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Structural transition between $$L^{p}(G)$$ and $$L^{p}(G/H)$$. (English) Zbl 1314.43005
This paper is devoted to present a systematic and straightforward extension of results derived in [H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups. 2nd ed. Oxford: Clarendon Press (2000; Zbl 0965.43001)], [A. G. Farashahi, Bull. Malays. Math. Sci. Soc. (2) 36, No. 4, 1109–1122 (2013; Zbl 1291.43003)] and [A. Ghaani Farashahi and R. A. Kamyabi-Gol, “Frames and homogeneous spaces” (English, Persian summary), J. Sci. Islam. Repub. Iran 22, No. 4, 355–361 (2011)], when the measure on the homogeneous space is not totally invariant under the group action. The idea of their work originated from a natural normalization of the linear operator $$T=T_H$$, which is called $$T_p$$, with respect to the rho-function and each $$L^p$$-space.
The only interesting results of the article, which are not straightforward from the results of the above mentioned papers, are Theorem 3.4 and Proposition 4.3. Theorem 3.4 presents $$L^\infty$$-extension of the linear operator $$P=P_H$$ and Proposition 4.3 gives a characterization for the adjoint operator $$T_p^\ast$$.
##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 46B25 Classical Banach spaces in the general theory
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##### References:
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