Some Intersections of the Weighted L p -Spaces

Abstract

Let $G$ be a locally compact group $\mathrm{\Omega }$ an arbitrary family of the weight functions on $\mathrm{G,}$ and . The locally convex space $I{L}_p(G,\mathrm{\Omega })$ as a subspace of ${\cap }_{\omega \in \mathrm{\Omega }}{L}^p(G,\omega )$ is defined. Also, some sufficient conditions for that space to be a Banach space are provided. Furthermore, for an arbitrary subset $J$ of $[\mathrm{1},\mathrm{\infty })$ and a positive submultiplicative weight function $\omega$ on $G$, Banach subspace $I{L}_J(G,\omega )$ of ${\cap }_{p\in J}{L}^p(G,\omega )$ is introduced. Then some algebraic properties of $I{L}_J(G,\omega )$, as a Banach algebra under convolution product, are investigated.