On the Boundedness of Singular Integral Operators. I

The boundedness of anisotropic singular integral operators with the domains of definition and ranges in various anisotropic spaces of Banach-valued functions is analyzed from the unified point of view. A number of sufficient parameterized classes of conditions are obtained that are expressed in terms of the approximation $\mathcal D$-functional and are sharp in a certain sense. Some classes of conditions are given with a simultaneous use of local and global approximations. The inhomogeneity of the dependence on certain parameters is revealed. The results obtained also apply to nonsingular (in the ordinary sense) integral operators, for example, to potential-type operators. The main results are presented in the style of the Calderón–Zygmund theory. The approach is based on the study of $p$-convex hulls, decompositions into a sum of atomic complexes, and other properties of function spaces.