The modified $\mathbf P$-integral and $\mathbf P$-derivative and their applications

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