Regular ArticleBoundedness and Compactness of Integral Operators on Spaces of Homogeneous Type and Applications, I☆

By Steven G. Krantz, S. Li. 2001

86 citations recorded by the source corpus.

Let X be a space of homogeneous type. For a function f on X, let Mf be the corresponding multiplication operator on some function space over X. Let TK be a standard Calderón–Zygmund (C-Z) operator that is bounded on L2 X . Let Cf = Mf TK be the commutator of Mf and TK . When X = N , a fundamental theorem of Coifman et al. [CRW] gives a characterization of boundedness of Cf when the TK are the Riesz transforms Rj j = 1 2 n . The characterization of compactness was given by Uchiyama [UCH] and Janson [JAN]. The boundedness result was generalized to other contexts and important applications to some non-linear PDEs were given by Coifman et al. [CLMS]. Boundedness and compactness for Cf on L space over X × 0 ∞ was proved by Beatrous and Li [BEL], and applications to Hankel-type operators on Bergman spaces were given there. More general operators—multilinear operator or bilinear forms rather than commutators on L X —were studied by several authors (see examples, [COG, GLY, CLMS]), and characterizations of boundedness were given in that context when X is N or the Heisenberg group. A sufficient condition on f so that Cf is bounded on L X 1 < p < ∞) was given in [BC].

Subfields: Applied Mathematics

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In Applied Mathematics

1. Ji Li
2. Xuan Thinh Duong
3. Peng Chen
4. Lixin Yan
5. Dachun Yang
6. P. Auscher
7. T. Coulhon
8. S. Hofmann

Citations in the corpus

• Compactness of Riesz transform commutator on stratified Lie groups (2018, background): “Later on and recently, there has been an intensive study of the compactness of commutators of singular integrals in many different settings, such as the Riesz transform associated with Bessel operator on the positive real line, the Cauchy’s integrals on the real line, the Calderón– Zygmund operator ”