ON FUNCTION SPACES CHARACTERIZED BY THE WIGNER TRANSFORM

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.