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

Let H and K be compact subgroups of locally compact group G. By considering the double coset space K\ G/H , which equipped with an N-strongly quasi invariant measure u, for 1 ⩽ p ⩽ +1, we make a norm decreasing linear map from L^p(G) onto Lp(K\ G;u) and demonstrate that it may be identified with a quotient space of L^p(G). In addition, we illustrate that L^p(K\ G/H ; u) 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.