Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators

Abstract

Anisotropic homogeneous mixed-norm Besov and Triebel–Lizorkin spaces are introduced and their properties are explored. A discrete adapted φ -transform decomposition is obtained. An associated class of almost diagonal operators is introduced and a boundedness result for such operators is obtained. Molecular decompositions for all the considered spaces are derived with the help of the algebra of almost diagonal operators. As an application, we obtain boundedness results on the considered spaces for Fourier multipliers and for pseudodifferential operators with suitable adapted homogeneous symbols using the molecular decomposition theory.