On fractional calculus associated to doubling and non-doubling measures

In this paper we present several proofs on the extension of M. Riesz fractional integration and di¤erentiation to the contexts of spaces of homogeneous type and measure metric spaces with non-doubling measures. 1. Introduction, some de…nitions, and a basic lemma Professor M. Ash asked me to write a survey article on some of the results that Stephen Vági and I obtained in the nineties on fractional calculus on spaces of homogeneous type. Since Professor Korányi has done an excellent job stating these results in his article in this volume, I thought it would be appropriate to add some proofs and extensions of these results to the recently open area of research of measure metric spaces associated to non-doubling measures. The notion of space of homogeneous type is due to Coifman and Weiss[CW] and it consists of the triple X a non-empty set; : X X ! R 0 a quasi-distance, that is (x; y) = (y; x) 0, (x; y) = 0 if and only if x = y, and (x; z) k ( (x; y) + (y; z)) where k is a positive constant; and a measure that satis…es the doubling condition, i.e. (B2r (x)) C Br (x) where Br (x) denotes the ball of radius r and center x and C is independent of x and r. Any space of homogeneous type can be normalized by introducing the measure quasidistance (x; y) = inf B f (B) : x; y 2 Bg where B denotes a -ball. The topologies induce by and are equivalent, but the measure quasidistance satis…es the Ahlfors-David property: Let now Br (x) be a -ball of radius r and center x; then there are constants c1 and c2 such that (1.1) c1r (Br (x)) c2r: Furthermore, in any space of homogeneous type there exists a quasidistance equivalent to that also satis…es: There is ; 0 < 1; and a constant M such 2000 Mathematics Subject Classi…cation. Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. Keyword one, keyword two, keyword three. Thanks for Author One. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1