arXiv subjects

Alejandro Claros

Publications and source records attributed to Alejandro Claros.

7 recordsLinked to original sources

arXiv2026

From generalized Poincar\'e to Poincar\'e-Sobolev inequalities via self-improving methods

We establish several improvements to the main results of [PR19] and [CP21], refining the seminal self-improving method for generalized Poincar\'e inequalities from [FPW98, MP98]. These results, together with various related applications, stem from a general self-improving property for functions satisfying the local inequality $$\frac{1}{|Q|}\int_Q |f(x)-f_Q|\,dx \le a(Q)$$ for all cubes $Q\subset\mathbb{R}^n$. The functional $a$ is assumed to obey a specific discrete geometric summability condition. By restricting our focus to axis-parallel cubes in $\mathbb{R}^n$, this geometric setting allows us to obtain sharper estimates than those available in more general metric measure spaces.

math.CA↗

arXiv2026

Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincar\'e-Sobolev inequalities

We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincar\'e-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$.

math.CA↗

arXiv2026

Bourgain-Brezis-Mironescu formula for Riesz Potentials

We identify the Bourgain-Brezis-Mironescu pointwise limit of the nonlocal potential operator $(1-\alpha)\, I_\alpha(\mathcal D^\alpha f)$, $0<\alpha<1$, where $I_\alpha$ denotes the Riesz potential and $\mathcal D^\alpha$ a nonlinear fractional differential operator. Specifically, for every $f\in C_c^\infty(\mathbb R^n)$ and every $x\in \mathbb R^n$, we show that \begin{equation*} \lim_{\alpha\to 1^-} (1-\alpha)\, I_\alpha(\mathcal D^\alpha f)(x) = K_n\, I_1(|\nabla f|)(x), \end{equation*} where $K_n$ is the geometric constant appearing in the well-known Bourgain-Brezis-Mironescu formula [BBM02]. By a density argument, we further extend this result to every $f\in W^{1,1}(\mathbb R^n)$, obtaining almost everywhere convergence along subsequences.

math.AP↗

arXiv2026

Commutators of Fractional Integrals with $\operatorname{BMO}^\beta$ Functions

We study commutators of the Riesz potential $I_\alpha$ with functions $b$ in the capacitary space $\mathrm{BMO}^\beta(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^\beta_\infty$. We prove a Chanillo-type theorem characterising $\mathrm{BMO}^\beta(\mathbb{R}^n)$ via the boundedness of the commutator $[b,I_\alpha]$ on capacitary Lebesgue spaces. In addition, we obtain the endpoint estimate in the form of a capacitary modular weak-type inequality. These results follow from a pointwise estimate for the $\beta$-dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].

math.CA↗

arXiv2025

Degenerate Poincar\'e-Sobolev inequalities via fractional integration

We present a local weighted estimate for the Riesz potential in $\mathbb{R}^n$, which improves the main theorem of Alberico, Cianchi, and Sbordone [C. R. Math. Acad. Sci. Paris \textbf{347} (2009)] in several ways. As a consequence, we derive weighted Poincar\'e-Sobolev inequalities with sharp dependence on the constants. We answer positively to a conjecture proposed by P\'erez and Rela [Trans. Amer. Math. Soc. 372 (2019)] related to the sharp exponent in the $A_1$ constant in the $(p^*,p)$ Poincar\'e-Sobolev inequality with $A_1$ weights. Our approach is versatile enough to prove Poincar\'e-Sobolev inequalities for high-order derivatives and fractional Poincar\'e-Sobolev inequalities with the BBM extra gain factor $(1-\delta)^{1/p}$. In particular, we improve one of the main results from Hurri-Syrj\"anen, Mart\'inez-Perales, P\'erez, and V\"ah\"akangas [Int. Math. Res. Not. 20 (2023)].

math.CA↗

arXiv2025

Quantitative BMO-BLO Estimates for the Hardy-Littlewood Maximal Function

In this note, we study a quantitative extension of the John-Nirenberg inequality for the Hardy-Littlewood maximal function of a $\operatorname{BMO}$ function. More precisely, for every nonconstant locally integrable function $f$ such that $Mf$ is not identically infinite, we prove the inequality \begin{equation*} \left( \frac{1}{w(Q)}\int_Q \left( \frac{Mf(x) - \operatorname{ess\,inf}_{Q} Mf }{M^\# f(x)} \right)^p w(x)\,dx\right)^\frac{1}{p} \le c_n \, [w]_{A_\infty}\, p \end{equation*} for every cube $Q$, every $1\le p<\infty$ and every weight $w\in A_\infty$, where $[w]_{A_\infty}$ denotes the Fujii-Wilson $A_\infty$ constant. This result extends the classical boundedness $\|Mf\|_{\operatorname{BLO}}\le C_n\|f\|_{\operatorname{BMO}}$ proved by Bennett, DeVore, and Sharpley (Ann. of Math. (2) 113 (1981)) and by Bennett (Proc. Amer. Math. Soc. 85 (1982)). Furthermore, we show that the class $A_\infty$ is both necessary and sufficient for this inequality to hold, providing a new characterization of $A_\infty$ in terms of the action of the maximal operator on bounded oscillation spaces.

math.CA↗

arXiv2024

$\beta$-dimensional sharp maximal function and applications

In this paper, we study $\beta$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _\beta f(x) := \sup_{Q} \inf_{c \in \mathbb{R}} \chi_{Q}(x) \frac{1}{\ell(Q)^\beta} \int_Q |f-c| \; d \mathcal{H}^{\beta}_\infty, \end{align*} where the supremum is taken over all cubes in $\mathbb{R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{H}^{\beta}_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _\beta f$ by giving a good lambda estimate for $\beta$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest.

math.FA↗