ON FUNCTION SPACES CHARACTERIZED BY THE WIGNER TRANSFORM

Authors : Öznur KULAK

Pages : 188-200

Doi:10.33773/jum.958029

View : 51 | Download : 12

Publication Date : 2021-07-31

Article Type : Research Paper

Abstract :Let $\omega _{i}$ be weight functions on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $, insert ignore into journalissuearticles values(i=1,2,3,4);. In this work, we define $CW_{\omega _{1},\omega _{2},\omega _{3},\omega _{4}}^{p,q,r,s,\tau }\leftinsert ignore into journalissuearticles values( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \right); $ to be vector space of $\leftinsert ignore into journalissuearticles values( f,g\right); \in \leftinsert ignore into journalissuearticles values( L_{\omega _{1}}^{p}\times L_{\omega _{2}}^{q}\right); \leftinsert ignore into journalissuearticles values( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \right); $ such that the $\tau -$Wigner transforms $W_{\tau }\leftinsert ignore into journalissuearticles values( f,.\right); $ and $W_{\tau }\leftinsert ignore into journalissuearticles values( .,g\right); $ belong to $L_{\omega _{3}}^{r}\leftinsert ignore into journalissuearticles values( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\right); $ and $L_{\omega _{4}}^{s}\leftinsert ignore into journalissuearticles values( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\right); $ respectively for $1\leq p,q,r,s<\infty $, $\tau \in \leftinsert ignore into journalissuearticles values( 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 }\leftinsert ignore into journalissuearticles values( %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 }\leftinsert ignore into journalissuearticles values( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \right); $ becomes an essential Banach module over $\leftinsert ignore into journalissuearticles values( L_{\omega _{1}}^{1}\times L_{\omega _{2}}^{1}\right); \leftinsert ignore into journalissuearticles values( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \right); $. We then consider approximate identities.
Keywords : Wigner transform, Essential Banach module, Approximate identity

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