Lipschitz spaces and Calderón-Zygmund operators associated to non-doubling measures

By J. García-cuerva, A. Gatto. 2002

54 citations recorded by the source corpus.

In the setting of a metric measure space $(\mathbb{X}, d, \mu)$ with an $n$-dimensional Radon measure $\mu$, we give a necessary and sufficient condition for the boundedness of Calderon-Zygmund operators associated to the measure $\mu$ on Lipschitz spaces on the support of $\mu$. Also, for the Euclidean space $\mathbb{R}^d$ with an arbitrary Radon measure $\mu$, we give several characterizations of Lipschitz spaces on the support of $\mu$, $\mathcal{L}ip(\alpha, \mu)$, in terms of mean oscillations involving $\mu$. This allows us to view the "regular" $\mathit{BMO}$ space of X. Tolsa as a limit case for $\alpha\to 0$ of the spaces $\mathcal{L}ip(\alpha, \mu)$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/6a7cb9b25bbf2db7eb522a5f7d857e9505f9be04

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In Applied Mathematics

Citations in the corpus

Extra cancellation of even Calderón–Zygmund operators and quasiconformal mappings (2008, methodology): “The general case follows from this by the T (1)-Theorem for Lipschitz spaces on spaces of homogeneous type [26] (see also [10] and [9] for the non-doubling case).”
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.”
Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces (2009, background): “e with measure.” Although a number of results for non-homogeneous singular integral operators (such as the weak-type L1 inequality under a priori L2 boundedness [NTV98], estimates on Lipschitz spaces [GCG05], and a certain restricted version of the T1 theorem [Bra09]—essentially with T1,T∗1 ∈ L∞, an”
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.”