Potential Operators with Mixed Homogeneity

Abstract

In 1966 Cora Sadosky introduced a number of results in a remarkable paper “A note on Parabolic Fractional and Singular Integrals”, see Sadosky (Studia Math 26:295–302, 1966), in particular, a quasi homogeneous version of Sobolev’s immersion theorem was discussed in the paper. Later, C. P. Calderón and T. Kwembe, following those ideas and incorporating the context of Fabes-Riviere homogeneity (Fabes and Riviere, Studia Math 27:19–38, 1966), proved a similar results for potential operators with kernels having mixed homogeneity. Calderón-Kwembe’s (Dispersal models. X Latin American School of Mathematics (Tanti, 1991). Rev Un Mat Argent 37(3–4):212–229, 1991/1992) basic theorem was very much in the spirit of Sadosky’s result. The natural extension of Sadosky’s paper is nevertheless the joint paper by C. Sadosky and M. Cotlar (On quasi-homogeneous Bessel potential operators. In: Singular integrals. Proceedings of symposia in pure mathematics, Chicago, 1966. American Mathematical Society, Providence, 1967, pp 275–287) which constitutes a true tour de force through, what is now considered, local properties of solutions of parabolic partial differential equations. The tools are the introduction of “Parabolic Bessel Potentials” combined with mixed homogeneity local smoothness estimates.

The aim of this paper is to extend Calderón-Kwembe’s theorem in two directions: (a) establish a corresponding result in terms of mixed norms in the Benedek-Panzone’s sense, see Benedek and Panzone (Duke Math J 28:3–21, 1961). (b) establish results for the case of unbounded characteristics (integrable to the power r on the unit sphere). Calderón-Kwembe’s theorem can also be estated in the frame of generalized homogeneity but that case will not be consider here, see N. Riviere (Arkiv för Math 9(2):243–278, 1971) and A. P. Calderón and A. Torchinsky (Adv Math 16:1–64, 1975).

Dedicated to the memory of Cora Sadosky