## Banach Journal of Mathematical Analysis

### Structural transition between $L^{p}(G)$ and $L^{p}(G/H)$

#### Abstract

Let $H$ be a compact subgroup of a locally compact group $G$. We consider the homogeneous space $G/H$ equipped with a strongly quasi-invariant Radon measure $\mu$. For $1\leq p\leq +\infty$, we introduce a norm decreasing linear map from $L^{p}(G)$ onto $L^{p}(G/H,\mu)$ and show that $L^{p}(G/H,\mu)$ may be identified with a quotient space of $L^{p}(G)$. Also, we prove that $L^{p}(G/H,\mu)$ is isometrically isomorphic to a closed subspace of $L^{p}(G)$. These help us study the structure of the classical Banach spaces constructed on a homogeneous space via those created on topological groups.

#### Article information

Source
Banach J. Math. Anal., Volume 9, Number 3 (2015), 194-205.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.bjma/1419001712

Digital Object Identifier
doi:10.15352/bjma/09-3-14

Mathematical Reviews number (MathSciNet)
MR3296134

Zentralblatt MATH identifier
1314.43005

#### Citation

Tavallaei, Narguess; Ramezanpour, Mohammad; Olfatian Gillan, Behrooz. Structural transition between $L^{p}(G)$ and $L^{p}(G/H)$. Banach J. Math. Anal. 9 (2015), no. 3, 194--205. doi:10.15352/bjma/09-3-14. https://projecteuclid.org/euclid.bjma/1419001712

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