On the extrapolation of quasi-subadditive singular operators from certain classes

Abstract

Dedicated to Academician S. M. Nikol'skiȋ on the occasion of his 100th anniversary A b s t r a c t . The boundedness of anisotropic quasi-subadditive singular operators with the domains in Bochner-Lebesgue and local approximation spaces of LizorkinTriebel-type of Banach-valued functions and the ranges in Bochner-Marcinkiewicz spaces is analyzed from the point of view of extrapolation. Parameterized classes of sufficient conditions generalizing author's AAD-conditions studied earlier are obtained that are expressed in terms of the approximation D-functional and ones being close to it. Our sufficient conditions are weaker then their known counterparts in the same settings. They are new in the isotropic case of linear scalar-valued operators too. The main results are presented in the style of the Calderón-Zygmund theory. The approach is based on decompositions of operators and the usage of counterparts of Calderón-Zygmund representations and some interpolation properties of the related function spaces established by the author.