arXiv subjects

Javier Parcet

Publications and source records attributed to Javier Parcet.

At least 19 recordsLinked to original sources

arXiv2025

The impact of Schur multipliers in harmonic analysis and operator algebras

Schur multipliers are basic linear maps on matrix algebras. Their close albeit still intriguing connection with Fourier multipliers establishes a powerful bridge between harmonic analysis and operator algebras. In this paper, we survey their growing impact over the past 15 years. Particular attention will be drawn to recent bounds on Schatten $p$-classes, with far-reaching applications in harmonic analysis on group von Neumann algebras and operator rigidity phenomena for higher-rank Lie groups and lattices. Key novelties arise from new insights into nonToeplitz Schur multipliers and unprecedented connections with highly singular operators from Euclidean harmonic analysis.

math.OA↗

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 H\"ormander-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 $(\Phi, \Phi)$ for a family of functions including $\Phi(t) = t \, \log^{2+\varepsilon} t$, 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 $(\Phi, \Phi)$, then $\Phi(s) \not\in o(s \, \log^2 s)$. 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↗

arXiv2023

The local geometry of idempotent Schur multipliers

A Schur multiplier is a linear map on matrices which acts on its entries by multiplication with some function, called the symbol. We consider idempotent Schur multipliers, whose symbols are indicator functions of smooth Euclidean domains. Given $1<p\neq 2<\infty$, we provide a local characterization (under some mild transversality condition) for the boundedness on Schatten $p$-classes of Schur idempotents in terms of a lax notion of boundary flatness. We prove in particular that all Schur idempotents are modeled on a single fundamental example: the triangular projection. As an application, we fully characterize the local $L_p$-boundedness of smooth Fourier idempotents on connected Lie groups. They are all modeled on one of three fundamental examples: the classical Hilbert transform, and two new examples of Hilbert transforms that we call affine and projective. Our results in this paper are vast noncommutative generalizations of Fefferman's celebrated ball multiplier theorem. They confirm the intuition that Schur multipliers share profound similarities with Euclidean Fourier multipliers |even in the lack of a Fourier transform connection| and complete, for Lie groups, a longstanding search of Fourier $L_p$-idempotents.

math.FA↗

arXiv2023

Schur multipliers in Schatten-von Neumann classes

We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1 < p < \infty$, a simple form our main result reads for $\mathbf{R}^n \times \mathbf{R}^n$ matrices as follows $$\big\| S_M: S_p \to S_p \big\|_{\mathrm{cb}} \lesssim \frac{p^2}{p-1} \sum_{|γ| \le [\frac{n}{2}] +1} \Big\| |x-y|^{|γ|} \Big\{ \big| \partial_x^γM(x,y) \big| + \big| \partial_y^γM(x,y) \big| \Big\} \Big\|_\infty.$$ In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders $σ> \frac{n}{2}$ as well. It trivially includes Arazy's conjecture for $S_p$-multipliers and extends it to $α$-divided differences. It also leads to new Littlewood-Paley characterizations of $S_p$-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.

math.FA↗

arXiv2022

Trigonometric chaos and $\mathrm{X}_p$ inequalities II -- $\mathrm{X}_p$ inequalities in group von Neumann algebras

In the line of previous work by Naor, we establish new forms of metric $\mathrm{X}_p$ inequalities in group algebras under very general assumptions. Our results' applicability goes beyond the previously known setting in two directions. In first place, we find continuous forms of the $\mathrm{X}_p$ inequality in the $n$-dimensional torus. Second, we consider transferred forms of the sharp scalar valued metric $\mathrm{X}_p$ inequality in the von Neumann algebra $\mathcal{L}(\mathrm{G})$ of a discrete group $\mathrm{G}$. As a byproduct of our results, some metric consequences and their relation with bi-Lipschitz nonembeddability of Banach spaces are explored in the context of noncommutative $L_p$ spaces.

math.FA↗

arXiv2022

Trigonometric chaos and $\mathrm{X}_p$ inequalities I -- Balanced Fourier truncations over discrete groups

We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of directional derivatives which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach $\mathrm{X}_p$ nature of noncommutative $L_p$-spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms.

math.FA↗

arXiv2022

Noncommutative Cotlar identities for groups acting on tree-like structures

Let $T_m$ be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity in order to prove that certain multipliers are bounded on the non-commutative $L_p$-spaces of a free group. Here, we study Cotlar type identities in full generality, giving a closed characterization for them in terms of $m$: \[ \big( m(g h) - m(g) \big) \, \big( m(g^{-1}) - m(h) \big) = 0, \; \forall g \in \mathrm{G} \setminus \{e\}, h \in \mathrm{G}. \] We manage to prove, using a geometric argument, that if $X$ is a tree -- or more generally an $\mathbb{R}$-tree -- on which $\mathrm{G}$ acts and $m$ lifts to a function $\widetilde{m}: X \to \mathbb{C}$ that is constant on the connected subsets of $X \setminus \{x_0\}$, then $m$ satisfies Cotlar's identity and thus $T_m$ is bounded in $L_p$ for $1 < p < \infty$. This result establishes a new connection between groups actions on $\mathbb{R}$-trees and Fourier multipliers. We show that $m$ is trivial when the action has global fixed points. This machinery allows us to simultaneously generalize the free group transforms of Mei and Ricard and the theory of Hilbert transforms in left-orderable groups, which follows from Arveson's subdiagonal algebras. Using Bass-Serre theory, we construct new examples of Fourier multipliers in groups. Those include new families like Baumslag-Solitar groups. We also show that a natural Hilbert transform in $\mathrm{PSL}_2(\mathbb{C})$ satisfies Cotlar's identity when restricted to the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-1}])$.

math.FA↗

arXiv2022

A H\"ormander-Mikhlin theorem for high rank simple Lie groups

We establish regularity conditions for $L_p$-boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural H\"ormander-Mikhlin criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard and de la Salle for $\SL$. Our approach is partly based on a sharp local H\"ormander-Mikhlin theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular nonToeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle-based approach to Fourier multipliers in group algebras by Junge, Mei and Parcet. A few related open problems are also discussed.

math.FA↗

arXiv2021

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak $L_1$ endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of $L_1$ endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length $ψ: \mathrm{G} \to \mathbb{R}_+$, we prove that Herz-Schur multipliers with symbol $m \circ ψ$ satisfying a Mikhlin condition in terms of the $ψ$-cocycle dimension are of weak type $(1,1)$. Our result extends to Fourier multipliers for amenable groups and impose sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This $L_1$ endpoint gives a very much expected inequality which complements the $L_\infty \to \mathrm{BMO}$ estimates proved in 2014 by Junge, Mei and Parcet.

math.CA↗

arXiv2021

Fourier multipliers in $\mathrm{SL}_n(\mathbf{R})$

We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is inspired by Hörmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Lafforgue/de la Salle's rigidity theorem fits here. The proof includes a new relation between Fourier and Schur $L_p$-multipliers for nonamenable groups. By transference, matters are reduced to a rather nontrivial $RC_p$-inequality for $SL_n(\mathbf{R})$-twisted forms of Riesz transforms associated to fractional laplacians. Our second result gives a new and much stronger rigidity theorem for radial multipliers in $SL_n(\mathbf{R})$. More precisely, additional regularity and Mikhlin type conditions are proved to be necessary up to an order $\sim |\frac12 - \frac1p| (n-1)$ for large enough $n$ in terms of $p$. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of $L_p$-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our Hörmander-Mikhlin sufficient conditions in rank $n$ and failing the rigidity conditions for ranks $m >> n$. We also prove automatic regularity and rigidity estimates for first and higher order derivatives of $\mathrm{K}$-biinvariant multipliers in the rank 1 groups $SO(n,1)$.

math.FA↗

arXiv2019

Noncommutative strong maximals and almost uniform convergence in several directions

Our first result is a noncommutative form of Jessen/Marcinkiewicz/Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$-norm of the $\limsup$ of a sequence of operators as a localized version of a $\ell_\infty/c_0$-valued $L_p$-space. In particular, our main result gives a strong $L_1$-estimate for the $\limsup$, as opposed to the usual weak $L_{1,\infty}$-estimate for the $\sup$. Let $\mathcal{L} \mathbf{F}_2$ denote the free group algebra and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F}_2)$ for which this semigroup converges to the initial data. Currently, the best known result is $L \log^2 L(\mathcal{L} \mathbf{F}_2)$. We improve this by adding to it the operators in $L_1(\mathcal{L} \mathbf{F}_2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set has exponential growth. The proof relies on our estimates for the noncommutative $\limsup$ together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal. More precisely, a weak $(Φ,Φ)$ inequality for noncommutative multiparametric martingales and $Φ(s) = s (1 + \log_+ s)^{2 + \varepsilon}$. This logarithmic power is an $\varepsilon$-perturbation of the expected optimal one. The proof combines a refinement of Cuculescu's construction with a quantum probabilistic interpretation of de Guzmán's argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu's projections.

math.FA↗

arXiv2019

Algebraic Calderón-Zygmund theory

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

math.FA↗

arXiv2018

The Hausdorff-Young inequality on Lie groups

We prove several results about the best constants in the Hausdorff-Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.

math.FA↗

arXiv2017

Singular integrals in quantum Euclidean spaces

In this paper, we establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce $L_p$-boundedness and Sobolev $p$-estimates for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$ level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove $L_p$-regularity of solutions for elliptic PDEs.

math.OA↗

arXiv2017

On spectral gaps of Markov maps

It is shown that if a Markov map $T$ on a noncommutative probability space $\mathcal{M}$ has a spectral gap on $L_2(\mathcal{M})$, then it also has one on $L_p(\mathcal{M})$ for $1<p<\infty$. For fixed $p$, the converse also holds if $T$ is factorizable. These results are also new for classical probability spaces.

math.PR↗

arXiv2016

Asymmetric Doob inequalities in continuous time

The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let $(\mathcal M,τ)$ be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras $(\mathcal M_t)_{0\leq t\leq1}$ whose union is weak-$*$ dense in $\mathcal{M}$. Let $\mathcal E_t$ denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for $1 < p < 2$ and $x \in L_p(\mathcal M,τ)$ one can find $a, b \in L_p(\mathcal M,τ)$ and contractions $u_t, v_t \in \mathcal M$ such that $$\mathcal E_t(x) = a u_t + v_t b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, $a u_t$ and $v_t b$ converge in the row/column Hardy spaces $\mathcal H_p^r(\mathcal M)$ and $\mathcal H_p^c(\mathcal M)$ respectively. We also confirm in the continuous setting the validity of related asymmetric maximal inequalities which we recently found for discrete filtrations, including $p=1$. As for other results in noncommutative martingale theory, the passage from discrete to continuous index is quite technical and requires genuinely new methods. Our approach towards asymmetric maximal inequalities is based on certain construction of conditional expectations for a sequence of projective systems of $L_p$-modules. The convergence in $\mathcal H_p^r(\mathcal M)$ and $\mathcal H_p^c(\mathcal M)$ also imposes new algebraic atomic decompositions.

math.PR↗

arXiv2016

Nondoubling Calderón-Zygmund theory -a dyadic approach-

Given a measure $μ$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $\mathrm{supp}(μ)$ which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric |which in turn are all doubling| and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions: i) A dyadic form of Tolsa's RBMO space which contains it. ii) Lerner's domination and $A_2$-type bounds for nondoubling measures. iii) A noncommutative form of nonhomogeneous Calderón-Zygmund theory. Our martingale RBMO space preserves the crucial properties of Tolsa's original definition and reveals its interpolation behavior with the $L_p$ scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative $L_p$ spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calderón-Zygmund decomposition which unifies those by Tolsa and López-Sánchez/Martell/Parcet.

math.CA↗