arXiv subjects

Jorge Pérez García

Publications and source records attributed to Jorge Pérez García.

4 recordsLinked to original sources

arXiv2026

Radial Schur multipliers on trees and the failure of Fourier-Schur transference

Given $1 < p < \infty$ and a function $m \colon \mathbf{N}_0 \to \mathbf{C}$, we provide a characterization of radial Schur multipliers $M(x,y) = m(d(x,y))$ on the Schatten $p$-class of a tree $(X,d)$ equipped with its graph metric. That is, the operator norm of the radial Schur multiplier $S_M$ is controlled by the norm of the Fourier multiplier $T_m$ on the torus: \[ \big\| S_M \colon S_p(X) \to S_p(X) \big\|_{\mathrm{cb}} \lesssim \frac{p^4}{(p-1)^2} \, \big\| T_m \colon L_p(\mathbf{T}) \to L_p(\mathbf{T}) \big\|_{\mathrm{cb}}, \] with both quantities being comparable when the tree $X$ has infinite diameter. This is a rather unexpected inequality, which necessarily fails at the endpoints $p=1,\infty$ due to the work of Haagerup, Steenstrup and Szwarc. As an application, we construct the first examples of completely $S_p$-bounded Herz-Schur multipliers in the free group that are unbounded as Fourier multipliers. This result disproves Fourier-Schur transference---a longstanding open problem since the work of Neuwirth and Ricard in 2011. In line with this, our main result also implies the surprising fact that free balls Schur multipliers are uniformly $S_p$-bounded, in sharp contrast to their Fourier behavior. In addition, we disprove Fourier-Schur transference over the algebraic groups $\mathrm{PGL}_2(\mathbf{Q}_q)$ and $\mathrm{PSL}_2(\mathbf{R})$ with $\mathrm{K}$-biinvariant counterexamples, the latter involving additional geometric and probabilistic techniques.

math.FA↗

arXiv2025

A note on Hilbert transform over lattices of $\mathrm{PSL}_2(\mathbb{C})$

González-Pérez, Parcet and Xia introduced recently a framework to study $L_p$-boundedness of certain families of idempotent multipliers on von Neumann algebras. It includes symbols $m\colon \mathrm{PSL}_2(\mathbb{C})\to \mathbb{R}$ arising from lifting the indicator function of a partition $\{Σ^+,Σ^+,Σ^-\}$ of the hyperbolic space $\mathbb{H}^3$ to its isometry group $\mathrm{PSL}_2(\mathbb{C})$. The boundedness of $T_m$ on $L_p(\mathcal{L} \mathrm{PSL}_2(\mathbb{C}))$ was disproved by Parcet, de la Salle and Tablate. Nevertheless, we will show that this Fourier multiplier is bounded when restricted to the arithmetic lattices $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-n}])$, solving a question left open by the first named authors.

math.FA↗

arXiv2024

Riesz-Schur transforms

We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of Hörmander-Mikhlin and Marcinkiewicz type.

math.FA↗

arXiv2024

Maximal weak Orlicz types and the strong maximal on von Neumann algebras

Let $\mathbf{E}_n: \mathcal{M} \to \mathcal{M}_n$ and $\mathbf{E}_m: \mathcal{N} \to \mathcal{N}_m$ be two sequences of conditional expectations on finite von Neumann algebras. The optimal weak Orlicz type of the associated strong maximal operator $\mathcal{E} = (\mathbf{E}_n\otimes \mathbf{E}_m)_{n,m}$ is not yet known. In a recent work of Jose Conde and the two first-named authors, it was show that $\mathcal{E}$ has weak type $(Φ, Φ)$ for a family of functions including $Φ(t) = t \, \log^{2+\varepsilon} t$, for every $\varepsilon > 0$. In this article, we prove that the weak Orlicz type of $\mathcal{E}$ cannot be lowered below $L \log^2 L$, meaning that if $\mathcal{E}$ is of weak type $(Φ, Φ)$, then $Φ(s) \not\in o(s \, \log^2 s)$. Our proof is based on interpolation. Namely, we use recent techniques of Cadilhac/Ricard to formulate a Marcinkiewicz type theorem for maximal weak Orlicz types. Then, we show that a weak Orlicz type lower than $L \log^2 L$ would imply a $p$-operator constant for $\mathcal{E}$ smaller than the known optimum as $p \to 1^{+}$.

math.OA↗