New distribution spaces associated to translation-invariant Banach spaces

Abstract

We introduce and study new distribution spaces, the test function space $\mathcal{D}_E$ and its strong dual $\mathcal{D}'_{E'_{\ast}}$. These spaces generalize the Schwartz spaces $\mathcal{D}_{L^{q}}$, $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$ and their weighted versions. The construction of our new distribution spaces is based on the analysis of a suitable translation-invariant Banach space of distributions $E$, which turns out to be a convolution module over a Beurling algebra $L^{1}_{\omega}$. The Banach space $E'_{\ast}$ stands for $L_{\check{\omega}}^1\ast E'$. We also study convolution and multiplicative products on $\mathcal{D}'_{E'_{\ast}}$

Translation–Modulation Invariant Banach Spaces of Ultradistributions

“…We mention that Definition 3.1 is intrinsically related to the notion of translation invariant Banach spaces of ultradistributions (TIB) studied by the authors in [6,7,8,9]. In fact, one readily verifies that E is a TMIB of class * − † if and only if E is a TIB of class * − † and F E is a TIB of class † − * (the latter readily follows from the identity F M ξ f = T ξ F f and the definition of the norm on the Banach space F E).…”

Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces

“…In this section we fix the notation and collect some notions that will be needed in the article. In particular, we give a short overview of properties of translation-invariant Banach spaces of tempered distributions and their associated distribution spaces of type D ′ E ′ * introduced in [9]. We use the standard notation from distribution theory [27,32].…”

“…Recall that the dual of L 1 ω is L ∞ ω , the Banach space of measurable functions satisfying ||u|| ∞,ω := ess sup x∈R n |u(x)|/ω(x) < ∞. We have proved in [9] that the convolution * : S(R n ) × S(R n ) → S(R n ) extends to * : L 1 ω × E → E in such a way that E is a Banach module over the Beurling algebra L 1 ω , i.e., ||u * g|| E ≤ ||u|| 1,ω ||g|| E .…”

Equivalent Conditions for Integrability of Distributions

“…The authors gave general functional definitions and proved fundamental results on convolvability and the convolution of Roumieu ultradistributions in a way analogous to the known general approaches of Chevalley and Schwartz in case of distributions. For other aspects of the theory see e. g. [2,4,6,7,22,24,27,28]. See also the recent article [23] for results concerning the quasianalytic case.…”