On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Abstract

Let 𝐺 be a locally compact group with a fixed left Haar measure 𝜆 and Ω be a system of weights on 𝐺. In this paper, we deal with locally convex space 𝐿𝑝(𝐺,Ω) equipped with the locally convex topology generated by the family of norms (‖.‖𝑝,𝜔)𝜔∈Ω. We study various algebraic and topological properties of the locally convex space 𝐿𝑝(𝐺,Ω). In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which 𝐿𝑝(𝐺,Ω) with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.