Convolution of vector-valued distributions: A survey and comparison

Abstract

We overview the literature concerning bilinear operations on vector-valued distributions in general, and more specifically the convolution of vector-valued functions or distributions. We compare and evaluate the different approaches to this problem of L. Schwartz on the one hand and of Y. Hirata and R. Shirarishi on the other. Moreover we discuss applications of the general existence and uniqueness results to different branches of mathematical analysis like partial differential equations or harmonic analysis.

New distribution spaces associated to translation-invariant Banach spaces

“…Fix ϕ ∈ S(R n ) and write again ϕ(x) = m∈Z n ψ(x − m)ϕ(x), where ψ is the partition of the unity used above. We use ρ m ∈ D [−1,1] n as in (2). Taking (6) into account and the fact that B = {ρ m : m ∈ Z n } is a bounded subset of D(R n ) (cf.…”

Convolution with the kernel $e^{s\langle x\rangle^q}, q\geq 1, s>0$ within ultradistribution spaces

“…Let x ∈ R d , |x| ≥ 2, be arbitrary but fixed. For every t ∈ B(−k 1 |x| −1+(q−1)/(q 1 −1) x, 1), we infer 1) .…”

“…When q 1 ≤ 1, this reduces to the definition of S ′{Mp} {p! 1/q 1 } -convolution of ultradistributions, see [17,Definition 5.7 and Theorem 5.8] (see also [6,14,9,4]) and is analogous to one of the equivalent formulations of S ′ -convolution in the distributional setting (see [19,20]; see also [1,5,12,13]).…”

Convolutors of translation-modulation invariant Banach spaces of ultradistributions

“…Convolution constitutes as one of the most important tools in mathematical analysis. In the theory of generalized functions it has been an extensively studied subject going back to Schwartz's work [41], where new results are still found until this day [1,17,33,34,35,43]. When looking at the theory of ultradistributions, a considerable amount of literature can be found on the existence of convolution in both the non-quasianalytic case [15,29,36,37] and the quasianalytic case [18,38].…”