Maximal inequalities and Fourier multipliers for spaces with mixed quasinorms. Applications

Abstract

The paper concerns with maximal inequalities and Fourier multipliers for systems of entire analytic functions of exponential type belonging to mixed quasinormed spaces. As an application 4 types of spaces of functions with dominating mixed smoothness properties are introduced. Relations to classical spaces of that type (spaces of S.M. Nikol’skij, T.I. Amanov, P.I. Lizorkin) are treated. As a special case the Sobolev spaces with a dominating mixed derivative are contained.