Besov, Sobolev and potentials type spaces on Chébli-Trimèche hypergroups

Abstract

R. S. Pathak and P. K. Pandey (1995, 1997) analyzed pseudo-differential operatorsand Sobolev Type spaces associated with the Bessel operators. Later on, N. Ben Salem andA. Dachraoui (1998, 2000) obtained similar results for the Jacobi differential operators. A.Dachraoui and K. Trim`eche (1999) studied pseudo-differential operators associated with asingular differential operator, and they generalized the results obtained by R.S. Pathak and P.K. Pandey (1995) and N. Ben Salem and A. Dachraoui (1998). The work begun by R.S. Pathakand P. K. Pandey(1997) is continued by D.I. Cruz-B´aez and J. Rodr´ıguez (2000, 2001) beingobtained new results on potentials, Sobolev type spaces, Besov and Triebel-Lizorkin type spacesassociated with the Bessel operators. In this paper, we generalize and continue the previouspapers. We prove a Calder´on’s type theorem, define Besov spaces Bsp,q,m, bsp,q,m on Ch´ebliTrim`eche hypergroups and give a characterization of Bsp,q,m in terms of bsp,q,m. Moreover wegive an embedding theorem between Bsp,q,m and Triebel-Lizorkin type spaces Fsp,q,m, establisha lifting property and finally, give some applications of these results.