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Abstract and Applied Analysis
VolumeΒ 2009, Article IDΒ 307404, 18 pages
http://dx.doi.org/10.1155/2009/307404
Research Article

Convolutions with the Continuous Primitive Integral

Department of Mathematics and Statistics, University of the Fraser Valley, Abbotsford, BC, Canada V2S 7M8,

Received 13 May 2009; Accepted 7 September 2009

Academic Editor: H. BevanΒ Thompson

Copyright Β© 2009 Erik Talvila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

If 𝐹 is a continuous function on the real line and 𝑓=πΉξ…ž is its distributional derivative, then the continuous primitive integral of distribution 𝑓 is βˆ«π‘π‘Žπ‘“=𝐹(𝑏)βˆ’πΉ(π‘Ž). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolution βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦 for 𝑓 an integrable distribution and 𝑔 a function of bounded variation or an 𝐿1 function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For 𝑔 of bounded variation, π‘“βˆ—π‘” is uniformly continuous and we have the estimate β€–π‘“βˆ—π‘”β€–βˆžβ‰€β€–π‘“β€–β€–π‘”β€–β„¬π’±, where ‖𝑓‖=sup𝐼|βˆ«πΌπ‘“| is the Alexiewicz norm. This supremum is taken over all intervals πΌβŠ‚β„. When π‘”βˆˆπΏ1, the estimate is β€–π‘“βˆ—π‘”β€–β‰€β€–π‘“β€–β€–π‘”β€–1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

1. Introduction and Notation

The convolution of two functions 𝑓 and 𝑔 on the real line is βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦. Convolutions play an important role in pure and applied mathematics in Fourier analysis, approximation theory, differential equations, integral equations, and many other areas. In this paper, we consider convolutions for the continuous primitive integral. This integral extends the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals on the real line and has a very simple definition in terms of distributional derivatives.

Some of the main results for Lebesgue integral convolutions are that the convolution defines a Banach algebra on 𝐿1 and βˆ—βˆΆπΏ1×𝐿1→𝐿1 such that β€–π‘“βˆ—π‘”β€–1≀‖𝑓‖1‖𝑔‖1. The convolution is commutative, associative, and commutes with translations. If π‘“βˆˆπΏ1 and π‘”βˆˆπΆπ‘›, then π‘“βˆ—π‘”βˆˆπΆπ‘› and (π‘“βˆ—π‘”)(𝑛)(π‘₯)=π‘“βˆ—π‘”(𝑛)(π‘₯). Convolutions also have the approximation property that if π‘“βˆˆπΏπ‘ (1≀𝑝<∞) and π‘”βˆˆπΏ1, then β€–π‘“βˆ—π‘”π‘‘βˆ’π‘Žπ‘“β€–π‘β†’0 as 𝑑→0, where 𝑔𝑑(π‘₯)=𝑔(π‘₯/𝑑)/𝑑 and βˆ«π‘Ž=βˆžβˆ’βˆžπ‘”. When 𝑓 is bounded and continuous, there is a similar result for 𝑝=∞. For these results see, for example, [1]; see [2] for related results with the Henstock-Kurzweil integral. Using the Alexiewicz norm, all of these results have generalizations to continuous primitive integrals that are proven in what follows.

We now define the continuous primitive integral. For this, we need some notation for distributions. The space of test functions is π’Ÿ=πΆβˆžπ‘(ℝ)={πœ™βˆΆβ„β†’β„βˆ£πœ™βˆˆπΆβˆž(ℝ)andsupp(πœ™)iscompact}. The support of function πœ™ is the closure of the set on which πœ™ does not vanish and is denoted supp(πœ™). Under usual pointwise operations, π’Ÿ is a linear space over field ℝ. In π’Ÿ, we have a notion of convergence. If {πœ™π‘›}βŠ‚π’Ÿ, then πœ™π‘›β†’0 as π‘›β†’βˆž if there is a compact set πΎβŠ‚β„ such that for each 𝑛, supp(πœ™π‘›)βŠ‚πΎ, and for each π‘šβ‰₯0, we have πœ™π‘›(π‘š)β†’0 uniformly on 𝐾 as π‘›β†’βˆž. The  distributions  are denoted π’Ÿξ…ž and are the continuous linear functionals on π’Ÿ. For π‘‡βˆˆπ’Ÿξ…ž and πœ™βˆˆπ’Ÿ, we write βŸ¨π‘‡,πœ™βŸ©βˆˆβ„. For πœ™,πœ“βˆˆπ’Ÿ and π‘Ž,π‘βˆˆβ„, we have βŸ¨π‘‡,π‘Žπœ™+π‘πœ“βŸ©=π‘ŽβŸ¨π‘‡,πœ™βŸ©+π‘βŸ¨π‘‡,πœ“βŸ©. Moreover, if πœ™π‘›β†’0 in π’Ÿ, then βŸ¨π‘‡,πœ™π‘›βŸ©β†’0 in ℝ. Linear operations are defined in π’Ÿξ…ž by βŸ¨π‘Žπ‘†+𝑏𝑇,πœ™βŸ©=π‘ŽβŸ¨π‘†,πœ™βŸ©+π‘βŸ¨π‘‡,πœ™βŸ© for 𝑆,π‘‡βˆˆπ’Ÿξ…ž; π‘Ž,π‘βˆˆβ„ and πœ™βˆˆπ’Ÿ. If π‘“βˆˆπΏ1loc, then βŸ¨π‘‡π‘“βˆ«,πœ™βŸ©=βˆžβˆ’βˆžπ‘“(π‘₯)πœ™(π‘₯)𝑑π‘₯ defines a distribution π‘‡π‘“βˆˆπ’Ÿξ…ž. The integral exists as a Lebesgue integral. All distributions have derivatives of all orders that are themselves distributions. For π‘‡βˆˆπ’Ÿξ…ž and πœ™βˆˆπ’Ÿ, the  distributional derivative of 𝑇 is π‘‡ξ…ž where βŸ¨π‘‡ξ…ž,πœ™βŸ©=βˆ’βŸ¨π‘‡,πœ™ξ…žβŸ©. This is also called the  weak derivative. If π‘βˆΆβ„β†’β„ is a function that is differentiable in the pointwise sense at π‘₯βˆˆβ„, then we write its derivative as π‘ξ…ž(π‘₯). If 𝑝 is a 𝐢∞ bijection such that π‘ξ…ž(π‘₯)β‰ 0 for  any π‘₯βˆˆβ„, then the composition  with  distribution 𝑇 is  defined  by βŸ¨π‘‡βˆ˜π‘,πœ™βŸ©=βŸ¨π‘‡,(πœ™βˆ˜π‘βˆ’1)/(π‘ξ…žβˆ˜π‘βˆ’1)⟩ for all πœ™βˆˆπ’Ÿ. Translations are a special case. For π‘₯βˆˆβ„, define the  translation 𝜏π‘₯ on  distribution π‘‡βˆˆπ’Ÿξ…ž by ⟨𝜏π‘₯𝑇,πœ™βŸ©=βŸ¨π‘‡,πœβˆ’π‘₯πœ™βŸ© for test function πœ™βˆˆπ’Ÿ, where 𝜏π‘₯πœ™(𝑦)=πœ™(π‘¦βˆ’π‘₯). All  of  the  results  on distributions we use can be found in [3].

The following Banach space will be of importance: ℬ𝐢={πΉβˆΆβ„β†’β„βˆ£πΉβˆˆπΆ0(ℝ),𝐹(βˆ’βˆž)=0,𝐹(∞)βˆˆβ„}. We use the notation 𝐹(βˆ’βˆž)=limπ‘₯β†’βˆ’βˆžπΉ(π‘₯) and 𝐹(∞)=limπ‘₯β†’βˆžπΉ(π‘₯). The extended real line is denoted by ℝ=[βˆ’βˆž,∞]. The space ℬ𝐢 then consists of functions continuous on ℝ with a limit of 0 at βˆ’βˆž. We denote the functions that are continuous on ℝ that have real limits at ±∞ by 𝐢0(ℝ). Hence, ℬ𝐢 is properly contained in 𝐢0(ℝ), which is itself properly contained in the space of uniformly continuous functions on ℝ. The space ℬ𝐢 is a Banach space under the uniform norm; β€–πΉβ€–βˆž=supπ‘₯βˆˆβ„|𝐹(π‘₯)|=maxπ‘₯βˆˆβ„|𝐹(π‘₯)| for πΉβˆˆβ„¬πΆ. The continuous primitive integral is defined by taking ℬ𝐢 as the space of primitives. The space of integrable distributions is π’œπΆ={π‘“βˆˆπ’Ÿξ…žβˆ£π‘“=πΉξ…žforπΉβˆˆβ„¬πΆ}. If π‘“βˆˆπ’œπΆ, then βˆ«π‘π‘Žπ‘“=𝐹(𝑏)βˆ’πΉ(π‘Ž) for π‘Ž,π‘βˆˆβ„. The distributional differential equation π‘‡ξ…ž=0 has only constant solutions so the primitive πΉβˆˆβ„¬πΆ satisfying πΉξ…ž=𝑓 is unique. Integrable distributions are then tempered and of order one. This integral, including a discussion of extensions to ℝ𝑛, is described in [4]. A more general integral is obtained by taking the primitives to be regulated functions, that is, functions with a left and right limit at each point, see [5].

Examples of distributions in π’œπΆ are 𝑇𝑓 for functions 𝑓 that have a finite Lebesgue, Henstock-Kurzweil, or wide Denjoy integral. We identify function 𝑓 with the distribution 𝑇𝑓. Pointwise function values can be recovered from 𝑇𝑓 at points of continuity of 𝑓 by evaluating the limit βŸ¨π‘‡π‘“,πœ™π‘›βŸ© for a delta sequence converging to π‘₯βˆˆβ„. This is a sequence of test functions {πœ™π‘›}βŠ‚π’Ÿ such that for each 𝑛, πœ™π‘›β‰₯0, βˆ«βˆžβˆ’βˆžπœ™π‘›=1, and the support of πœ™π‘› tends to {π‘₯} as π‘›β†’βˆž. Note that if 𝐹∈𝐢0(ℝ) is an increasing function with πΉξ…ž(π‘₯)=0 for almost all π‘₯βˆˆβ„, then the Lebesgue integral βˆ«π‘π‘ŽπΉξ…ž(π‘₯)𝑑π‘₯=0 but πΉξ…žβˆˆπ’œπΆ and βˆ«π‘π‘ŽπΉξ…ž=𝐹(𝑏)βˆ’πΉ(π‘Ž). For another example of a distribution in π’œπΆ, let 𝐹∈𝐢0(ℝ) be continuous and nowhere differentiable in the pointwise sense. Then πΉξ…žβˆˆπ’œπΆ and βˆ«π‘π‘ŽπΉξ…ž=𝐹(𝑏)βˆ’πΉ(π‘Ž) for all π‘Ž,π‘βˆˆβ„.

The space π’œπΆ is a Banach space under the Alexiewicz norm; ‖𝑓‖=supπΌβŠ‚β„|βˆ«πΌπ‘“|, where the supremum is taken over all intervals πΌβŠ‚β„. An equivalent norm is β€–π‘“β€–ξ…ž=supπ‘₯βˆˆβ„|∫π‘₯βˆ’βˆžπ‘“|. The continuous primitive integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals since their primitives are continuous functions. These three spaces of functions are not complete under the Alexiewicz norm and in fact π’œπΆ is their completion. The lack of a Banach space has hampered application of the Henstock-Kurzweil integral to problems outside of real analysis. As we will see in what follows, the Banach space π’œπΆ is a suitable setting for applications of nonabsolute integration.

We will also need to use functions of bounded variation. Let π‘”βˆΆβ„β†’β„. The variation of 𝑔 is βˆ‘π‘‰π‘”=sup|𝑔(π‘₯𝑖)βˆ’π‘”(𝑦𝑖)| where the supremum is taken over all disjoint intervals {(π‘₯𝑖,𝑦𝑖)}. The functions of bounded variation are denoted ℬ𝒱={π‘”βˆΆβ„β†’β„βˆ£π‘‰π‘”<∞}. This is a Banach space under the norm ‖𝑔‖ℬ𝒱=|𝑔(βˆ’βˆž)|+𝑉𝑔. Equivalent norms are β€–π‘”β€–βˆž+𝑉𝑔 and |𝑔(π‘Ž)|+𝑉𝑔 for each π‘Žβˆˆβ„. Functions of bounded variation have a left and right limit at each point in ℝ and limits at ±∞, so, as above, we will define 𝑔(±∞)=limπ‘₯β†’Β±βˆžπ‘”(π‘₯).

If π‘”βˆˆπΏ1loc, then the essential variation of 𝑔 is ∫essvar𝑔=supβˆžβˆ’βˆžπ‘”πœ™ξ…ž, where the supremum is taken over all πœ™βˆˆπ’Ÿ with β€–πœ™β€–βˆžβ‰€1. Then ℰℬ𝒱={π‘”βˆˆπΏ1loc∣essvar𝑔<∞}. This is a Banach space under the norm ‖𝑔‖ℰℬ𝒱=esssup|𝑔|+essvar𝑔. Let 0≀𝛾≀1. For π‘”βˆΆβ„β†’β„, define 𝑔𝛾(π‘₯)=(1βˆ’π›Ύ)𝑔(π‘₯βˆ’)+𝛾𝑔(π‘₯+). For left continuity, 𝛾=0 and for right continuity 𝛾=1. The functions  of  normalized bounded variation are 𝒩ℬ𝒱𝛾={π‘”π›Ύβˆ£π‘”βˆˆβ„¬π’±}. If π‘”βˆˆβ„°β„¬π’±, then essvar𝑔=infπ‘‰β„Ž such that β„Ž=𝑔 almost everywhere. For each 0≀𝛾≀1, there is exactly one function β„Žβˆˆπ’©β„¬π’±π›Ύ such that 𝑔=β„Ž almost everywhere. In this case, essvar𝑔=π‘‰β„Ž. Changing 𝑔 on a set of measure zero does not affect its essential variation. Each function of essential bounded variation has a distributional derivative that is a signed Radon measure. This will be denoted πœ‡π‘” where βŸ¨π‘”ξ…ž,πœ™βŸ©=βˆ’βŸ¨π‘”,πœ™ξ…žβˆ«βŸ©=βˆ’βˆžβˆ’βˆžπ‘”πœ™ξ…ž=βˆ«βˆžβˆ’βˆžπœ™π‘‘πœ‡π‘” for all πœ™βˆˆπ’Ÿ.

We will see that βˆ—βˆΆπ’œπΆΓ—β„¬π’±β†’πΆ0(ℝ) and that β€–π‘“βˆ—π‘”β€–βˆžβ‰€β€–π‘“β€–β€–π‘”β€–β„¬π’±. Similarly for π‘”βˆˆβ„°β„¬π’±. Convolutions for π‘“βˆˆπ’œπΆ and π‘”βˆˆπΏ1 will be defined using sequences in β„¬π’±βˆ©πΏ1 that converge to 𝑔 in the 𝐿1 norm. It will be shown that βˆ—βˆΆπ’œπΆΓ—πΏ1β†’π’œπΆ and that β€–π‘“βˆ—π‘”β€–β‰€β€–π‘“β€–β€–π‘”β€–1.

Convolutions can be defined for distributions in several different ways.

Definition 1.1. Let 𝑆,π‘‡βˆˆπ’Ÿξ…ž and πœ™,πœ“βˆˆπ’Ÿ. Define ξ‚πœ™(π‘₯)=πœ™(βˆ’π‘₯): (i) βŸ¨π‘‡βˆ—πœ“,πœ™βŸ©=βŸ¨π‘‡,πœ™βˆ—ξ‚πœ“βŸ©, (ii) for each π‘₯βˆˆβ„, let π‘‡βˆ—πœ“(π‘₯)=βŸ¨π‘‡,𝜏π‘₯ξ‚πœ“βŸ©; (iii) βŸ¨π‘†βˆ—π‘‡,πœ™βŸ©=βŸ¨π‘†(π‘₯),βŸ¨π‘‡(𝑦),πœ™(π‘₯+𝑦)⟩⟩.

In (i), βˆ—βˆΆπ’Ÿξ…žΓ—π’Ÿβ†’π’Ÿξ…ž. This definition also applies to other spaces of test functions and their duals, such as the Schwartz space of rapidly decreasing functions or the compactly supported distributions. In (ii), βˆ—βˆΆπ’Ÿξ…žΓ—π’Ÿβ†’πΆβˆž. In [1], it is shown that definitions (i) and (ii) are equivalent. In (iii), βˆ—βˆΆπ’Ÿξ…žΓ—π’Ÿξ…žβ†’π’Ÿξ…ž. However, this definition requires restrictions on the supports of 𝑆 and 𝑇. It suffices that one of these distributions has compact support. Other conditions on the supports can be imposed (see [3, 6]). This definition is an instance of the tensor product, βŸ¨π‘†βŠ—π‘‡,Φ⟩=βŸ¨π‘†(π‘₯),βŸ¨π‘‡(𝑦),Ξ¦(π‘₯,𝑦)⟩⟩, where now Ξ¦βˆˆπ’Ÿ(ℝ2).

Under (i), π‘‡βˆ—πœ“ is in 𝐢∞. It satisfies (π‘‡βˆ—πœ“)βˆ—πœ™=π‘‡βˆ—(πœ“βˆ—πœ™), 𝜏π‘₯(π‘‡βˆ—πœ“)=(𝜏π‘₯𝑇)βˆ—πœ“=π‘‡βˆ—(𝜏π‘₯πœ“), and (π‘‡βˆ—πœ“)(𝑛)=π‘‡βˆ—πœ“(𝑛)=𝑇(𝑛)βˆ—πœ“. Under (iii), with appropriate support restrictions, π‘†βˆ—π‘‡ is in π’Ÿξ…ž. It is commutative and associative, commutes with translations, and satisfies (π‘†βˆ—π‘‡)(𝑛)=𝑆(𝑛)βˆ—π‘‡=π‘†βˆ—π‘‡(𝑛). It is weakly continuous in π’Ÿξ…ž, that is, if 𝑇𝑛→𝑇 in π’Ÿξ…ž, then π‘‡π‘›βˆ—πœ“β†’π‘‡βˆ—πœ“ in π’Ÿξ…ž see [1, 3, 6, 7] for additional properties of convolutions of distributions.

Although elements of π’œπΆ are distributions, we show in this paper that their behavior as convolutions is more like that of integrable functions.

An appendix contains the proof of a type of Fubini theorem.

2. Convolution in π’œπΆΓ—β„¬π’±

In this section, we prove basic results for the convolution when π‘“βˆˆπ’œπΆ and π‘”βˆˆβ„¬π’±. Under these conditions, π‘“βˆ—π‘” is commutative, continuous on ℝ, and commutes with translations. It can be estimated in the uniform norm in terms of the Alexiewicz and ℬ𝒱 norms. There is also an associative property. We first need the result that ℬ𝒱 forms the space of multipliers for π’œπΆ, that is, if π‘“βˆˆπ’œπΆ, then π‘“π‘”βˆˆπ’œπΆ for all π‘”βˆˆβ„¬π’±. The integral βˆ«πΌπ‘“π‘” is defined using the integration by parts formula in the appendix. The HΓΆlder inequality (A.5) shows that ℬ𝒱 is the dual space of π’œπΆ.

We define the convolution of π‘“βˆˆπ’œπΆ and π‘”βˆˆβ„¬π’± as βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆž(π‘“βˆ˜π‘Ÿπ‘₯)𝑔, where π‘Ÿπ‘₯(𝑑)=π‘₯βˆ’π‘‘. We write this as βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦.

Theorem 2.1. Let π‘“βˆˆπ’œπΆ and let π‘”βˆˆβ„¬π’±. Then (a)β€‰β€‰π‘“βˆ—π‘” exists on ℝ.  (b) Let π‘“βˆ—π‘”=π‘”βˆ—π‘“.  (c)  Letβ€‰β€‰β€–π‘“βˆ—π‘”β€–βˆžβˆ«β‰€|βˆžβˆ’βˆžπ‘“|infℝ|𝑔|+‖𝑓‖𝑉𝑔≀‖𝑓‖‖𝑔‖ℬ𝒱.   (d)   Assume π‘“βˆ—π‘”βˆˆπΆ0(ℝ), limπ‘₯β†’Β±βˆžβˆ«π‘“βˆ—π‘”(π‘₯)=𝑔(±∞)βˆžβˆ’βˆžπ‘“.   (e) If β„ŽβˆˆπΏ1, then π‘“βˆ—(π‘”βˆ—β„Ž)=(π‘“βˆ—π‘”)βˆ—β„ŽβˆˆπΆ0(ℝ).    (f) Let π‘₯,π‘§βˆˆβ„,   then πœπ‘§(π‘“βˆ—π‘”)(π‘₯)=(πœπ‘§π‘“)βˆ—π‘”(π‘₯)=(π‘“βˆ—πœπ‘§π‘”)(π‘₯).   (g)   For eachβ€‰β€‰π‘“βˆˆπ’œπΆ,   define Ξ¦π‘“βˆΆβ„¬π’±β†’πΆ0(ℝ) by Φ𝑓[𝑔]=π‘“βˆ—π‘”.   Then Φ𝑓 is a bounded linear operator and ‖Φ𝑓‖≀‖𝑓‖.   There exists a nonzero distributionβ€‰β€‰π‘“βˆˆπ’œπΆ such that ‖Φ𝑓‖=‖𝑓‖. For each π‘”βˆˆβ„¬π’±, define Ξ¨π‘”βˆΆπ’œπΆβ†’πΆ0(ℝ) by Ψ𝑔[𝑓]=π‘“βˆ—π‘”. Then Ψ𝑔 is a bounded linear operator and ‖Ψ𝑔‖≀‖𝑔‖ℬ𝒱. There exists a nonzero function π‘”βˆˆβ„¬π’± such that ‖Ψ𝑔‖=‖𝑔‖ℬ𝒱.   (h)  supp(π‘“βˆ—π‘”)βŠ‚supp(𝑓)+supp(𝑔).

Proof. (a) Existence is given via the integration by parts formula (A.1) in the appendix. (b) See [4, Theorem  11] for a change of variables theorem that can be used with 𝑦↦π‘₯βˆ’π‘¦. (c) This inequality follows from the HΓΆlder inequality (A.5). (d) Let π‘₯,π‘‘βˆˆβ„. From (c), we have ||||π‘“βˆ—π‘”(𝑑)βˆ’π‘“βˆ—π‘”(π‘₯)≀‖𝑓(π‘‘βˆ’β‹…)βˆ’π‘“(π‘₯βˆ’β‹…)‖‖𝑔‖ℬ𝒱=‖𝑓(π‘‘βˆ’π‘₯βˆ’β‹…)βˆ’π‘“(β‹…)β€–β€–π‘”β€–β„¬π’±βŸΆ0asπ‘‘βŸΆπ‘₯.(2.1) The last line follows from continuity in the Alexiewicz norm [4, Theorem  22]. Hence, π‘“βˆ—π‘” is uniformly continuous on ℝ. Also, it follows that limπ‘₯β†’βˆžβˆ«βˆžβˆ’βˆžβˆ«π‘“(𝑦)𝑔(π‘₯βˆ’π‘¦)𝑑𝑦=βˆžβˆ’βˆžπ‘“(𝑦)limπ‘₯β†’βˆžβˆ«π‘”(π‘₯βˆ’π‘¦)𝑑𝑦=𝑔(∞)βˆžβˆ’βˆžπ‘“. The limit π‘₯β†’βˆž can be taken under the integral sign since 𝑔(π‘₯βˆ’π‘¦) is of uniform bounded variation, that is, π‘‰π‘¦βˆˆβ„π‘”(π‘₯βˆ’π‘¦)=𝑉𝑔. Theorem  22 in [4] then applies. Similarly, as π‘₯β†’βˆ’βˆž. (e) First show π‘”βˆ—β„Žβˆˆβ„¬π’±. Let {(𝑠𝑖,𝑑𝑖)} be disjoint intervals in ℝ. Then ||ξ€·π‘ π‘”βˆ—β„Žπ‘–ξ€Έξ€·π‘‘βˆ’π‘”βˆ—β„Žπ‘–ξ€Έ||β‰€ξ“ξ€œβˆžβˆ’βˆž||π‘”ξ€·π‘ π‘–ξ€Έξ€·π‘‘βˆ’π‘¦βˆ’π‘”π‘–ξ€Έ||||||=ξ€œβˆ’π‘¦β„Ž(𝑦)π‘‘π‘¦βˆžβˆ’βˆžξ“||π‘”ξ€·π‘ π‘–ξ€Έξ€·π‘‘βˆ’π‘¦βˆ’π‘”π‘–ξ€Έ||||||βˆ’π‘¦β„Ž(𝑦)𝑑𝑦.(2.2) Hence, 𝑉(π‘”βˆ—β„Ž)β‰€π‘‰π‘”β€–β„Žβ€–1. The interchange of sum and integral follows from the Fubini-Tonelli theorem. Now (d) shows π‘“βˆ—(π‘”βˆ—β„Ž)∈𝐢0(ℝ). Write ξ€œπ‘“βˆ—(π‘”βˆ—β„Ž)(π‘₯)=βˆžβˆ’βˆžξ€œπ‘“(𝑦)βˆžβˆ’βˆž=ξ€œπ‘”(π‘₯βˆ’π‘¦βˆ’π‘§)β„Ž(𝑧)π‘‘π‘§π‘‘π‘¦βˆžβˆ’βˆžξ€œβ„Ž(𝑧)βˆžβˆ’βˆžπ‘“(𝑦)𝑔(π‘₯βˆ’π‘¦βˆ’π‘§)𝑑𝑦𝑑𝑧=(π‘“βˆ—π‘”)βˆ—β„Ž(π‘₯).(2.3) We can interchange orders of integration using Proposition A.3. For (ii) in Proposition A.3, the function π‘§β†¦π‘‰π‘¦βˆˆβ„π‘”(π‘₯βˆ’π‘¦βˆ’π‘§)β„Ž(𝑧)=π‘‰π‘”β„Ž(𝑧) is in 𝐿1 for each fixed π‘₯βˆˆβ„. Since 𝑔 is of bounded variation, it is bounded so |𝑔(π‘₯βˆ’π‘¦βˆ’π‘§)β„Ž(𝑧)|β‰€β€–π‘”β€–βˆž|β„Ž(𝑧)| and condition (iii) is satisfied. (f) This follows from a linear change of variables as in (a). (g) From (c), we have ‖Φ𝑓‖=sup‖𝑔‖ℬ𝒱=1β€–π‘“βˆ—π‘”β€–βˆžβ‰€sup‖𝑔‖ℬ𝒱=1‖𝑓‖‖𝑔‖ℬ𝒱=‖𝑓‖. Let 𝑓>0 be in 𝐿1. If 𝑔=1, then ‖𝑔‖ℬ𝒱=1 and βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“ so ‖Φ𝑓‖=‖𝑓‖=‖𝑓‖1. To prove ‖Ψ𝑔‖≀‖𝑔‖ℬ𝒱, note that ‖Ψ𝑔‖=sup‖𝑓‖=1β€–π‘“βˆ—π‘”β€–βˆžβ‰€sup‖𝑓‖=1‖𝑓‖‖𝑔‖ℬ𝒱=‖𝑔‖ℬ𝒱. Let 𝑔=πœ’(0,∞). Then ‖Ψ𝑔‖=sup‖𝑓‖=1β€–π‘“βˆ—π‘”β€–βˆž=sup‖𝑓‖=1supπ‘₯βˆˆβ„|∫π‘₯βˆ’βˆžπ‘“|=1=‖𝑔‖ℬ𝒱. (h) Suppose π‘₯βˆ‰supp(𝑓)+supp(𝑔). Note that we can write βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘”(π‘₯βˆ’π‘¦)𝑑𝐹(𝑦) in terms of a Henstock-Stieltjes integral, see [4] for details. This integral is approximated by Riemann sums βˆ‘π‘π‘›=1𝑔(π‘₯βˆ’π‘§π‘›)[𝐹(𝑑𝑛)βˆ’πΉ(π‘‘π‘›βˆ’1)] where π‘§π‘›βˆˆ[π‘‘π‘›βˆ’1,𝑑𝑛], βˆ’βˆž=𝑑0<𝑑1<β‹―<𝑑𝑁=∞ and there is a gauge function 𝛾 mapping ℝ to the open intervals in ℝ such that [π‘‘π‘›βˆ’1,𝑑𝑛]βŠ‚π›Ύ(𝑧𝑛). If π‘§π‘›βˆ‰supp(𝑓), then since ℝ⧡supp(𝑓) is open, there is an open interval π‘§π‘›βŠ‚πΌβŠ‚β„β§΅supp(𝑓). We can take 𝛾 such that [π‘‘π‘›βˆ’1,𝑑𝑛]βŠ‚πΌ for all 1≀𝑛≀𝑁. Also, 𝐹 is constant on each interval in ℝ⧡supp(𝑓). Therefore, 𝑔(π‘₯βˆ’π‘§π‘›)[𝐹(𝑑𝑛)βˆ’πΉ(π‘‘π‘›βˆ’1)]=0 and only tags π‘§π‘›βˆˆsupp(𝑓) can contribute to the Riemann sum. However, for all π‘§π‘›βˆˆsupp(𝑓), we have π‘₯βˆ’π‘§π‘›βˆ‰supp(𝑔) so 𝑔(π‘₯βˆ’π‘§π‘›)[𝐹(𝑑𝑛)βˆ’πΉ(π‘‘π‘›βˆ’1)]=0. It follows that π‘“βˆ—π‘”(π‘₯)=0.

Similar results are proven for π‘“βˆˆπΏπ‘ in [1, Section  8.2].

If we use the equivalent norm β€–π‘“β€–ξ…ž=supπ‘₯βˆˆβ„|∫π‘₯βˆ’βˆžπ‘“|, then ‖Φ𝑓‖=β€–π‘“β€–ξ…ž. Also, integration by parts gives β€–Ξ¦π‘“β€–β‰€β€–π‘“β€–ξ…ž. Now, given π‘“βˆˆπ’œπΆ, let 𝑔=πœ’(0,∞). Then ‖𝑔‖ℬ𝒱=1,  and βˆ«π‘“βˆ—π‘”(π‘₯)=π‘₯βˆ’βˆžπ‘“. Hence, β€–π‘“βˆ—π‘”β€–βˆž=β€–π‘“β€–ξ…ž and ‖Φ𝑓‖=β€–π‘“β€–ξ…ž. We can have strict inequality in ‖Ψ𝑔‖≀‖𝑔‖ℬ𝒱. For example, let 𝑔=πœ’{0},  then ‖𝑔‖ℬ𝒱=2 but integration by parts shows π‘“βˆ—π‘”=0 for each π‘“βˆˆπ’œπΆ.

Remark 2.2. If π‘“βˆˆπ’œπΆ and π‘”βˆˆβ„°β„¬π’±, one can use Definition A.2 to define π‘“βˆ—π‘”(π‘₯)=π‘“βˆ—π‘”π›Ύ(π‘₯) where 𝑔𝛾=𝑔 almost everywhere and π‘”π›Ύβˆˆπ’©β„¬π’±π›Ύ. All of the results in Theorem 2.1 and the rest of this paper have analogues. Note that π‘“βˆ—π‘”(π‘₯)=𝐹(∞)𝑔𝛾(βˆ’βˆž)+πΉβˆ—πœ‡π‘”.

Proposition 2.3. The three definitions of convolution for distributions in Definition 1.1 are compatible with π‘“βˆ—π‘” for π‘“βˆˆπ’œπΆ and π‘”βˆˆβ„¬π’±.

Proof. Let π‘“βˆˆπ’œπΆ, π‘”βˆˆβ„¬π’±, and πœ™,πœ“βˆˆπ’Ÿ. Definition 1.1(i) gives ξ€œβŸ¨π‘“,ξ‚πœ“βˆ—πœ™βŸ©=βˆžβˆ’βˆžξ€œπ‘“(π‘₯)βˆžβˆ’βˆžξ€œπœ“(π‘¦βˆ’π‘₯)πœ™(𝑦)𝑑𝑦𝑑π‘₯=βˆžβˆ’βˆžξ€œβˆžβˆ’βˆžπ‘“(π‘₯)πœ“(π‘¦βˆ’π‘₯)πœ™(𝑦)𝑑π‘₯𝑑𝑦=βŸ¨π‘“βˆ—πœ“,πœ™βŸ©.(2.4) Since πœ“βˆˆβ„¬π’± and πœ™βˆˆπΏ1, Proposition A.3 justifies the interchange of integrals. Definition 1.1(ii) gives βŸ¨π‘“,𝜏π‘₯ξ€œξ‚πœ“βŸ©=βˆžβˆ’βˆžπ‘“(𝑦)πœ“(π‘₯βˆ’π‘¦)𝑑𝑦=π‘“βˆ—πœ“(π‘₯).(2.5) Definition 1.1(iii) gives ξ€œβŸ¨π‘“(𝑦),βŸ¨π‘”(π‘₯),πœ™(π‘₯+𝑦)⟩⟩=βˆžβˆ’βˆžξ€œπ‘“(𝑦)βˆžβˆ’βˆž=ξ€œπ‘”(π‘₯)πœ™(π‘₯+𝑦)𝑑π‘₯π‘‘π‘¦βˆžβˆ’βˆžξ€œπ‘“(𝑦)βˆžβˆ’βˆž=ξ€œπ‘”(π‘₯βˆ’π‘¦)πœ™(π‘₯)𝑑π‘₯π‘‘π‘¦βˆžβˆ’βˆžξ€œπœ™(π‘₯)βˆžβˆ’βˆžπ‘“(𝑦)𝑔(π‘₯βˆ’π‘¦)𝑑π‘₯𝑑π‘₯=βŸ¨π‘“βˆ—π‘”,πœ™βŸ©.(2.6) The interchange of integrals is accomplished using Proposition A.3 since π‘”βˆˆβ„¬π’± and πœ™βˆˆπΏ1.

The locally integrable distributions are defined as π’œπΆ(loc)={π‘“βˆˆπ’Ÿξ…žβˆ£π‘“=πΉξ…žforsome𝐹∈𝐢0(ℝ)}. Let π‘“βˆˆπ’œπΆ(loc) and let π‘”βˆˆβ„¬π’± with support in the compact interval [π‘Ž,𝑏]. By the Hake theorem [4, Theorem  25], π‘“βˆ—π‘”(π‘₯) exists if and only if the limits of βˆ«π›½π›Όπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦 exist as π›Όβ†’βˆ’βˆž and π›½β†’βˆž. This gives

ξ€œπ‘“βˆ—π‘”(π‘₯)=π‘π‘Žξ€œπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦=π‘₯βˆ’π‘Žπ‘₯βˆ’π‘π‘“(𝑦)𝑔(π‘₯βˆ’π‘¦)𝑑𝑦.(2.7)

There are analogues of the results in Theorem 2.1. For example, ∫|π‘“βˆ—π‘”(π‘₯)|≀|π‘₯βˆ’π‘Žπ‘₯βˆ’π‘π‘“|inf[π‘Ž,𝑏]|𝑔|+β€–π‘“πœ’[π‘₯βˆ’π‘,π‘₯βˆ’π‘Ž]‖𝑉[π‘Ž,𝑏]𝑔. There are also versions where the supports are taken to be semi-infinite intervals.

We can also define the distributions with bounded primitive as π’œπΆ(𝑏𝑑)={π‘“βˆˆπ’Ÿξ…žβˆ£π‘“=πΉξ…žforsomebounded𝐹∈𝐢0(ℝ)with𝐹(0)=0}. Let π‘“βˆˆπ’œπΆ(𝑏𝑑) and let 𝐹 be its unique primitive. If π‘”βˆˆβ„¬π’± such that 𝑔(±∞)=0, then

π‘“βˆ—π‘”(π‘₯)=limπ›Όβ†’βˆ’βˆžπ›½β†’βˆžξ€œπ›½π›Όπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦=limπ›Όβ†’βˆ’βˆžπ›½β†’βˆžξ‚Έξ€œπΉ(π‘₯βˆ’π›Ό)𝑔(𝛼)βˆ’πΉ(π‘₯βˆ’π›½)𝑔(𝛽)+𝛽𝛼=ξ€œπΉ(π‘₯βˆ’π‘¦)𝑑𝑔(𝑦)βˆžβˆ’βˆžξ€œπΉ(π‘₯βˆ’π‘¦)𝑑𝑔(𝑦)=βˆžβˆ’βˆžπΉ(𝑦)𝑑𝑔(π‘₯βˆ’π‘¦).(2.8)

It follows that β€–π‘“βˆ—π‘”β€–βˆžβ‰€β€–πΉβ€–βˆžπ‘‰π‘”.

It is possible to formulate other existence criteria. For example, if 𝑓(π‘₯)=log|π‘₯|sin(π‘₯) and 𝑔(π‘₯)=|π‘₯|βˆ’π›Ό for some 0<𝛼<1, then 𝑓 and 𝑔 are not in π’œπΆ, ℬ𝒱 or 𝐿𝑝 for any 1β‰€π‘β‰€βˆž but π‘“βˆ—π‘” exists on ℝ because 𝑓,π‘”βˆˆπΏ1loc and if ∫𝐹(π‘₯)=π‘₯0𝑓, then lim|π‘₯|β†’βˆžπΉ(π‘₯)𝑔(π‘₯)=0.

The following example shows that π‘“βˆ—π‘” needs not to be of bounded variation and hence not absolutely continuous. Let 𝑔=πœ’(0,∞). For π‘“βˆˆπ’œπΆ, we have βˆ«π‘“βˆ—π‘”(π‘₯)=π‘₯βˆ’βˆžπ‘“=𝐹(π‘₯), where πΉβˆˆβ„¬πΆ is the primitive of 𝑓. However, 𝐹 needs not to be of bounded variation or even of local bounded variation. For example, let 𝑓(π‘₯)=sin(π‘₯βˆ’2)βˆ’2π‘₯βˆ’2cos(π‘₯βˆ’2) and let 𝐹 be its primitive in ℬ𝐢. Finally, although π‘“βˆ—π‘” is continuous, it needs not to be integrable over ℝ. For example, let 𝑔=1, then βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“ and βˆ«βˆžβˆ’βˆžπ‘“βˆ—π‘” only exists if βˆ«βˆžβˆ’βˆžπ‘“=0.

3. Convolution in π’œπΆΓ—πΏ1

We now extend the convolution π‘“βˆ—π‘” to π‘“βˆˆπ’œπΆ and π‘”βˆˆπΏ1. Since there are functions in 𝐿1 that are not of bounded variation, there are distributions π‘“βˆˆπ’œπΆ and functions π‘”βˆˆπΏ1 such that the integral βˆ«βˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦 does not exist. The convolution is then defined as the limit in β€–β‹…β€– of a sequence π‘“βˆ—π‘”π‘› for π‘”π‘›βˆˆβ„¬π’±βˆ©πΏ1 such that 𝑔𝑛→𝑔 in the 𝐿1 norm. This is possible since β„¬π’±βˆ©πΏ1 is dense in 𝐿1. We also give an equivalent definition using the fact that 𝐿1 is dense in π’œπΆ. Take a sequence {𝑓𝑛}βŠ‚πΏ1 such that β€–π‘“π‘›βˆ’π‘“β€–β†’0. Then π‘“βˆ—π‘” is the limit in β€–β‹…β€– of π‘“π‘›βˆ—π‘”. In this more general setting of convolution defined in π’œπΆΓ—πΏ1, we now have an Alexiewicz norm estimate for π‘“βˆ—π‘” in terms of estimates of 𝑓 in the Alexiewicz norm and 𝑔 in the 𝐿1 norm. There is associativity with 𝐿1 functions and commutativity with translations.

Definition 3.1. Let π‘“βˆˆπ’œπΆ and let π‘”βˆˆπΏ1. Let {𝑔𝑛}βŠ‚β„¬π’±βˆ©πΏ1 such that β€–π‘”π‘›βˆ’π‘”β€–1β†’0. Define π‘“βˆ—π‘” as the unique element in π’œπΆ such that β€–π‘“βˆ—π‘”π‘›βˆ’π‘“βˆ—π‘”β€–β†’0.

To see that the definition makes sense, first note that β„¬π’±βˆ©πΏ1 is dense in 𝐿1 since step functions are dense in 𝐿1. Hence, the required sequence {𝑔𝑛} exists. Let [𝛼,𝛽]βŠ‚β„ be a compact interval. Let πΉβˆˆβ„¬πΆ be the primitive of 𝑓. Then

ξ€œπ›½π›Όπ‘“βˆ—π‘”π‘›ξ€œ(π‘₯)𝑑π‘₯=π›½π›Όξ€œβˆžβˆ’βˆžπ‘“(𝑦)𝑔𝑛=ξ€œ(π‘₯βˆ’π‘¦)𝑑𝑦𝑑π‘₯βˆžβˆ’βˆžξ€œπ‘“(𝑦)𝛽𝛼𝑔𝑛(=ξ€œπ‘₯βˆ’π‘¦)𝑑π‘₯𝑑𝑦(3.1)βˆžβˆ’βˆžπ‘“ξ€œ(𝑦)π›½βˆ’π‘¦π›Όβˆ’π‘¦π‘”π‘›ξ€œ(π‘₯)𝑑π‘₯𝑑𝑦=βˆ’βˆžβˆ’βˆžπΉξ‚Έξ€œ(𝑦)π‘‘π›½βˆ’π‘¦π›Όβˆ’π‘¦π‘”π‘›ξ‚Ή=ξ€œ(3.2)βˆžβˆ’βˆžξ€Ίπ‘”πΉ(𝑦)𝑛(π›½βˆ’π‘¦)βˆ’π‘”π‘›ξ€»=ξ€œ(π›Όβˆ’π‘¦)π‘‘π‘¦βˆžβˆ’βˆžξ‚΅ξ€œπ›½βˆ’π‘¦π›Όβˆ’π‘¦π‘“ξ‚Άπ‘”π‘›(𝑦)𝑑𝑦.(3.3) The interchange of orders of integration in (3.1) is accomplished with Proposition A.3 using 𝑔(π‘₯,𝑦)=𝑔𝑛(π‘₯βˆ’π‘¦)πœ’[𝛼,𝛽](π‘₯). Integration by parts gives (3.2) since limπ‘¦β†’βˆžβˆ«π›½βˆ’π‘¦π›Όβˆ’π‘¦π‘”π‘›=0. As 𝐹 is continuous and the function βˆ«π‘¦β†¦π›½βˆ’π‘¦π›Όβˆ’π‘¦π‘”π‘› is absolutely continuous, we get (3.3). Taking the supremum over 𝛼,π›½βˆˆβ„ gives

β€–β€–π‘“βˆ—π‘”π‘›β€–β€–β‰€β€–β€–π‘”β€–π‘“β€–π‘›β€–β€–1.(3.4) We now have

β€–β€–π‘“βˆ—π‘”π‘šβˆ’π‘“βˆ—π‘”π‘›β€–β€–=β€–β€–ξ€·π‘”π‘“βˆ—π‘šβˆ’π‘”π‘›ξ€Έβ€–β€–β€–β€–π‘”β‰€β€–π‘“β€–π‘šβˆ’π‘”π‘›β€–β€–1(3.5)

and {π‘“βˆ—π‘”π‘›} is a Cauchy sequence in π’œπΆ. Since π’œπΆ is complete, this sequence has a limit in π’œπΆ which we denote π‘“βˆ—π‘”. The definition does not depend on the choice of sequence {𝑔𝑛}, thus if {β„Žπ‘›}βŠ‚β„¬π’±βˆ©πΏ1 such that β€–β„Žπ‘›βˆ’π‘”β€–1β†’0, then β€–π‘“βˆ—π‘”π‘›βˆ’π‘“βˆ—β„Žπ‘›β€–β‰€β€–π‘“β€–(β€–π‘”π‘›βˆ’π‘”β€–1+β€–β„Žπ‘›βˆ’π‘”β€–1)β†’0 as π‘›β†’βˆž. The previous calculation also shows that if π‘”βˆˆβ„¬π’±βˆ©πΏ1, then the integral definition βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦 and the limit definition agree.

Definition 3.2. Let π‘“βˆˆπ’œπΆ and let π‘”βˆˆπΏ1. Let {𝑓𝑛}βŠ‚πΏ1 such that β€–π‘“π‘›βˆ’π‘“β€–β†’0. Define π‘“βˆ—π‘” as the unique element in π’œπΆ such that β€–π‘“π‘›βˆ—π‘”βˆ’π‘“βˆ—π‘”β€–β†’0.

To show this definition makes sense, first show 𝐿1 is dense in π’œπΆ.

Proposition 3.3. 𝐿1 is dense in π’œπΆ.

Proof. Let 𝐴𝐢(ℝ) be the functions that are absolutely continuous on each compact interval and which are of bounded variation on the real line. Then, π‘“βˆˆπΏ1 if and only if there exists 𝐹∈𝐴𝐢(ℝ) such that πΉξ…ž(π‘₯)=𝑓(π‘₯) for almost all π‘₯βˆˆβ„. Let π‘“βˆˆπ’œπΆ be given. Let πΉβˆˆβ„¬πΆ be its primitive. For πœ–>0, take 𝑀>0 such that |F(π‘₯)|<πœ– for π‘₯<βˆ’π‘€ and |𝐹(π‘₯)βˆ’πΉ(∞)|<πœ– for π‘₯>𝑀. Due to the Weierstrass approximation theorem, there is a continuous function π‘ƒβˆΆβ„β†’β„ such that 𝑃(π‘₯)=𝐹(βˆ’π‘€) for π‘₯β‰€βˆ’π‘€, 𝑃(π‘₯)=𝐹(𝑀) for π‘₯β‰₯𝑀, |𝑃(π‘₯)βˆ’πΉ(π‘₯)|<πœ– for |π‘₯|≀𝑀 and 𝑃 is a polynomial on [βˆ’π‘€,𝑀]. Hence, π‘ƒβˆˆπ΄πΆ(ℝ) and β€–π‘ƒξ…žβˆ’π‘“β€–<3πœ–.

In Definition 3.2, the required sequence {𝑓𝑛}βŠ‚πΏ1 exists. Let [𝛼,𝛽]βŠ‚β„ be a compact interval. Then, by the usual Fubini-Tonelli theorem in 𝐿1,

ξ€œπ›½π›Όπ‘“π‘›ξ€œβˆ—π‘”(π‘₯)𝑑π‘₯=π›½π›Όξ€œβˆžβˆ’βˆžπ‘“π‘›=ξ€œ(π‘₯βˆ’π‘¦)𝑔(𝑦)𝑑𝑦𝑑π‘₯βˆžβˆ’βˆžξ€œπ‘”(𝑦)𝛽𝛼𝑓𝑛(π‘₯βˆ’π‘¦)𝑑π‘₯𝑑𝑦.(3.6)

Take the supremum over 𝛼,π›½βˆˆβ„ and use the 𝐿1βˆ’πΏβˆž HΓΆlder inequality to get

β€–β€–π‘“π‘›β€–β€–β‰€β€–β€–π‘“βˆ—π‘”π‘›β€–β€–β€–π‘”β€–1.(3.7) It now follows that {π‘“π‘›βˆ—π‘”} is a Cauchy sequence. It then converges to an element of π’œπΆ. However, (3.7) also shows that this limit is independent of the choice of {𝑓𝑛}. To see that Definitions 3.1 and 3.2 agree, take {𝑓𝑛}βŠ‚πΏ1 with β€–π‘“π‘›βˆ’π‘“β€–β†’0 and {𝑔𝑛}βŠ‚β„¬π’±βˆ©πΏ1 with β€–π‘”π‘›βˆ’π‘”β€–1β†’0. Then

β€–β€–π‘“π‘›βˆ—π‘”βˆ’π‘“βˆ—π‘”π‘›β€–β€–=β€–β€–ξ€·π‘“π‘›ξ€Έξ€·π‘”βˆ’π‘“βˆ—π‘”βˆ’π‘“βˆ—π‘›ξ€Έβ€–β€–β‰€β€–β€–ξ€·π‘“βˆ’π‘”π‘›ξ€Έβ€–β€–+β€–β€–ξ€·π‘”βˆ’π‘“βˆ—π‘”π‘“βˆ—π‘›ξ€Έβ€–β€–β‰€β€–β€–π‘“βˆ’π‘”π‘›β€–β€–βˆ’π‘“β€–π‘”β€–1‖‖𝑔+β€–π‘“β€–π‘›β€–β€–βˆ’π‘”1(3.8)

Letting π‘›β†’βˆž shows that the limits of π‘“π‘›βˆ—π‘” in Definition 3.2 and π‘“βˆ—π‘”π‘› in Definition 3.1 are the same.

Theorem 3.4. Let π‘“βˆˆπ’œπΆ and π‘”βˆˆπΏ1. Define π‘“βˆ—π‘” as in Definition 3.1. Then (a)β€–π‘“βˆ—π‘”β€–β‰€β€–π‘“β€–β€–π‘”β€–1.  (b)  Let β„ŽβˆˆπΏ1.  Then (π‘“βˆ—π‘”)βˆ—β„Ž=π‘“βˆ—(π‘”βˆ—β„Ž)βˆˆπ’œπΆ.  (c) For each π‘§βˆˆβ„, πœπ‘§(π‘“βˆ—π‘”)=(πœπ‘§π‘“)βˆ—π‘”=(π‘“βˆ—πœπ‘§π‘”).  (d) For each π‘“βˆˆπ’œπΆ, define Ξ¦π‘“βˆΆπΏ1β†’π’œπΆ by Φ𝑓[𝑔]=π‘“βˆ—π‘”. Then Φ𝑓 is a bounded linear operator and ‖Φ𝑓‖≀‖𝑓‖. There exists a nonzero distribution π‘“βˆˆπ’œπΆ such that ‖Φ𝑓‖=‖𝑓‖. For each π‘”βˆˆπΏ1, define Ξ¨π‘”βˆΆπ’œπΆβ†’π’œπΆ by Ψ𝑔[𝑓]=π‘“βˆ—π‘”. Then Ψ𝑔 is a bounded linear operator and ‖Ψ𝑔‖≀‖𝑔‖1. There exists a nonzero function π‘”βˆˆπΏ1 such that ‖Ψ𝑔‖=‖𝑔‖ℬ𝒱.  (e) Define 𝑔𝑑(π‘₯)=𝑔(π‘₯/𝑑)/𝑑 for 𝑑>0. We have βˆ«π‘Ž=βˆžβˆ’βˆžπ‘”π‘‘βˆ«(π‘₯)𝑑π‘₯=βˆžβˆ’βˆžπ‘”. Then β€–π‘“βˆ—π‘”π‘‘βˆ’π‘Žπ‘“β€–β†’0 as 𝑑→0.  (f)  Let supp(π‘“βˆ—π‘”)βŠ‚supp(𝑓)+supp(𝑔).

Proof. Let {𝑔𝑛} be as in Definition 3.1. (a) Since β€–π‘“βˆ—π‘”π‘›β€–β†’β€–π‘“βˆ—π‘”β€–, (3.4) shows β€–π‘“βˆ—π‘”β€–β‰€β€–π‘“β€–β€–π‘”β€–1. (b) Let {β„Žπ‘›}βŠ‚β„¬π’±βˆ©πΏ1 such that β€–β„Žπ‘›βˆ’β„Žβ€–1β†’0. Then (π‘“βˆ—π‘”)βˆ—β„ŽβˆΆ=πœ‰βˆˆπ’œπΆ such that β€–(π‘“βˆ—π‘”)βˆ—β„Žπ‘›βˆ’πœ‰β€–β†’0. Since π‘”βˆ—β„ŽβˆˆπΏ1, there is {𝑝𝑛}βŠ‚β„¬π’±βˆ©πΏ1 such that β€–π‘π‘›βˆ’π‘”βˆ—β„Žβ€–1β†’0. Then π‘“βˆ—(π‘”βˆ—β„Ž)∢=πœ‚βˆˆπ’œπΆ such that β€–π‘“βˆ—π‘π‘›βˆ’πœ‚β€–β†’0. Now, β€–β€–β€–πœ‰βˆ’πœ‚β€–β‰€(π‘“βˆ—π‘”)βˆ—β„Žπ‘›β€–β€–+β€–β€–βˆ’πœ‰π‘“βˆ—π‘π‘›β€–β€–+β€–β€–βˆ’πœ‚(π‘“βˆ—π‘”)βˆ—β„Žπ‘›βˆ’ξ€·π‘“βˆ—π‘”π‘›ξ€Έβˆ—β„Žπ‘›β€–β€–+β€–β€–ξ€·π‘“βˆ—π‘”π‘›ξ€Έβˆ—β„Žπ‘›βˆ’π‘“βˆ—π‘π‘›β€–β€–.(3.9) Using (3.4), β€–β€–(π‘“βˆ—π‘”)βˆ—β„Žπ‘›βˆ’ξ€·π‘“βˆ—π‘”π‘›ξ€Έβˆ—β„Žπ‘›β€–β€–=β€–β€–ξ€Ίξ€·π‘“βˆ—π‘”βˆ’π‘”π‘›ξ€Έξ€»βˆ—β„Žπ‘›β€–β€–β€–β€–π‘”β‰€β€–π‘“β€–π‘›β€–β€–βˆ’π‘”1β€–β€–β„Žπ‘›β€–β€–1⟢0asπ‘›βŸΆβˆž.(3.10) Finally, use Theorem 2.1(e) and (3.4) to write β€–β€–ξ€·π‘“βˆ—π‘”π‘›ξ€Έβˆ—β„Žπ‘›βˆ’π‘“βˆ—π‘π‘›β€–β€–=β€–β€–ξ€·π‘”π‘“βˆ—π‘›βˆ—β„Žπ‘›βˆ’π‘π‘›ξ€Έβ€–β€–ξ€·β€–β€–π‘”β‰€β€–π‘“β€–π‘›β€–β€–βˆ’π‘”1β€–β€–β„Žπ‘›β€–β€–1+‖𝑔‖1β€–β€–β„Žπ‘›β€–β€–βˆ’β„Ž1+β€–β€–π‘π‘›β€–β€–βˆ’π‘”βˆ—β„Ž1ξ€ΈβŸΆ0asπ‘›βŸΆβˆž.(3.11)(c) The Alexiewicz norm is invariant under translation [4, Theorem  28] so πœπ‘§(π‘“βˆ—π‘”)βˆˆπ’œπΆ. Use Theorem 2.1(f) to write β€–πœπ‘§(π‘“βˆ—π‘”)βˆ’πœπ‘§(π‘“βˆ—π‘”π‘›)β€–=β€–π‘“βˆ—π‘”βˆ’π‘“βˆ—π‘”π‘›β€–=β€–πœπ‘§(π‘“βˆ—π‘”)βˆ’(πœπ‘§π‘“)βˆ—π‘”π‘›)β€–=β€–πœπ‘§(π‘“βˆ—π‘”)βˆ’π‘“βˆ—(πœπ‘§π‘”π‘›)β€–. Translation invariance of the 𝐿1 norm completes the proof. (d) From (a), we have ‖Φ𝑓‖=sup‖𝑔‖1=1β€–π‘“βˆ—π‘”β€–β‰€sup‖𝑔‖1=1‖𝑓‖‖𝑔‖1=‖𝑓‖. We get equality by considering 𝑓 and 𝑔 to be positive functions in 𝐿1. To prove ‖Ψ𝑔‖≀‖𝑔‖1, note that ‖Ψ𝑔‖=sup‖𝑓‖=1β€–π‘“βˆ—π‘”β€–β‰€sup‖𝑓‖=1‖𝑓‖‖𝑔‖1=‖𝑔‖1. We get equality by considering 𝑓 and 𝑔 to be positive functions in 𝐿1. (e) First consider π‘”βˆˆβ„¬π’±βˆ©πΏ1. We have π‘“βˆ—π‘”π‘‘ξ€œ(π‘₯)=βˆžβˆ’βˆžξ‚€π‘¦π‘“(π‘₯βˆ’π‘¦)𝑔𝑑𝑑𝑦𝑑=ξ€œβˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘‘π‘¦)𝑔(𝑦)𝑑𝑦.(3.12) For βˆ’βˆž<𝛼<𝛽<∞, ||||ξ€œπ›½π›Όξ€Ίπ‘“βˆ—π‘”π‘‘ξ€»||||=||||ξ€œ(π‘₯)βˆ’π‘Žπ‘“(π‘₯)𝑑π‘₯π›½π›Όξ€œβˆžβˆ’βˆž[𝑓]𝑔||||=||||ξ€œ(π‘₯βˆ’π‘‘π‘¦)βˆ’π‘“(π‘₯)(𝑦)𝑑𝑦𝑑π‘₯βˆžβˆ’βˆžξ€œπ›½π›Ό[]||||β‰€ξ€œπ‘“(π‘₯βˆ’π‘‘π‘¦)βˆ’π‘“(π‘₯)𝑔(𝑦)𝑑π‘₯𝑑𝑦(3.13)βˆžβˆ’βˆžβ€–β€–πœπ‘‘π‘¦β€–β€–||||π‘“βˆ’π‘“π‘”(𝑦)𝑑𝑦≀2‖𝑓‖‖𝑔‖1.(3.14) By dominated convergence, we can take the limit 𝑑→0 inside the integral (3.14). Continuity of 𝑓 in the Alexiewicz norm then shows β€–π‘“βˆ—π‘”π‘‘βˆ’π‘Žπ‘“β€–β†’0 as 𝑑→0.
Now take a sequence {𝑔(𝑛)}βŠ‚β„¬π’±βˆ©πΏ1 such that ‖𝑔(𝑛)βˆ’π‘”β€–1β†’0. Define 𝑔𝑑(𝑛)(π‘₯)=𝑔(𝑛)(π‘₯/𝑑)/𝑑 and π‘Ž(𝑛)=βˆ«βˆžβˆ’βˆžπ‘”(𝑛)(π‘₯)𝑑π‘₯. We have
β€–β€–π‘“βˆ—π‘”π‘‘β€–β€–β‰€β€–β€–βˆ’π‘Žπ‘“π‘“βˆ—π‘”π‘‘(𝑛)βˆ’π‘Ž(𝑛)𝑓‖‖+β€–β€–π‘“βˆ—π‘”π‘‘(𝑛)βˆ’π‘“βˆ—π‘”π‘‘β€–β€–+β€–β€–π‘Ž(𝑛)β€–β€–π‘“βˆ’π‘Žπ‘“.(3.15) By the inequality in (a), β€–π‘“βˆ—π‘”π‘‘(𝑛)βˆ’π‘“βˆ—π‘”π‘‘β€–β‰€β€–π‘“β€–β€–π‘”π‘‘(𝑛)βˆ’π‘”π‘‘β€–1. Whereas, ‖‖𝑔𝑑(𝑛)βˆ’π‘”π‘‘β€–β€–1=ξ€œβˆžβˆ’βˆž|||𝑔(𝑛)ξ‚€π‘₯𝑑π‘₯βˆ’π‘”π‘‘ξ‚|||𝑑π‘₯𝑑=‖‖𝑔(𝑛)β€–β€–βˆ’π‘”1⟢0asπ‘›βŸΆβˆž,(3.16) and β€–π‘Ž(𝑛)π‘“βˆ’π‘Žπ‘“β€–=|π‘Ž(𝑛)βˆ’π‘Ž|‖𝑓‖=‖𝑔(𝑛)βˆ’π‘”β€–1‖𝑓‖. Given πœ–>0 fix 𝑛 large enough so that β€–π‘“βˆ—π‘”π‘‘(𝑛)βˆ’π‘“βˆ—π‘”π‘‘β€–+β€–π‘Ž(𝑛)π‘“βˆ’π‘Žπ‘“β€–<πœ–. Now let 𝑑→0 in (3.15).
The interchange of order of integration in (3.13) is justified as follows. A change of variables and Proposition A.3 give
ξ€œπ›½π›Όξ€œβˆžβˆ’βˆžξ€œπ‘“(π‘₯βˆ’π‘‘π‘¦)𝑔(𝑦)𝑑𝑦𝑑π‘₯=π›½π›Όξ€œβˆžβˆ’βˆžξ‚€π‘“(𝑦)𝑔π‘₯βˆ’π‘¦π‘‘ξ‚π‘‘π‘¦π‘‘=ξ€œπ‘‘π‘₯βˆžβˆ’βˆžξ€œπ›½π›Όξ‚€π‘“(𝑦)𝑔π‘₯βˆ’π‘¦π‘‘ξ‚π‘‘π‘₯𝑑𝑦𝑑,ξ€œβˆžβˆ’βˆžξ€œπ›½π›Όπ‘“ξ€(π‘₯βˆ’π‘‘π‘¦)𝑔(𝑦)𝑑π‘₯𝑑𝑦=βˆžβˆ’βˆžπ‘“ξ‚€π‘¦(π‘₯)π‘”π‘‘ξ‚πœ’(π›Όβˆ’π‘¦,π›½βˆ’π‘¦)(π‘₯)𝑑π‘₯𝑑𝑦𝑑=ξ€βˆžβˆ’βˆžξ‚€π‘¦π‘“(π‘₯)π‘”π‘‘ξ‚πœ’(π›Όβˆ’π‘¦,π›½βˆ’π‘¦)(π‘₯)𝑑𝑦𝑑=ξ€œπ‘‘π‘₯βˆžβˆ’βˆžξ€œπ›½π›Όπ‘“ξ‚€(π‘₯)π‘”π‘¦βˆ’π‘₯𝑑𝑑𝑦𝑑𝑑π‘₯.(3.17) Note that βˆ«π›½π›Όβˆ«βˆžβˆ’βˆžβˆ«π‘“(π‘₯)𝑔(𝑦)𝑑𝑦𝑑π‘₯=βˆžβˆ’βˆžβˆ«π›½π›Όπ‘“(π‘₯)𝑔(𝑦)𝑑π‘₯𝑑𝑦 by Corollary A.4. (f) This follows from the equivalence of Definitions 1.1 and 3.1, proved in Proposition 3.5, see [6, Theorems  5.4-2 and 5.3-1].

Young's inequality states that β€–π‘“βˆ—π‘”β€–π‘β‰€β€–π‘“β€–π‘β€–π‘”β€–1 when π‘“βˆˆπΏπ‘ for some 1β‰€π‘β‰€βˆž and π‘”βˆˆπΏ1. Part (a) of Theorem 3.4 extends this to π‘“βˆˆπ’œπΆ. see [1] for other results when π‘“βˆˆπΏπ‘.

The fact that convolution is linear in both arguments, together with (b), shows that π’œπΆ is an 𝐿1-module over the 𝐿1 convolution algebra, see [8] for the definition. It does not appear that π’œπΆ is a Banach algebra under convolution.

We now show that Definition 1.1(iii) and the aforementioned definitions agree.

Proposition 3.5. Let π‘“βˆˆπ’œπΆ, π‘”βˆˆπΏ1, and πœ™βˆˆπ’Ÿ. Define ∫𝐹(𝑦)=π‘¦βˆ’βˆžπ‘“ and ∫𝐺(π‘₯)=π‘₯βˆ’βˆžπ‘”. Definitions 1.1 and 3.1 both give ξ€œβŸ¨π‘“βˆ—π‘”,πœ™βŸ©=βˆžβˆ’βˆžξ€œπ‘“(𝑦)βˆžβˆ’βˆž=𝑔(π‘₯)πœ™(π‘₯+𝑦)𝑑π‘₯π‘‘π‘¦βˆžβˆ’βˆžπΉ(𝑦)𝐺(π‘₯)πœ™ξ…ž(π‘₯+𝑦)𝑑π‘₯𝑑𝑦.(3.18)

Proof. Let ∫Φ(𝑦)=βˆžβˆ’βˆžπ‘”(π‘₯)πœ™(π‘₯+𝑦)𝑑π‘₯. Then Φ∈𝐢∞(ℝ) and Ξ¦ξ…žβˆ«(𝑦)=βˆžβˆ’βˆžπ‘”(π‘₯)πœ™ξ…ž(π‘₯+𝑦)𝑑π‘₯. Also, βˆ«βˆžβˆ’βˆž|Ξ¦ξ…žβˆ«(𝑦)|π‘‘π‘¦β‰€βˆžβˆ’βˆžβˆ«|𝑔(π‘₯)|βˆžβˆ’βˆž|πœ™ξ…ž(π‘₯+𝑦)|𝑑𝑦𝑑π‘₯≀‖𝑔‖1β€–πœ™ξ…žβ€–1, so Φ∈𝐴𝐢(ℝ). Dominated convergence then shows lim|𝑦|β†’βˆžΞ¦(𝑦)=0. Integration by parts now gives (3.18).
Let {𝑔𝑛}βŠ‚β„¬π’±βˆ©πΏ1 such that β€–π‘”π‘›βˆ’π‘”β€–1β†’0. Since convergence in β€–β‹…β€– implies convergence in π’Ÿξ…ž, we have
βŸ¨π‘“βˆ—π‘”,πœ™βŸ©=limπ‘›β†’βˆžβŸ¨π‘“βˆ—π‘”π‘›,πœ™βŸ©=limπ‘›β†’βˆžξ€βˆžβˆ’βˆžπ‘“(𝑦)𝑔𝑛(π‘₯βˆ’π‘¦)πœ™(π‘₯)𝑑𝑦𝑑π‘₯=limπ‘›β†’βˆžξ€œβˆžβˆ’βˆžπ‘“ξ€œ(𝑦)βˆžβˆ’βˆžπ‘”π‘›(π‘₯βˆ’π‘¦)πœ™(π‘₯)𝑑π‘₯𝑑𝑦.(3.19) Proposition A.3 allows interchange of the iterated integrals. Define Ξ¦π‘›βˆ«(𝑦)=βˆžβˆ’βˆžπ‘”π‘›(π‘₯)πœ™(π‘₯+𝑦)𝑑π‘₯. Then, 𝑉Φ𝑛≀‖𝑔𝑛‖1β€–πœ™ξ…žβ€–1≀(‖𝑔‖1+1)β€–πœ™ξ…žβ€–1 for large enough 𝑛. Hence, Φ𝑛 is of uniform bounded variation. Theorem  22 in [4], then gives βˆ«βŸ¨π‘“βˆ—π‘”,πœ™βŸ©=βˆžβˆ’βˆžπ‘“(𝑦)limπ‘›β†’βˆžΞ¦π‘›βˆ«(𝑦)𝑑𝑦=βˆžβˆ’βˆžβˆ«π‘“(𝑦)βˆžβˆ’βˆžπ‘”(π‘₯βˆ’π‘¦)πœ™(π‘₯)𝑑π‘₯𝑑𝑦. The last step follows since β€–π‘”π‘›βˆ’π‘”β€–1β†’0.

If π‘”βˆˆπΏ1⧡ℬ𝒱, then π‘“βˆ—π‘” needs not to be continuous or bounded. For example, take 1/2≀𝛼<1 and let 𝑓(π‘₯)=𝑔(π‘₯)=π‘₯βˆ’π›Όπœ’(0,1)(π‘₯). Then, π‘“βˆˆπΏ1βŠ‚π’œπΆ and π‘”βˆˆπΏ1⧡ℬ𝒱. We have π‘“βˆ—π‘”(π‘₯)=0 for π‘₯≀0. For 0<π‘₯≀1, we have βˆ«π‘“βˆ—π‘”(π‘₯)=π‘₯0π‘¦βˆ’π›Ό(π‘₯βˆ’π‘¦)βˆ’π›Όπ‘‘π‘¦=π‘₯1βˆ’2π›Όβˆ«10π‘¦βˆ’π›Ό(1βˆ’π‘¦)βˆ’π›Όπ‘‘π‘¦=π‘₯1βˆ’2𝛼Γ2(1βˆ’π›Ό)/Ξ“(2βˆ’2𝛼). Hence, π‘“βˆ—π‘” is not continuous at 0. If 1/2<𝛼<1, then π‘“βˆ—π‘” is unbounded at 0.

As another example, consider 𝑓(π‘₯)=sin(πœ‹π‘₯)/log|π‘₯| and 𝑔(π‘₯)=πœ’(0,1)(π‘₯). Then π‘“βˆˆπ’œπΆ and for each 1β‰€π‘β‰€βˆž, we have π‘”βˆˆβ„¬π’±βˆ©πΏπ‘. And,

ξ€œπ‘“βˆ—π‘”(π‘₯)=π‘₯π‘₯βˆ’1sin(πœ‹π‘¦)=log(𝑦)𝑑𝑦forπ‘₯β‰₯2cos(πœ‹(π‘₯βˆ’1))βˆ’πœ‹log(π‘₯βˆ’1)cos(πœ‹π‘₯)βˆ’1πœ‹log(π‘₯)πœ‹ξ€œπ‘₯π‘₯βˆ’1cos(πœ‹π‘¦)𝑦log2(𝑦)π‘‘π‘¦βˆΌβˆ’2cos(πœ‹π‘₯)πœ‹log(π‘₯)asπ‘₯⟢∞.(3.20)

Therefore, by Theorem 2.1(d), π‘“βˆ—π‘”βˆˆπΆ0(ℝ), and lim|π‘₯|β†’βˆžπ‘“βˆ—π‘”(π‘₯)=0 but for each 1≀𝑝<∞, we have π‘“βˆ—π‘”βˆ‰πΏπ‘.

4. Differentiation and Integration

If 𝑔 is sufficiently smooth, then the pointwise derivative is (π‘“βˆ—π‘”)ξ…ž(π‘₯)=π‘“βˆ—π‘”ξ…ž(π‘₯). Recall the definition 𝐴𝐢(ℝ) of primitives of 𝐿1 functions given in the proof of Proposition 3.3. In the following theorem, we require pointwise derivatives of 𝑔 to exist at each point in ℝ.

Theorem 4.1. Let π‘“βˆˆπ’œπΆ, π‘›βˆˆβ„•, and 𝑔(π‘˜)∈𝐴𝐢(ℝ) for each 0β‰€π‘˜β‰€π‘›. Then π‘“βˆ—π‘”βˆˆπΆπ‘›(ℝ) and (π‘“βˆ—π‘”)(𝑛)(π‘₯)=π‘“βˆ—π‘”(𝑛)(π‘₯) for each π‘₯βˆˆβ„.

Proof. First consider 𝑛=1. Let π‘₯βˆˆβ„. Then (π‘“βˆ—π‘”)ξ…ž(π‘₯)=limβ„Žβ†’0ξ€œβˆžβˆ’βˆžξ‚Έπ‘“(𝑦)𝑔(π‘₯+β„Žβˆ’π‘¦)βˆ’π‘”(π‘₯βˆ’π‘¦)β„Žξ‚Ήπ‘‘π‘¦.(4.1) To take the limit inside the integral we can show that the bracketed term in the integrand is of uniform bounded variation for 0<|β„Ž|≀1. Let β„Žβ‰ 0. Since π‘”βˆˆπ΄πΆ(ℝ) it follows that the variation is given by the Lebesgue integrals π‘‰π‘¦βˆˆβ„ξ‚Έπ‘”(π‘₯+β„Žβˆ’π‘¦)βˆ’π‘”(π‘₯βˆ’π‘¦)β„Žξ‚Ή=ξ€œβˆžβˆ’βˆž||||π‘”ξ…ž(π‘₯+β„Žβˆ’π‘¦)βˆ’π‘”ξ…ž(π‘₯βˆ’π‘¦)β„Ž||||β‰€ξ€œπ‘‘π‘¦βˆžβˆ’βˆž||π‘”ξ…žξ…ž||ξ€œ(𝑦)𝑑𝑦+βˆžβˆ’βˆž||||π‘”ξ…ž(π‘₯+β„Žβˆ’π‘¦)βˆ’π‘”ξ…ž(π‘₯βˆ’π‘¦)β„Žβˆ’π‘”ξ…žξ…ž||||(π‘₯βˆ’π‘¦)𝑑𝑦.(4.2) Since π‘”ξ…žβˆˆπ΄πΆ(ℝ), we have π‘”ξ…žξ…žβˆˆπΏ1. The second integral on the right of (4.2) gives the 𝐿1 derivative of π‘”ξ…ž in the limit β„Žβ†’0; see [1, page 246]. Hence, in (4.1), we can use [4, Theorem  22] to take the limit under the integral sign. This then gives (π‘“βˆ—π‘”)ξ…ž(π‘₯)=π‘“βˆ—π‘”ξ…ž(π‘₯). Theorem 2.1(d) now shows (π‘“βˆ—π‘”)ξ…žβˆˆπΆ0(ℝ). Induction on 𝑛 completes the proof.

For similar results when π‘“βˆˆπΏ1, see [1, Proposition  8.10].

Note that π‘”ξ…žβˆˆπ΄πΆ(ℝ) does not imply π‘”βˆˆπ΄πΆ(ℝ). For example, 𝑔(π‘₯)=π‘₯. The conditions 𝑔(π‘˜)βˆˆβ„¬π’± for 0β‰€π‘˜β‰€π‘›+1 imply those in Theorem 4.1. To see this, it suffices to consider 𝑛=1. If π‘”ξ…ž,π‘”ξ…žξ…žβˆˆβ„¬π’±, then π‘”ξ…žξ…ž exists at each point and is bounded. Hence, the Lebesgue integral π‘”ξ…ž(π‘₯)=π‘”ξ…žβˆ«(0)+π‘₯0π‘”ξ…žξ…ž(𝑦)𝑑𝑦 exists for each π‘₯βˆˆβ„ and π‘”ξ…ž is absolutely continuous. Since π‘”ξ…žβˆˆβ„¬π’±, we then have π‘”ξ…žβˆˆπ΄πΆ(ℝ). Similarly, for 𝑛>1. The example 𝑔(π‘₯)=|π‘₯|1.5sin(1/[1+x2]) shows that the 𝐴𝐢(ℝ) condition in the theorem is weaker than the aforementioned ℬ𝒱 condition since 𝑔,π‘”ξ…žβˆˆπ΄πΆ(ℝ) but π‘”ξ…žξ…ž(0) does not exist so π‘”ξ…žξ…žβˆ‰β„¬π’±.

We found that when π‘”βˆˆβ„¬π’±βˆ©πΏ1, then π‘“βˆ—π‘”βˆˆπ’œπΆ. We can compute the distributional derivative (πΉβˆ—π‘”)ξ…ž=π‘“βˆ—π‘”, where 𝐹 is a primitive of 𝑓.

Proposition 4.2. Let 𝐹∈𝐢0(ℝ) and write 𝑓=πΉξ…žβˆˆπ’œπΆ. Let π‘”βˆˆβ„¬π’±βˆ©πΏ1. Then πΉβˆ—π‘”βˆˆπΆ0(ℝ) and (πΉβˆ—π‘”)ξ…ž=π‘“βˆ—π‘”βˆˆπ’œπΆ.

Proof. Let π‘₯,π‘‘βˆˆβ„. Then by the usual HΓΆlder inequality, ||||=||||ξ€œπΉβˆ—π‘”(π‘₯)βˆ’πΉβˆ—π‘”(𝑑)βˆžβˆ’βˆž[]||||𝐹(π‘₯βˆ’π‘¦)βˆ’πΉ(π‘‘βˆ’π‘¦)𝑔(𝑦)𝑑𝑦≀‖𝐹(π‘₯βˆ’β‹…)βˆ’πΉ(π‘‘βˆ’β‹…)β€–βˆžβ€–π‘”β€–1⟢0asπ‘‘βŸΆπ‘₯since𝐹isuniformlycontinuousonℝ.(4.3) Hence, πΉβˆ—π‘” is continuous on ℝ. Dominated convergence shows that limπ‘₯β†’Β±βˆžβˆ«πΉβˆ—π‘”(π‘₯)=𝐹(±∞)βˆžβˆ’βˆžπ‘”. Therefore, πΉβˆ—π‘”βˆˆπΆ0(ℝ).
Let πœ™βˆˆπ’Ÿ. Then
(πΉβˆ—π‘”)ξ…žξ¬ξ«,πœ™=βˆ’πΉβˆ—π‘”,πœ™ξ…žξ¬ξ€=βˆ’βˆžβˆ’βˆžπΉ(π‘₯βˆ’π‘¦)𝑔(𝑦)πœ™ξ…žξ€œ(π‘₯)𝑑𝑦𝑑π‘₯=βˆ’βˆžβˆ’βˆžξ€œπ‘”(𝑦)βˆžβˆ’βˆžπΉ(π‘₯βˆ’π‘¦)πœ™ξ…ž(π‘₯)𝑑π‘₯𝑑𝑦(Fubini-Tonellitheorem).(4.4) Integrate by parts and use the change of variables π‘₯↦π‘₯+𝑦 to get (πΉβˆ—π‘”)ξ…žξ¬=ξ€œ,πœ™βˆžβˆ’βˆžξ€œπ‘”(𝑦)βˆžβˆ’βˆž=ξ€œπ‘“(π‘₯)πœ™(π‘₯+𝑦)𝑑π‘₯π‘‘π‘¦βˆžβˆ’βˆžξ€œπ‘“(π‘₯)βˆžβˆ’βˆž=ξ€œπ‘”(𝑦)πœ™(π‘₯+𝑦)𝑑𝑦𝑑π‘₯(byPropositionA.3)βˆžβˆ’βˆžξ€œπ‘“(π‘₯)βˆžβˆ’βˆž=ξ€œπ‘”(π‘¦βˆ’π‘₯)πœ™(𝑦)𝑑𝑦𝑑π‘₯βˆžβˆ’βˆžξ€œπœ™(𝑦)βˆžβˆ’βˆžπ‘“(π‘₯)𝑔(π‘¦βˆ’π‘₯)𝑑π‘₯𝑑𝑦(byPropositionA.3)=βŸ¨π‘“βˆ—π‘”,πœ™βŸ©.(4.5)

This gives an alternate definition of π‘“βˆ—π‘” for π‘“βˆˆπ’œπΆ and π‘”βˆˆπΏ1.

Theorem 4.3. Let π‘“βˆˆπ’œπΆ, let πΉβˆˆβ„¬πΆ be the primitive of 𝑓 and let π‘”βˆˆπΏ1. Define π‘“βˆ—π‘” as in Definition 3.1. Then (πΉβˆ—π‘”)ξ…ž=π‘“βˆ—π‘”βˆˆπ’œπΆ.

Proof. Let βˆ’βˆž<𝛼<𝛽<∞. Let {𝑔𝑛}βŠ‚β„¬π’±βˆ©πΏ1 such that β€–π‘”π‘›βˆ’π‘”β€–1β†’0. By Proposition 4.2, we have ξ€œπ›½π›Ό(πΉβˆ—π‘”)ξ…žξ€œ=πΉβˆ—π‘”(𝛽)βˆ’πΉβˆ—π‘”(𝛼)=βˆžβˆ’βˆž[]𝐹(𝑦)𝑔(π›½βˆ’π‘¦)βˆ’π‘”(π›Όβˆ’π‘¦)𝑑𝑦.(4.6) As in (3.3), βˆ«π›½π›Όπ‘“βˆ—π‘”π‘›=βˆ«βˆžβˆ’βˆžπΉ(𝑦)[𝑔𝑛(π›½βˆ’π‘¦)βˆ’π‘”π‘›(π›Όβˆ’π‘¦)]𝑑𝑦. Hence, ||||ξ€œπ›½π›Όξ€Ί(πΉβˆ—π‘”)ξ…žβˆ’π‘“βˆ—π‘”π‘›ξ€»||||=||||ξ€œβˆžβˆ’βˆžπΉ(𝑦)𝑔(π›½βˆ’π‘¦)βˆ’π‘”π‘›(ξ€Έβˆ’ξ€·π›½βˆ’π‘¦)𝑔(π›Όβˆ’π‘¦)βˆ’π‘”π‘›(||||π›Όβˆ’π‘¦)ξ€Έξ€»π‘‘π‘¦β‰€β€–πΉβ€–βˆžξ€·β€–β€–π‘”(π›½βˆ’β‹…)βˆ’π‘”π‘›β€–β€–(π›½βˆ’β‹…)1+‖‖𝑔(π›Όβˆ’β‹…)βˆ’π‘”π‘›β€–β€–(π›Όβˆ’β‹…)1‖‖𝑔=2β€–π‘“β€–π‘›β€–β€–βˆ’π‘”1.(4.7) Therefore, β€–(πΉβˆ—π‘”)ξ…žβˆ’π‘“βˆ—π‘”π‘›β€–β‰€2β€–π‘“β€–β€–π‘”π‘›βˆ’π‘”β€–1β†’0 as π‘›β†’βˆž.

The following theorem and its corollary give results on integrating convolutions.

Theorem 4.4. Let π‘“βˆˆπ’œπΆ and let π‘”βˆˆπΏ1. Define ∫𝐹(π‘₯)=π‘₯βˆ’βˆžπ‘“ and ∫𝐺(π‘₯)=π‘₯βˆ’βˆžg. Then, π‘“βˆ—πΊβˆˆπΆ0(ℝ) and π‘“βˆ—πΊ(π‘₯)=πΉβˆ—π‘”(π‘₯) for all π‘₯βˆˆβ„.

Proof. Since 𝐺∈𝐴𝐢(ℝ), Theorem 2.1(d) shows π‘“βˆ—πΊβˆˆπΆ0(ℝ). We have ξ€œπ‘“βˆ—πΊ(π‘₯)=βˆžβˆ’βˆžξ€œπ‘“(𝑦)π‘₯βˆ’π‘¦βˆ’βˆž=𝑔(𝑧)π‘‘π‘§π‘‘π‘¦βˆžβˆ’βˆžπ‘“(𝑦)πœ’(βˆ’βˆž,π‘₯βˆ’π‘¦)=(𝑧)𝑔(𝑧)π‘‘π‘§π‘‘π‘¦βˆžβˆ’βˆžπ‘“(𝑦)πœ’(βˆ’βˆž,π‘₯βˆ’π‘¦)=ξ€œ(𝑧)𝑔(𝑧)π‘‘π‘¦π‘‘π‘§βˆžβˆ’βˆžξ€œπ‘”(𝑧)π‘₯βˆ’π‘§βˆ’βˆžπ‘“(𝑦)𝑑𝑦𝑑𝑧=πΉβˆ—π‘”(π‘₯).(4.8) Proposition A.3 justifies the interchange of orders of integration.

Corollary 4.5. Note that the following hold: (a)π‘“βˆ—π‘”=(πΉβˆ—π‘”)ξ…ž=(π‘“βˆ—πΊ)ξ…ž,(b) for all βˆ’βˆžβ‰€π›Ό<π›½β‰€βˆž, one has βˆ«π›½π›Όπ‘“βˆ—π‘”=πΉβˆ—π‘”(𝛽)βˆ’πΉβˆ—π‘”(𝛼)=π‘“βˆ—πΊ(𝛽)βˆ’π‘“βˆ—πΊ(𝛽).

Hence, the convolution π‘“βˆ—π‘” can be evaluated by taking the distributional derivative of the Lebesgue integral πΉβˆ—π‘”. Since π‘“βˆ—πΊβˆˆπΆ0(ℝ), when π‘“βˆˆπ’œπΆ and πΊβˆˆβ„¬π’±, we can use the equation π‘“βˆ—π‘”=(π‘“βˆ—πΊ)ξ…ž to define π‘“βˆ—π‘” for π‘“βˆˆπ’œπΆ and 𝑔=πΊξ…ž for πΊβˆˆβ„¬π’±. In this case, 𝑔 will be a signed Radon measure. As ∫𝐺(π‘₯)=π‘₯βˆ’βˆžπ‘” and this integral is a regulated primitive integral [5], we will save this case for discussion elsewhere.

Appendix

The integration by parts formula is as follows. If π‘“βˆˆπ’œπΆ and π‘”βˆˆβ„¬π’±, it gives the integral of 𝑓𝑔 in terms of a Henstock-Stieltjes integral:

ξ€œβˆžβˆ’βˆžξ€œπ‘“π‘”=𝐹(∞)𝑔(∞)βˆ’βˆžβˆ’βˆžπΉπ‘‘π‘”,(A.1) see [4] and [9, page 199].

We have the following corollary for functions of essential bounded variation.

Corollary A.1. Let 𝐹∈𝐢0(ℝ). Let π‘”βˆˆβ„°β„¬π’±. Fix 0≀𝛾≀1. Take π‘”π›Ύβˆˆπ’©β„¬π’±π›Ύ such that 𝑔𝛾=𝑔 almost everywhere. Let πœ‡π‘” be the signed Radon measure given by π‘”ξ…ž. Then βˆ«βˆžβˆ’βˆžπΉπ‘‘π‘”π›Ύ=βˆ«βˆžβˆ’βˆžπΉπ‘‘πœ‡π‘”.

Proof. The distributional derivative of 𝑔 is βŸ¨π‘”ξ…ž,πœ™βŸ©=βˆ’βŸ¨π‘”,πœ™ξ…žβˆ«βŸ©=βˆ’βˆžβˆ’βˆžπ‘”πœ™ξ…ž=βˆ«βˆžβˆ’βˆžπœ™π‘‘πœ‡π‘” for all πœ™βˆˆπ’Ÿ. Note that 𝑔𝛾 is unique and πœ‡π‘”=πœ‡π‘”π›Ύ. Suppose πœ™βˆˆπ’Ÿ with supp(πœ™)βŠ‚[𝐴,𝐡]βŠ‚β„. Then, using integration by parts for the Henstock-Stieltjes integral: 𝑔𝛾,πœ™ξ…žξ¬=ξ€œπ΅π΄π‘”π›Ύπœ™ξ…ž=𝑔𝛾(𝐡)πœ™(𝐡)βˆ’π‘”π›Ύξ€œ(𝐴)πœ™(𝐴)βˆ’π΅π΄πœ™π‘‘π‘”π›Ύξ€œ=βˆ’βˆžβˆ’βˆžπœ™π‘‘π‘”π›Ύ=βˆ’βŸ¨π‘”ξ…žπ›Ύξ€œ,πœ™βŸ©=βˆ’βˆžβˆ’βˆžπœ™π‘‘πœ‡π‘”π›Ύξ€œ=βˆ’βˆžβˆ’βˆžπœ™π‘‘πœ‡π‘”.(A.2)
Let 𝐹∈𝐢0(ℝ). There is a uniformly bounded sequence {πœ™π‘›}βŠ‚π’Ÿ such that πœ™π‘›β†’πΉ pointwise on ℝ. By dominated convergence,
limπ‘›β†’βˆžξ€œβˆžβˆ’βˆžπœ™π‘›π‘‘π‘”π›Ύ=ξ€œβˆžβˆ’βˆžπΉπ‘‘π‘”π›Ύ=limπ‘›β†’βˆžξ€œβˆžβˆ’βˆžπœ™π‘›π‘‘πœ‡π‘”=ξ€œβˆžβˆ’βˆžπΉπ‘‘πœ‡π‘”.(A.3)

Corollary A.1 now justifies the following definition.

Definition A.2. Let π‘“βˆˆπ’œπΆ and let πΉβˆˆβ„¬πΆ be its primitive. Let π‘”βˆˆβ„°β„¬π’±. Fix 0≀𝛾≀1 and take π‘”π›Ύβˆˆπ’©β„¬π’±π›Ύ such that 𝑔𝛾=𝑔 almost everywhere. Define ξ€œβˆžβˆ’βˆžπ‘“π‘”=π‘”π›Ύξ€œ(∞)𝐹(∞)βˆ’βˆžβˆ’βˆžπΉπ‘‘πœ‡π‘”=ξ€œβˆžβˆ’βˆžπ‘“π‘”π›Ύ.(A.4)

Since limits at infinity are not affected by the choice of 𝛾, the definition is independent of 𝛾.

The HΓΆlder inequality is

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