‎Some‎ relations between ‎$‎L^p‎$‎-spaces on locally compact group ‎$‎G‎$ ‎and‎ double coset $K\setminus G/H‎$

Subject Areas : Abstract harmonic analysis

R. A.  Kamyabi Gol ¹ (Department of Pure Mathematics, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebric Structures (CEAAS), P. O. Box 1159-91775, Mashhad, Iran)
F.  Fahimian ² (Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran)
F.  Esmaeelzadeh ³ (Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran)

Received: 2020-02-29 Accepted : 2020-06-24 Published : 2020-06-01

Keywords: Double coset space, $L^p(K\setminus G/H, \mu )$, quotient space of $L^p(G)$, duality of $L^p(K/G, \mu)$‎,

Abstract :

Let $H$ and $K$ be compact subgroups of locally compact group $G$. By considering the double coset space $K\setminus G/H$, which equipped with an $N$-strongly quasi invariant measure $\mu$, for $1\leq p\leq +\infty$, we make a norm decreasing linear map from $L^p(G)$ onto $L^p(K\setminus G/H,\mu)$ and demonstrate that it may be identified with a quotient space of $L^p(G)$. In addition, we illustrate that $L^p(K\setminus G/H, \mu)$ is isometrically isomorphic to a closed subspace of $L^p(G)$. These assist us to study the structure of the classical Banach space created on a double coset space by those produced on topological space.

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