The space Ω m p (R d ) and some properties

Let m be a v-moderate function defined on R ^d and let g ∈ L ²(R ^d ). In this work, we defineΩ _m ^p (R ^d ) to be the vector space of f ∈ L _m 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 _m ^p (R ^d ). We also discuss inclusion properties and obtain the dual space of Ω _m ^p (R ^d ). At the end of this work, we study multipliers from L _w ¹ (R ^d ) into Ω _w ^p (R ^d ) and from Ω _w ^p (R ^d ) into L _w−1 ^∞ (R ^d ), where w is the Beurling weight function. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 139–145, January, 2006.