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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 \(T_p^\ast\).
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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