Product rule and chain rule estimates for fractional derivatives on spaces that satisfy the doubling condition

Abstract

The purpose of this paper is to prove some classical estimates for fractional derivatives of functions defined on a Coifman-Weiss space of homogeneous type, in particular the product rule and chain rule estimates in (T. Kato and G. Ponce, 1988, Comm. Pure Appl. Math. 41, 891-907) and (F.M. Christ and M.I. Weinstein, 1991, J. Funct. Anal. 100, 87-109). The fractional calculus of M. Riesz was extended to these spaces in (A.E. Gatto, C. Segovia, and S. Vági, 1996, Rev. Mat. Iberoamericana 12). Our main tools are fractional difference quotients and the square fractional derivative of R. Strichartz in (1967, J. Math. Mech. 16, 9) extended to this context. For the particular case of ℝn, our approach unifies the proofs of these estimates and clarifies the role of Calderón's formula for these results. Since the square fractional derivative can be easily discretized, we also show that the discrete and continuous Triebel-Lizorkin norms for fractional Sobolev spaces on spaces of homogeneous type are equivalent. © 2002 Elsevier Science (USA).