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.
