INTEGRALES SINGULIÈRES. Doctorat thesis (2011) , Université de Batna 2.

Abstract

On the localized Besov space (Bs p;q(Rn))`r we study the boundedness of the singular integral operators defined by pseudo differential operators of order m with symbols satisfying a condition of Dini-type. Then we deduce the continuity on pointwise multipliers Besov algebra space M(Bs p;q(Rn)) when p < q: We are interested in the superposition operators Tf (g) := f ◦g on vector valued Besov and Lizorkin-Triebel spaces of positive smoothness exponent s. We establish that the local Lipschitz continuity of f is necessary if Bs p;q(Rn;Rm) (or Fs p;q(Rn;Rm)) is imbedded into L1(Rn;Rm), and that the uniform Lipschitz continuity of f is necessary if not. We study also the regularity of Tf