arXiv subjects

Gustavo Garrigós

Publications and source records attributed to Gustavo Garrigós.

14 recordsLinked to original sources

arXiv2026

Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals

We study differentiability conditions on a complex measure $ν$ at a point $x_0\in\mathbb{R}^d$, in relation with the boundary convergence at that point of the Poisson-type integral $P_tν=e^{-t\sqrt L}ν$, where $L=-Δ+|x|^2$ is the Hermite operator. In particular, we show that $x_0$ is a Lebesgue point for $ν$ iff a slightly stronger notion than non-tangential convergence holds for $P_tν$ at $x_0$. We also show non-tangential convergence when $x_0$ is a $σ$-point of $ν$, a weaker notion than Lebesgue point, which for $d=1$ coincides with the classical Fatou condition.

math.AP↗

arXiv2024

Local cone multipliers and Cauchy-Szego projections in bounded symmetric domains

We show that the cone multiplier satisfies local $L^p$-$L^q$ bounds only in the trivial range $1\leq q\leq 2\leq p\leq\infty$. To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we answer negatively a question by Békollé and Bonami (Colloq. Math. 68, 1995, 81-100), regarding the continuity from $L^p\to L^q$ of the Cauchy-Szegö projections associated with a class of bounded symmetric domains in $\mathbb{C}^n$ with rank $r\geq2$.

math.AP↗

arXiv2023

Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts

We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on $\mathbb{R}$, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness $s<1$, in which the spaces $F^s_{p,q}$ and $B^s_{p,q}$ are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that $1/p<s<1$ and $f\in B^s_{p,q}$, we actually prove that the usual Haar coefficient norm, $\|\{2^j\langle f, h_{j,μ}\rangle\}_{j,μ}\|_{b^s_{p,q}}$ remains equivalent to $\|f\|_{B^s_{p,q}}$, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case $s=1$ and $q=\infty$, we show that such an expression gives an equivalent norm for the Sobolev space $W^{1}_p(\mathbb{R})$, $1<p<\infty$, which is related to a classical result by Bočkarev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.

math.FA↗

arXiv2023

A sufficient condition for Haar multipliers in Triebel-Lizorkin spaces

We consider Haar multiplier operators $T_m$ acting on Sobolev spaces, and more generally Triebel-Lizorkin spaces $F^s_{p,q}(\mathbb{R})$, for indices in which the Haar system is not unconditional. When $m$ depends only on the Haar frequency, we give a sufficient condition for the boundedness of $T_m$ in $F^s_{p,q}$, in terms of the variation norms $\|m\|_{V_u}$, which is optimal in $u$ (up to endpoints) when $p, q> 1$.

math.CA↗

arXiv2022

The Weak Chebyshev Greedy Algorithm (WCGA) in $L^p (\log L)^α$ spaces

We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces $\mathbb{X}$. First, we generalize a result of Temlyakov to cover situations in which the modulus of smoothness and the so called A3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces $\mathbb{X}=L^p(\log L)^α$, with $1 0$, takes the form $ϕ(N)\approx \log(\log N)$.

math.FA↗

arXiv2020

The Haar System in Triebel-Lizorkin Spaces: Endpoint Results

We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(\Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=\infty$. Together with the earlier results in [10], [4], this completes the picture for such properties in the Triebel-Lizorkin scale, and complements a similar study for the Besov spaces given in [5].

math.CA↗

arXiv2019

Mean value formulas for Ornstein-Uhlenbeck and Hermite temperatures

We obtain explicit mean value formulas for the solutions of the diffusion equations associated with the Ornstein-Uhlenbeck and Hermite operators. From these, we derive various useful properties, such as maximum principles, uniqueness theorems and Harnack-type inequalities.

math.AP↗

arXiv2019

Basis properties of the Haar system in limiting Besov spaces

We study Schauder basis properties for the Haar system in Besov spaces $B^s_{p,q}(\mathbb{R}^d)$. We give a complete description of the limiting cases, obtaining various positive results for $q\leq \min\{1,p\}$, and providing new counterexamples in other situations. The study is based on suitable estimates of the dyadic averaging operators $\mathbb{E}_N$; in particular we find asymptotically optimal growth rates for the norms of these operators in global and local situations.

math.CA↗

arXiv2018

Lebesgue inequalities for Chebyshev Thresholding Greedy Algorithms

We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.

math.FA↗

arXiv2017

The Haar system as a Schauder basis in spaces of Hardy-Sobolev type

We show that, for suitable enumerations, the multivariate Haar system is a Schauder basis in the classical Sobolev spaces on $\mathbb R^d$ with integrability $1<p<\infty$ and smoothness $1/p-1<s<1/p$. This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a {conditional} Schauder basis for a nonempty open subset of the $(1/p,s)$-diagram. The results extend to (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type in the range of parameters $\frac{d}{d+1}<p<\infty$ and $\max\{d(1/p-1),1/p-1\}<s<\min\{1,1/p\}$, which is optimal except perhaps at the end-points.

math.CA↗

arXiv2017

On uniform boundedness of dyadic averaging operators in spaces of Hardy-Sobolev type

We give an alternative proof of recent results by the authors on uniform boundedness of dyadic averaging operators in (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type. This result served as the main tool to establish Schauder basis properties of suitable enumerations of the univariate Haar system in the mentioned spaces. The rather elementary proof here is based on characterizations of the respective spaces in terms of orthogonal compactly supported Daubechies wavelets.

math.FA↗

arXiv2016

Lebesgue inequalities for the greedy algorithm in general bases

We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.

math.FA↗

arXiv2012

Lebesgue type inequalities for quasi-greedy bases

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases.

math.FA↗

arXiv2009

Democracy functions and optimal embeddings for approximation spaces

We prove optimal embeddings for nonlinear approximation spaces in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for $N$-term wavelet approximation in Lebesgue, Orlicz, and Lorentz norms. We also study the "greedy classes" introduced by Gribonval and Nielsen.

math.FA↗