arXiv subjects

José M. Conde Alonso

Publications and source records attributed to José M. Conde Alonso.

7 recordsLinked to original sources

arXiv2026

Minimizing measures for the doubling condition

We study those measures whose doubling constant is the least possible among doubling measures on a given metric space. It is shown that such measures exist on every complete metric space supporting at least one doubling measure. In addition, a connection between minimizers for the doubling constant and superharmonic functions is exhibited. This allows us to show that for the particular case of the Euclidean space $\mathbb R^d$, Lebesgue measure is the only minimizer for the doubling constant (up to constant multiples) precisely when $d=1$ or $d=2$, while for $d\geq3$ there are infinitely many independent minimizers when considering the Euclidean metric. Analogously, in the discrete setting, we can show uniqueness of the counting measure as a minimizer for regular graphs where the standard random walk is a recurrent Markov chain. The counting measure is also shown to be a minimizer in every infinite graph where the cardinality of balls depends solely on their radii.

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arXiv2026

Almost sparse operators and Bergman projectors on trees

We develop a sparse domination framework adapted to nested families in general measure spaces, possibly including point masses. We show that the Carleson condition is equivalent to a slight generalization of the usual sparse condition, and use this characterization to obtain one- and two-weight estimates for the associated averaging operators, together with converse testing conditions. We apply our general results to integral operators on locally finite trees. Trees come naturally equipped with two nested geometries: sectors and pruned sectors. We prove pointwise sparse domination of natural versions of the harmonic Bergman projector by the corresponding averaging operators, depending on whether the measure on the tree is doubling or not, and show that the distinction is necessary in general. For the positive Bergman projector, we obtain pointwise comparability. As a consequence, we obtain new weighted inequalities and boundedness results for the Bergman projector and a class of Toeplitz-like multipliers.

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arXiv2026

Cancellative sparse domination

We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the $H^1$ norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calderón-Zygmund operators acting on $H^p$ spaces.

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arXiv2024

Endpoint estimates for Haar shift operators with balanced measures

We prove $\mathrm{H}^1$ and $\mathrm{BMO}$ endpoint inequalities for generic cancellative Haar shifts defined with respect to a possibly non-homogeneous Borel measure $μ$ satisfying a weak regularity condition. This immediately yields a new, highly streamlined proof of the $L^p$-results for the same operators due to López-Sanchez, Martell, and Parcet. We also prove regularity properties for the Haar shift operators on the natural martingale Lipschitz spaces defined with respect to the underlying dyadic system, and show that the class of measures that we consider is sharp.

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arXiv2024

Commutator estimates for Haar shifts with general measures

We study $L^p(μ)$ estimates for the commutator $[H,b]$, where the operator $H$ is a dyadic model of the classical Hilbert transform introduced in \cite{arXiv:2012.10201,arXiv:2212.00090} and is adapted to a non-doubling Borel measure $μ$ satisfying a dyadic regularity condition which is necessary for $H$ to be bounded on $L^p(μ)$. We show that $\|[H, b]\|_{L^p(μ) \rightarrow L^p(μ)} \lesssim \|b\|_{\mathrm{BMO}(μ)}$, but to {\it characterize} martingale BMO requires additional commutator information. We prove weighted inequalities for $[H, b]$ together with a version of the John-Nirenberg inequality adapted to appropriate weight classes $\widehat{A}_p$ that we define for our non-homogeneous setting. This requires establishing reverse Hölder inequalities for these new weight classes. Finally, we revisit the appropriate class of nonhomogeneous measures $μ$ for the study of different types of Haar shift operators.

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arXiv2021

A metric approach to sparse domination

We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hytönen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of $\mathbb R^n$.

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arXiv2018

BMO from dyadic BMO for nonhomogeneous measures

The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. We consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional difficulties which forces us to consider martingale filtrations that are not regular. The dyadic covering that we find can be used to clarify the relationship between martingale BMO spaces and the most natural BMO space in this setting, which is the space RBMO introduced by Tolsa.

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