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Paco Villarroya

Publications and source records attributed to Paco Villarroya.

10 recordsLinked to original sources
arXiv2024

Sparse Domination of Singular Bilinear Forms on Non-Homogeneous spaces

We introduce a new sparse $T1$ theorem that estimates the dual pair associated with a Calderon-Zygmund operator by a sub-bilinear form supported on a sparse family of cubes. The main result in the paper improves previous sparse $T1$ theorems in several ways: it applies to non-homogeneous measures of power growth, it only requires a numerable family of testing conditions, and it can be used to prove boundedness of Calderon-Zygmund operators on weighted spaces for a class of weights larger than the Muckenhoupt $A_p$ weights.

math.CA↗
arXiv2021

New Local T1 Theorems on non-homogeneous spaces

We develop new local $T1$ theorems to characterize Calder\'on-Zygmund operators that extend boundedly or compactly on $L^{p}(\mathbb R^{n},\mu)$ with $\mu$ a measure of power growth. The results, whose proofs do not require random grids, allow the use of a countable collection of testing functions. As a corollary, we describe the measures $\mu$ of the complex plane for which the Cauchy integral defines a compact operator on $L^p(\mathbb C,\mu)$.

math.CA↗
arXiv2019

Sparse domination results for compactness on weighted spaces

By means of appropriate sparse bounds, we deduce compactness on weighted $L^p(w)$ spaces, $1<p<\infty$, for all Calder\'on-Zygmund operators having compact extensions on $L^2(\mathbb{R}^n)$. Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $A_p$ class.

math.CA↗
arXiv2017

A global Tb Theorem for compactness and boundedness

We prove a Tb Theorem that characterizes all Calderon-Zygmund operators that extend compactly on L^p(R^n), 1<p<\infty . The result, whose proof does not require the property of accretivity, can be used to prove compactness of the Double Layer Potential operator on a wide class of domains. The study also provides conditions for boundedness of singular integral operators by means of non-accretive testing functions.

math.CA↗
arXiv2017

Endpoint compactness of singular integrals and perturbations of the Cauchy Integral

We prove sufficient and necessary conditions for compactness of Calderón-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with $1<p<\infty $ of certain perturbations of the Cauchy integral on curves with normal derivatives satisfying a ${\rm CMO}$-condition.

math.CA↗