Generalized Lipschitz conditions and Riesz derivatives on the space of Bessel potentials

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Abstract
In this paper, the first of a series, the space of n-dimensional Bessel potentials Lρ α, 0 < α ≦2, is considered with the aim of describing smoothness properties of its elements. This is achieved by forming norms involving the existence of derivatives or the order of Lipschitz conditions of f or its Riesz transform, and by showing these to be equivalent to the Lα ρ- The method of proof, inspired by Sunouchi and Shapiro, consists in interpreting the characterization itself as a saturation problem with Favard class Lα ρ; thus, the characterizations have only to satisfy the conditions of a general saturation theorem, established in Lρ,1≦ρ≦∞ To obtain more specific results in case 1 < ρ < ∞ the Marcinkiewicz–Mikhlin multiplier theorem is applied. Our general results contain particular ones due to Berens–Nessel, Butzer, Butzer–Trebels, Calderón, Cooper, Görlich, Nessel–Trebels, and Trebels.

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