Boundedness of Multi-Parameter Fourier Multiplier Operators with Characterizations of Logarithmic Besov and Triebel–Lizorkin Spaces

Abstract

Some developments in the theory of function spaces involving differences are shown. Difference and new characterizations of Besov, Sobolev and Triebel-Lizorkin spaces on metric measure spaces, on the Euclidean space and on averages balls are studied. We obtain directly and perfectly the dual multi-parameter, singular integrals and boundedness of the composition operators on Triebel-Lizorkin and Besov spaces associated with singular integrals with different homogeneities and of regular distributions. We introduce the method of Ho ̈rmander type theorems for multi-linear and boundednessof multi-parameter Fourier multiplier operators with limited smoothness and on Triebel-Lizorkin and Besov-Lipschitz spaces of logarithmic smoothness, approximation spaces and limiting interpolation. We explain the treatments of the characterizations of generalized and logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets bases and semi-groups.