arXiv subjects

Gonzalo H. Ibañez-Firnkorn

Publications and source records attributed to Gonzalo H. Ibañez-Firnkorn.

5 recordsLinked to original sources

arXiv2022

Mixed weak type inequalities in euclidean spaces and in spaces of homogeneous type

In this paper we provide mixed weak type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente, Martell, Pérez and Riveros. One of the main novelties is that, besides obtaining estimates in the euclidean setting, results are provided as well in spaces of homogeneous type, being the first mixed weak type estimates that we are aware of in that setting.

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arXiv2021

Sharp bounds for fractional operator with $L^{α,r'}$-Hörmander conditions

In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{α,r'}$-Hörmander conditions and a fractional size condition, where $0<α<n$ and $1< r'\leq \infty$. To prove this result we use a new appropriate sparse domination which we provide in this work. For the case $r'=\infty$ we recover the sharp boundedness for the fractional integral, $I_α$, proved in [Lacey, M. T., Moen, K., Pérez, C., Torres, R. H. (2010). Sharp weighted bounds for fractional integral operators. Journal of Functional Analysis, 259(5), 1073-1097.]

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arXiv2020

Necessary condition on the weight for maximal and integral operators with rough kernels

Let $0\leq α<n$, $m\in \mathbb{N}$ and let consider $T_{α,m}$ be a of integral operator, given by kernel of the form $$K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),$$ where $A_i$ are invertible matrices and each $k_i$ satisfies a fractional size and generalized fractional Hörmander condition. In [Ibañez-Firnkorn, G. H., and Riveros, M. S. (2018). Certain fractional type operators with Hörmander conditions. To appear in Ann. Acad. Sci. Fenn. Math.] it was proved that $T_{α,m}$ is controlled in $L^p(w)$-norms, $w\in A_{\infty}$, by the sum of maximal operators $M_{A_i^{-1},α}$. In this paper we present the class of weights $\mathcal{A}_{A,p,q}$, where $A$ is an invertible matrix. This class are the good weights for the weak-type estimate of $M_{A^{-1},α}$. For certain kernels $k_i$ we can characterize the weights for the strong-type estimate of $T_{α,m}$. Also, we give a the strong-type estimate using testing conditions.

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arXiv2018

Sparse and weighted estimates for generalized Hörmander operators and commutators

In this paper we obtain a pointwise sparse domination for generalized Hörmander operators and also for iterated commutators with those operators. As a particular case of our result we obtain a extension of the sparse domination for commutators obtained by A. Lerner, S. Ombrosi and the second author to iterated commutators. Relying upon that sparse domination a number of quantitative estimates such as Coifman-Fefferman estimates, strong type estimates, and endpoint estimates that improve and complete results in papers by M. Lorente, J.M. Martell, C. Pérez, M.S. Riveros and A. de la Torre are obtained. We also provide a new local decay estimate and we also extend results in a paper due to J.M. Martell, C. Perez and R. Trujillo-Gonzalez to kernels satisfying generalized Hörmander conditions. Among other applications, as a particular case of our result for endpoint estimates, we extend the endpoint estimate obtained in a work by A. Lerner, S. Ombrosi and the second author to iterated commutators.

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arXiv2018

Commutators of certain fractional type operators with Hörmander conditions, one-weighted and two-weighted inequalities

In this paper we study the commutators of fractional type integral operators. This operators are given by kernels of theform $$K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),$$ where $A_i$ are invertibles matrices and each $k_i$ satisfies a fractional size condition and generalized fractional Hörmander condition. We obtain weighted Coifman estimates, weighted $L^p(w^p)$ - $L^q(w^q)$ estimates and weighted BMO estimates. We also give a two weight strong estimate for pair of weights of the form $(u,Su)$ where $u$ is an arbitrary non-negative function and $S$ is a maximal operator depending on the smoothness of the kernel $K$. For the singular case we also give a two-weighted endpoint estimate.

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