### The Canonical Mapping $T_H$ for Weighted $L^p$-Spaces the General Case

#### Abstract

Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. It is well-known that $G/H$ as a homogeneous space admits a strongly quasi invariant measure and the linear mapping $T_H$ of $L^1(G)$ into $L^1(G/H)$ is bounded and surjective. In this note it is shown that by means of complex interpolation theorem, that under restrictions on weight function $\omega$, the mapping $T_H$ of weighted spaces $L^p(G,\omega)$ into $L^p(G/H,\varpi)$ is well-defined, bounded linear and surjective , for $1\leq p \leq \infty$.

#### Keywords

homogeneous space, weighted space, rho-function, quasi invariant measure, relatively invariant measure

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