Multipliers of the Hölder classes

Abstract

Following the lines of [13] we introduce the classes of mixed smoothness L α p( R n), L α p( Z n) for a multi-index α = ( α 1,…, α n ). Such classes are naturally tied up with the study of semi-elliptic differential and difference equations. Besides a brief presentation of such classes, we concentrate our research on the study of mixed homogeneous multipliers with homogeneity β = ( β 1, …, β n ) and their preservation of mixed homogeneous Hölder classes L α ∞, for a different multi-index α. In the last paragraph we apply the results to produce various improvements of the classical Schauder's estimates, for differential and difference equations, in the parabolic and elliptic case.