Boundedness properties of fractional integral operators associated to non-doubling measures

Abstract

Abstract. The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, also play a basic role. For a Euclidean space equipped with one of these measures, we also consider the so-called regular BMO space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the regular BMO space is contained in a suitable Lipschitz space.

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

“…[d(x,y)] n−β(x) of a constant is a bounded function, which is known, see for instance [15], Lemma 2.1, and is easily verified via the standard decomposition…”

“…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.…”

“…In the case of constant exponents p and α, the statement of Theorem 3.2 is known to be valid without the doubling condition and with optimal Sobolev exponent q instead of the "quasi-Sobolev" exponent q, see Theorem 3.2 and Corollary 3.3 in [15]. The progress in [15] …”

“…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…”

Besov spaces with non-doubling measures

“…Recently more attention has been paid to non-doubling measures. It has been shown that many results of this theory still hold without assuming the doubling property; see [16,17,18,19,23,24,25,29,5,6] for some results on Calderón-Zygmund operators, [15,26,27,28] for some other results related to the spaces BM O(µ) and H 1 (µ), and [7,8,20] for the vector-valued inequalities on the Calderón-Zygmund operators and weights. …”