On fractional calculus associated to doubling and non-doubling measures

Abstract

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. (B 2r (x)) C B r (x) where B r (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 B r (x) be a-ball of radius r and center x; then there are constants c 1 and c 2 such that (1.1) c 1 r (B r (x)) c 2 r: 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

Boundedness on inhomogeneous Lipschitz spaces of fractional integrals singular integrals and hypersingular integrals associated to non-doubling measures

“…On the other hand, the case of homogeneous Lipschitz spaces was treated in [2,3], and [4]. The letter C or c will denote a constant not necessarily the same at each ocurrence.…”

Fractional and Hypersingular Operators in Variable Exponent Spaces on Metric Measure Spaces

“…We prove two versions of Sobolev-type theorems with variable exponents. Various versions of such theorems for constant p were proved in [18], [15], [16], [17], [30], [31]. We also give boundedness statements for corresponding fractional maximal operators.…”

“…the book [18] for I α , and the book [13] and papers [15], [16], [17] for I α n . In the next theorem, for functions on doubling measure spaces with upper bound (2.6), we deal with the "quasi-Sobolev" exponent q = q(n, N ) defined by 1…”

“…A detailed information about hypersingular integrals (of constant order) of functions defined in R m can be found in [47] and [49]; variable order hypersingular integrals were studied in [46]. Hypersingular integrals of constant order on metric measure spaces were considered in [16], [17] within the frameworks of Lipschitz (Hölder) function spaces.…”

“…In the case of constant exponents, hypersingular integrals on metric spaces, jointly with potential operators, were considered in [16], [17]. This paper is structured as follows.…”