The modified P-integral and P-derivative and their applications

The paper is concerned with properties of the modified P-integral and P-derivative, which are defined as multipliers with respect to the generalized Walsh-Fourier transform. Criteria for a function to have a representation as the P-integral or P-derivative of an Lp-function are given, and direct and inverse approximation theorems for P-differentiable functions are established. A relation between the approximation properties of a function and the behaviour of P-derivatives of the appropriate approximate identity is obtained. Analogues of Lizorkin and Taibleson's results on embeddings between the domain of definition of the P-derivative and Hölder-Besov classes are established. Some theorems on embeddings into BMO, Lipschitz and Morrey spaces are proved. Bibliography: 40 titles.