Searcharxiv⌕ Search

arXiv subjects

Ángel D. Martínez

Publications and source records attributed to Ángel D. Martínez.

14 recordsLinked to original sources
arXiv2026

Aomoto interpolation and Coxeter systems

In this paper, we construct a Lagrange-type basis for the Aomoto space $AO(\mathcal A)$, naturally indexed by the chambers of the hyperplane arrangement $\mathcal A$. The construction relies on a dimension theorem of Orlik and Terao and yields an interpolation formula for elements of $AO(\mathcal A)$. We use this formula to characterize the extremal configurations in the strong polarization inequality as those arising from finite Coxeter reflection systems. We further show that the interpolation formula gives rise to a family of \emph{chamber identities}, including identities that were central to our earlier proof of the strong polarization problem and the Gaussian product inequality. Finally, we adapt the recent breakthrough of Ouimet and Greaves to prove a generalized Gaussian Product Inequality for completely monotone functions.

math.MG↗
arXiv2026

Polarization problems and Coxeter systems

In this paper we provide a proof of the strong polarization conjecture due to Ball and Frenkel. We observe that Coxeter systems produce unexpected new examples of extremizers. Furthermore, as a by-product of our work we solve the real polarization problem and completely characterize its extremal cases.

math.FA↗
arXiv2025

On monotonicity of heat kernels: a new example and counterexamples

We discover a new, non-radial example of a manifold whose heat kernel decreases monotonically along all minimal geodesics. We also classify the flat tori with this monotonicity property. Furthermore, we show that for a generic metric on any smooth manifold the monotonicity property fails at large times. This answers a recent question of Alonso-Orán, Chamizo, Martínez, and Mas.

math.DG↗
arXiv2024

$C^{1,1-ε}$ Isometric embeddings

In this paper we use the convex integration technique enhanced by an extra iteration originally due to Källén and revisited by Kröner to provide a local $h$-principle for isometric embeddings in the class $C^{1,1-ε}$ for $n$-dimensional manifolds in codimension $\frac{1}{2}n(n+1)$.

math.DG↗
arXiv2022

Eigenfunction maxima and spherical means

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that Hörmander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic spheres for a certain class of manifolds.

math.CA↗
arXiv2022

Distribution symmetry of toral eigenfunctions

In this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we provide a counterexample for higher dimensional tori, which relies on a computer assisted argument. Moreover we prove a theorem on the distribution symmetry of a certain class of trigonometric polynomials that might be of independent interest.

math.CA↗
arXiv2022

Sign equidistribution of Legendre polynomials

We prove {\em sign equidistribution} of Legendre polynomials: the ratio between the lengths of the regions in the interval $[-1, 1]$ where the Legendre polynomial assumes positive versus negative values, converges to one as the degree grows. The proof method also has application to the symmetry conjecture for a basis of eigenfunctions in the sphere.

math.CA↗
arXiv2022

Córdoba's differentation theorem: revisited

In this paper we prove an exponential covering lemma implying the three dimensional case of a well-known conjecture formulated by A. Zygmund circa 1935 and solved by A. Córdoba in 1978. Our approach avoids a subtle argument involving the power series of the exponential function.

math.CA↗
arXiv2022

On the blow up of a non-local transport equation in compact manifolds

In this note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi's method.

math.AP↗
arXiv2020

An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω\subset \mathbb{R}^N$, $\mathcal{H}^β_{\infty}$ is the Hausdorff content, $L^{N/α,q}(Ω)$ is a Lorentz space with $q \in (1,\infty]$, $q'=q/(q-1)$ is the Hölder conjugate to $q$, and $I_αf$ denotes the Riesz potential of $f$ of order $α\in (0,N)$.

math.FA↗
arXiv2019

Pointwise monotonicity of heat kernels

In this paper the authors present a proof of a pointwise radial monotonicity property of heat kernels that is shared by the euclidean spaces, spheres and hyperbolic spaces. The main result deals with monotonicity from special points on revolution hypersurfaces from which the aforementioned are deduced. The proof relies on a non straightforward but elementary application of the parabolic maximum principle.

math.CA↗
arXiv2018

A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets

A set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ is called $\textit{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\textit{transverse}$ when it is stable under vertical and horizontal sums. A theorem of the first author provides a generalization of Bogolyubov's theorem to the bilinear setting. Roughly speaking, it implies that any dense transverse set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ contains a large bilinear set. In this paper, we elucidate the extent to which a transverse set is forced to be (and not only contain) a bilinear set.

math.CO↗
arXiv2018

A conjecture of Antoni Zygmund

In this paper we establish an exponential covering theorem implying a conjecture formulated by A. Zygmund circa 1935 whose three-dimensional case was obtained by the first named author in 1978.

math.CA↗