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

S. S. Volosivets

Sbornik. Mathematics, Tome 203 (2012) no. 5, pp. 613-644

Résumé

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.

Détail

Keywords: modified $\mathbf P$-integral, modified $\mathbf P$-derivative, direct and inverse approximation theorems
Mots-clés : multiplicative Fourier transform, Hölder-Besov spaces.

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