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

By J. García-cuerva, A. Gatto. 2002

143 citations recorded by the source corpus.

The main purpose of this paper is to investigate the behaviour 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, play a basic role too. 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 space "regular"BMO is contained in the adequate Lipschitz space.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8d08c85595c3206ade2e018e36f56e85645446ba

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

SHARP MAXIMAL INEQUALITIES AND COMMUTATORS ON MORREY SPACES WITH NON-DOUBLING MEASURES (2007, unknown): “It is well-known that T is a bounded operator on L(μ) if 1 < p < ∞ (see [10]) and Iα is a bounded operator from Lp(μ) to Lq(μ) if 1 < p < q ≤ ∞ and 1/q = 1/p− α/n (see [4]).”
GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, background): “During the past ten to fifteen years, considerable attention has been paid to the study of the classical theory of harmonic analysis on Euclidean spaces with non-doubling measures only satisfying the polynomial growth condition (see, for example, [11, 10, 37, 38, 39, 40, 41, 42, 5, 29, 14, 15, 16, 1”
The extremal solution for the fractional Laplacian (2013, background): “L p)) ˆI 2s n=p(BMO) ˆC 2s n p; which are commented in [34, p.164]. In the more general framework of spaces with non-doubling n-dimensional measures, a short proof of this result can also be found in [19]. THE EXTREMAL SOLUTION FOR THE FRACTIONAL LAPLACIAN 7 Having Theorem 1.6 available, to prove Pr”
Morrey Spaces for Non–doubling Measures (2005, unknown): “9.1 Definition of the kernel of regularity of fi We adopt the definition of [ 4 ].”
Generalized Morrey Spaces for Non-doubling Measures (2008, background): “We also remark that Garćıa-Cuerva and Gatto showed that Iα raises the regularity of functions [3].”
Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure (2010, background): “The boundedness of potential operators in classical Lebesgue spaces on nonhomogeneous spaces was established in [13] (for Euclidean spaces), [14,15,18] (see also [17, Chapter 6]).”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, background): “(A notable exception to the Euclidean restriction consists of the papers of Garćıa-Cuerva and Gatto [7], [8] and Gatto [9], where some results concerning fractional, singular and hypersingular integrals on Lipschitz spaces are obtained in abstract metric spaces.”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, background): “However, when Gar ía-Cuerva and Gatto ome to a limiting ase of their results on erning the regularised BMO spa e, both [6, 7℄ restri t themselves to the original set-up of Tolsaon Rn, and Bramanti's work ompletely bypasses the BMO aspe ts, whi h would be essential forobtaining Tb theorems in full ge”