The Space Ωmp(Rd)) and Some Properties

Abstract

Let m be a v-moderate function defined on R d and let g L 2(R d ). In this work, we defineΩ <inf>m</inf> p (R d ) to be the vector space of f L <inf>m</inf> 2 (R d ) such that the Gabor transform V f belongs to L p (R 2d ), where 1 ≤ p < ∞. We equip it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of Ω <inf>m</inf> p (R d ). We also discuss inclusion properties and obtain the dual space of Ω <inf>m</inf> p (R d ). At the end of this work, we study multipliers from L <inf>w</inf> 1 (R d ) into Ω <inf>w</inf> p (R d ) and from Ω <inf>w</inf> p (R d ) into L <inf>w-1</inf> ∞ (R d ), where w is the Beurling weight function. © 2006 Springer Science+Business Media, Inc.