)(ℝd).For more details concerning the definitions and properties of the aforementioned spaces of test functions and the spaces𝒟′(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 thatS∈𝒮′(Mp)(ℝd). Then there are functions Fα in L2(ℝd) for α∈ℕ0d and positive constants λ, L, and c such that
(23)S∣U=∑α∈ℕ0d‍(EλFα)(α),∥Fα∥2≤cLαMαforα∈ℕ0d,
where Eλ(x)=eM(λ|x|) for x∈ℝd.There are various general definitions of the convolution in𝒟′(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 in𝒟′(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(ηn) of elements of 𝒟(Mp)(ℝd) is said to be a strong 𝒟(Mp)-approximate unit on ℝd if for every K⊏ℝd there exists an n0∈ℕ such that ηn(x)=1 for x∈K and n≥n0 (hence ηn→1 in ℰ(Mp)(ℝd) as n→∞) and, in addition, if there exists a positive constant h such that
(24)supn∈ℕsupk∈ℕ0d(hkMk∥ηn(k)∥∞)<∞.One denotes the set of all strong𝒟(Mp)-approximate units on ℝd by 𝕌¯(Mp)(ℝd).Definition 10.
If in the previous definition the assumption thatηn∈𝒟(Mp)(ℝd) for n∈ℕ is replaced by ηn∈𝒮(Mp)(ℝd) for n∈ℕ and the remaining assumptions are preserved, then the sequence (ηn) is called a strong 𝒮(Mp)-approximate unit. One denotes the set of all strong 𝒮(Mp)-approximate units on ℝd by 𝕌¯s(Mp)(ℝd).Remark 11.
According to the known Denjoy-Carleman-Mandelbrojt theorem, the previously defined class𝕌¯(Mp)(ℝd) of strong 𝒟(Mp)-approximate units as well as the class 𝕌¯s(Mp)(ℝd) of strong 𝒮(Mp)-approximate units contains sufficiently many sequences, for example, all sequences of the form
(25)ηn(x)=η(xn),x∈ℝd,n∈ℕ,
where η is a function of class 𝒟(Mp)(ℝd) or 𝒮(Mp)(ℝd), respectively, such that η=1 in some neighbourhood of 0.Vladimirov’s version of the definition of the convolution in𝒟′(Mp) of Beurling ultradistributions has the following form.Definition 12.
For given Beurling ultradistributionsS,T∈𝒟′(Mp)(ℝd) the convolution S*T in 𝒟′(Mp) is defined by
(26)〈S*T,φ〉:=limn→∞〈S⊗T,ηnφΔ〉,φ∈𝒟(Mp)(ℝd),
whenever the limit in (26) exists for every strong approximate unit (ηn)∈𝕌¯(Mp)(ℝ2d) and φ∈𝒟(Mp)(ℝd).Analogously, the convolution in𝒮′(Mp) of Beurling tempered ultradistributions can be defined as follows.Definition 13.
For given two Beurling tempered ultradistributionsS,T∈𝒮′(Mp)(ℝd) we define the convolution S*T in 𝒮′(Mp) by
(27)〈S*T,ψ〉:=limn→∞〈S⊗T,ηnψΔ〉,ψ∈𝒮(Mp)(ℝd),
whenever the limit in (27) exists for every strong approximate unit (ηn)∈𝕌¯s(Mp)(ℝ2d) and ψ∈𝒮(Mp)(ℝd).


## 5. Compatibility Conditions

Let us present various forms of the condition imposed on setsA,B⊆ℝd 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 setsA and B are closed in ℝd (see, e.g., [5, p. 383]).Proposition 14.
LetA,B⊆ℝd be arbitrary closed sets. The following conditions are equivalent:(Σ)
(A×B)∩K▵⊏ℝ2dfor everyK⊏ℝd;(Σ′)
A∩(K-B)⊏ℝdfor everyK⊏ℝd;(Σ′′)
(K-A)∩B⊏ℝdfor everyK⊏ℝd.The meaning of the conditions ford=1 can be seen in Figure 1 (where X and Y mean ℝd).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 ℝd. In general, we have the following equivalence (see [76]).Proposition 15.
LetA,B⊆ℝd be arbitrary sets. The following conditions are equivalent:(Σb)
(A×B)∩KΔis bounded inℝ2dfor every bounded setKinℝd;(Σb′)
A∩(K-B)is bounded inℝdfor every bounded setKinℝd;(Σb′′)
(K-A)∩Bis bounded inℝdfor every bounded setKinℝd;(M)
ifxn∈Aandyn∈Bforn∈ℕ, thenlimn→∞(|xn|+|yn|)=∞implieslimn→∞|xn+yn|=∞.
If the setsA,B⊆ℝd are closed, then each of the above conditions is equivalent to each of conditions (Σ)–(Σ′′).Definition 16.
Two setsA, B⊆ℝd are called compatible if one of equivalent conditions (Σb)–(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 setsA and B in ℝ1:(1∘)
at least one of the setsA and B is bounded;(2∘)
bothA and B are bounded from the left or both from the right.Case(1∘) extends clearly to ℝd for d>1 and case (2∘) can be described in ℝd in the following form: A,B⊂ℝd 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 inℝ1 is the following:(3∘)
both setsA 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 (pn)n∈ℕ0 be a sequence of positive integers. First denote the sets Pn of indices Pn:={0,1,…,pn-1} for n∈ℕ0. Consider the (doubly indexed) intervals In,k and Jn,k in (-∞,0) and (0,∞), respectively, defined in the following way:
(28)In,k:=[an,k,an,k+1),Jn,k:=(bn,k,bn,k+1],
where
(29)an,k:=-∑j=0n-1‍(2j+1)pj-(2n+1)k-(n+1),(30)bn,k:=∑j=0n-1‍(2j+1)pj+(2n+1)k
for any k∈Pn 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
(31)A:=⋃n∈ℕ0‍(An∪Bn),whereAn:=⋃k∈Pn‍In,k,Bn:=⋃k∈Pn‍Jn,k.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 ℝd in various ways, for example, constructing sets contained in certain infinite spirals in ℝd (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 ℝd can be obtained in this way (see [63]).Figure 3
Spiral.


## 6. Compatibility of Support and Existence of Convolution

The condition of compatibility (see Definition16) of the supports of locally integrable functions is sufficient for existence of their convolution in Lloc1 (see, e.g., [7, p. 124]; see also [5, 6]).Theorem 18.
LetF,G∈Lloc1(ℝd). If the supports of the functions F and G are compatible sets, then the convolution F*G exists in Lloc1.The condition of compatibility of supports is also sufficient for existence of the convolution in the space𝒟′(ℝd) of distributions. The following result is very well known (see, e.g., [7, p. 156]; see also [5, 6]).Theorem 19.
LetS,T∈𝒟′(ℝd). If the supports of the distributions S and T are compatible sets, then S*T exists in 𝒟′ and S*T∈𝒟′(ℝd).However, it follows from Theorem3 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(ℝd) 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 setsA,B⊆ℝd are polynomially compatible, if there is a positive polynomial P on [0,∞) such that the following implication holds:
(32)x∈A,y∈B⟹|x|+|y|≤P(|x+y|).The assumption of polynomial compatibility of supports is sufficient for the convolution of two measurable slowly increasing functions to exist inLloc1 and to represent a function of the same class (see [56–58]):Theorem 21.
LetF,G∈𝒫(ℝd). If the supports of the functions F and G are polynomially compatible sets, then F*G exists in Lloc1 and F*G∈𝒫(ℝd).Under the assumption of polynomial compatibility of supports the result concerning the convolution in𝒮′ 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.
LetS,T∈𝒮′(ℝd). 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′(ℝd).The last result was extended in different ways to the case of distributions in the Gelfand-Shilov spaces𝒦′(Mp) in [60–62] (see also [76]).There exist results on the existence of the convolution in𝒟′(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.
LetS,T∈𝒟′(Mp)(ℝd). 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)(ℝd).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 (Mp). We present it below in a slightly relaxed form (see [77]).Definition 24.
Two setsA,B⊆ℝd 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.
LetS,T∈𝒮′(Mp)(ℝd). 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)(ℝd).


## 7. Inverse Results

It should be noted that the conditions of compatibility and polynomial compatibility of supports of distributions and tempered distributions, assumed in Theorems19 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.
LetA,B⊆ℝd. Suppose that the convolution S*T exists in 𝒟′ for each pair S, T∈𝒟′(ℝd) of distributions such that suppS⊆A and suppT⊆B. Then A and B are compatible.Theorem 27.
LetA,B⊆ℝd. Suppose that the convolution S*T exists in 𝒮′ for each pair S,T∈𝒮′(ℝd) of tempered distributions such that suppS⊆A and suppT⊆B. Then A and B are polynomially compatible.We prove next an analogue of Theorem26 for the case of Beurling ultradistributions.Theorem 28.
LetA,B⊆ℝd. Assume that the convolution S*T exists in 𝒟′(Mp) for each pair S,T∈𝒟′(Mp)(ℝd) of Beurling ultradistributions with the supports contained in A and B, respectively. Then A and B are compatible.Proof.
Assume thatA and B are not compatible. This assumption implies that there exist sequences (xn) and (yn) of elements of sets A and B, respectively, such that
(33)limn→∞|xn|=limn→∞|yn|=∞,limn→∞(xn+yn)=zhhhhhhhhhhhhhihhhhiiinℝdforsomez∈ℝd.
It is clear that we may assume, in addition, that(34)|xn+1|-|xn|>1,|yn+1|-|yn|>1,|xn+yn-z|<1,iiiiiiiiiiiiforn∈ℕ.
Let us defineS,T∈𝒟′(Mp)(ℝd) as measures of the form
(35)S:=∑i=1∞‍δxi,T:=∑i=1∞‍δyi,
where δx denotes the Dirac measure in 𝒟′(Mp)(ℝd) concentrated at the point x∈ℝd, that is, the Beurling ultradistribution defined by 〈δx,φ〉:=φ(x) for all φ∈𝒟(Mp)(ℝd). Obviously, supports of S and T are contained in A and B, respectively.
Choose nonnegative functionsη∈𝒟(Mp)(ℝ2d) and ω∈𝒟(Mp)(ℝd), identically equal to 1 on the balls B¯2d(0,2) and B¯d(z,1), respectively. Then define the functions ηk by
(36)ηk(x,y):=η(xk,yk)forx,y∈ℝd,k∈ℕ.
Hence
(37)ω(u)=1if|u-z|≤1,ηk(x,y)=1if|x|≤k,|y|≤k,
for x,y, and u∈ℝd and k∈ℕ. Clearly, (ηk)∈𝕌¯(ℝd).
For sufficiently largek, we have the inclusion B¯d(z,1)⊂B¯d(0,k). For such indices k, due to (34)–(37), we get
(38)〈S⊗T,ηkωΔ〉≥∑i=1∞‍ηk(xi,yi)ω(xi+yi)≥sk,
where sk is the number of the indices i for which xi,yi∈B¯d(0,k). Since sk→∞ 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 Theorem27 has its counterpart for Beurling tempered ultradistributions remains an open problem.Problem 29.
LetA,B⊆ℝd. Assume that the convolution S*T exists in 𝒮′(Mp) for each pair S, T∈𝒮′(Mp)(ℝd) 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 inLloc1 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.
LetA and B be measurable sets in ℝd. The following conditions are equivalent.(i)
For eachK⊏ℝdthere exists anL⊏ℝdsuch thatℓd((x-A)∩B∩Lc)=0for allx∈K.(ii)
For eachK⊏ℝdthere is anL⊏ℝdsuch that(39)ℓd({x∈K:ℓd((x-A)∩B∩Lc)>0})=0.(iii)
For arbitrary sequences(An)and(Bn)of positively bounded subsets ofAandB, respectively, the following implication holds:(40)limn→∞(|An|∞+|Bn|∞)=∞implieslimn→∞|An+Bn|∞=∞.(iv)
For arbitrary sequences(xn)and(yn)of massive points of the setsAandB, respectively, the following implication holds:(41)limn→∞(|xn|+|yn|)=∞implieslimn→∞|xn+yn|=∞.Definition 31.
One calls the setsA,B⊂ℝd quasi-compatible if one of conditions (i)–(iv) is satisfied.It is clear that if measurable setsA,B⊂ℝd are compatible, then they are quasi-compatible, but not conversely.The proof of Proposition30 is based on the two following lemmas (see [79]).Lemma 32.
If a measurable setA⊂ℝd is positively bounded, then for every ɛ with 0<ɛ<|A|∞ there exists a massive point x of A such that |x|≥|A|∞-ɛ.Lemma 33.
For any two measurable setsA and B in ℝd such that ℓd(A)>0 and ℓd(B)>0 there exists a pair (x,C), where x is a point in ℝd and C is a measurable subset of B with ℓd(C)>0, having the following property: for every y∈C there exists a measurable subset Dy of B with ℓd(Dy)>0 such that the inclusions x+y-Dy⊆A hold for all y∈C.The following assertion is an easy corollary from Theorem18.Theorem 34.
IfF,G∈Lloc1(ℝd) 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 inℝd can be described in the following way in the context of the existence of the convolutions in the space Lloc1(ℝd) of locally integrable functions (cf. [79]).Theorem 35.
LetA and B be measurable sets in ℝd. Suppose that convolution F*G exists and represents a locally integrable function in ℝd for any F,G∈Lloc1(ℝd) whose supports are contained in A and B, respectively. Then A and B are quasi-compatible.Proof.
Suppose thatA and B are not quasi-compatible sets in ℝd. Then, by Proposition 30, one can find two sequences (xn) and (yn) of massive points of the sets A and B, respectively, such that
(42)limn→∞|xn|=limn→∞|yn|=∞,ρ:=supn∈ℕ|xn+yn|<∞.
We may assume, in addition, that
(43)|xn+1|-|xn|>2,|yn+1|-|yn|>2,forn∈ℕ.
LetB¯(xn) and B¯(yn) be the closed balls in ℝd of radius 1 with centers at xn and yn, respectively, for n∈ℕ. Let C¯ be the closed ball with the center at 0 and radius ρ+2. Put
(44)An:=A∩B¯(xn),Bn:=B∩B¯(yn),forn∈ℕ.
By (42), (43), and Proposition 30, we have An+Bn⊆C¯ for all n∈ℕ and, moreover,  An∩Am=∅ and Bn∩Bm=∅ for n≠m; in addition, ℓd(An)>0 and ℓd(Bn)>0 for all n∈ℕ.
Define the functionsF,G, and ψ in the following way:
(45)F=∑n=1∞‍αnχAn,G=∑n=1∞‍βnχBn,ψ=χC¯,
where αn=[ℓd(An)]-1 and βn=[ℓd(Bn)]-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∈Lloc1 and suppψ=C¯⊏ℝd. Hence
(46)∫ℝd(|F|*|G|)(x)ψ(x)dx≥∑n=1∞∫An∫Bn[F(s)G(t)χC¯(s+t)]dsdt≥∑n=1∞αnβn∫Ands∫Bndt=∞,
which contradicts our assumption that F*G exists in Lloc1 and thus the proof of our assertion is completed.



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