Some relations between $L^p$-spaces on locally compact group $G$ and double coset $K\setminus G/H$
الموضوعات :R. A. Kamyabi Gol 1 , F. Fahimian 2 , F. Esmaeelzadeh 3
1 - Department of Pure Mathematics, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebric Structures (CEAAS), P. O. Box 1159-91775, Mashhad, Iran
2 - Department of Pure Mathematics, Ferdowsi University of Mashhad,
P. O. Box 1159-91775, Mashhad, Iran
3 - Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran
الکلمات المفتاحية: Double coset space, $L^p(K\setminus G/H, \mu )$, quotient space of $L^p(G)$, duality of $L^p(K/G, \mu)$,
ملخص المقالة :
Let $H$ and $K$ be compact subgroups of locally compact group $G$. By considering the double coset space $K\setminus G/H$, which equipped with an $N$-strongly quasi invariant measure $\mu$, for $1\leq p\leq +\infty$, we make a norm decreasing linear map from $L^p(G)$ onto $L^p(K\setminus G/H,\mu)$ and demonstrate that it may be identified with a quotient space of $L^p(G)$. In addition, we illustrate that $L^p(K\setminus G/H, \mu)$ 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.
[1] O. Christensen, Frames an Bases; An Introductory Course, Springer, Boston, 2008.
[2] A. Deitmar, A First Course in Harmonic Analysis, Springer-Verlag, New York, 2005.
[3] F. Fahimian, R. A. Kamyabi-Gol, F. Esmaeelzadeh, N-relatively invariant and N-invariant measure on double coset spaces, Bull. Iran. Math. Soc. 45 (2) (2019), 515-525.
[4] F. Fahimian, R. A. Kamyabi-Gol, F. Esmaeelzadeh N-strongly quasi invariant measure on double coset space, arXiv:1807.00132v1.[math.RT].
[5] R. I. Jewett, Space with an abstract convolution of measures, Adv. Math. 18 (1) (1975), 1-101.
[6] E. Kaniuth, K. F. Taylor, Induced Representation of Locally Compact Groups, Cambridge University Press, New York, 2013.
[7] T. S. Liu, Invariant measure on double coset spaces, university of Pennsylvania and university of Massachusetts, 1965.
[8] N. Tavallaei, M, Ramezanpour, B. Golfatian, Structural transition between Lp(G) and Lp(G/H), Banach J. Math. Anal. 9 (3) (2015), 194-205.
[9] M. Toomanian, M. Amini, A. Heydari, Representation of double coset Lie hypergroups, IJMSI. 11 (2) (2016), 87-96.
