ON FUNCTION SPACES CHARACTERIZED BY THE WIGNER TRANSFORM

Year 2021, , 188 - 200, 31.07.2021

Öznur Kulak

https://doi.org/10.33773/jum.958029

Cited By: 1

Abstract

Let $\omega _{i}$ be weight
functions on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, (i=1,2,3,4). In this work, we define $CW_{\omega _{1},\omega _{2},\omega _{3},\omega
_{4}}^{p,q,r,s,\tau }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ to be vector space of $\left( f,g\right) \in \left( L_{\omega
_{1}}^{p}\times L_{\omega _{2}}^{q}\right) \left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ such that the $\tau -$Wigner transforms $W_{\tau }\left(
f,.\right) $ and $W_{\tau }\left( .,g\right) $ belong to $L_{\omega
_{3}}^{r}\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}\right) $ and $L_{\omega _{4}}^{s}\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}\right) $ respectively for $1\leq p,q,r,s<\infty $, $\tau \in \left(
0,1\right) $. We endow this space with a sum norm and prove that $%
CW_{\omega _{1},\omega _{2},\omega _{3},\omega _{4}}^{p,q,r,s,\tau }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ is a Banach space. We also show that $CW_{\omega _{1},\omega
_{2},\omega _{3},\omega _{4}}^{p,q,r,s,\tau }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ becomes an essential Banach module over $\left( L_{\omega
_{1}}^{1}\times L_{\omega _{2}}^{1}\right) \left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $. We then consider approximate identities.

Keywords

Wigner transform, Essential Banach module, Approximate identity

Supporting Institution

Giresun University

Project Number

FEN-BAP-C-150219-01

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