Salah Ben Mahmoud2022-03-032022-03-032022http://dspace.univ-msila.dz:8080//xmlui/handle/123456789/28105In this thesis we introduce new equivalent characterizations and certain commutator estimations on Besov spaces, Besov-type spaces and Triebel-Lizorkin spaces of variable smoothness and integrability. We prove the continuous characterization of the space B ( ) p( ),q( ) and characterization via Peetre-type maximal functions of local means of variable Besov spaces B ( ) p( ),q( ) based on the continuous version of Calderón reproducing formula and and of variable Besov-type spaces B ( ), p( ),q( ). We prove certain estimates on Triebel-Lizorkin spaces of variable smoothness and integrability F ( ) p( ),q( ) for the commutator [V r, j ](f ), these estimates are obtained under no vanishing assumptions on the divergence of the vector fieldBesov space, Triebel-Lizorkin space, variable exponent,Calderón reproducing formulaThesis