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The local geometry of idempotent Schur multipliers

Javier Parcet, Mikael de la Salle, Eduardo Tablate

TL;DR

The paper analyzes the local geometry of idempotent Schur multipliers, proving that $S_p$-boundedness is equivalent to a zero-curvature condition and to a triangular truncation representation, thereby showing that all Schur idempotents are locally modeled by the triangular projection. It then extends these local models to idempotent Fourier multipliers on Lie groups, establishing a transference framework that reduces local cb-$L_p$-multipliers to three fundamental Hilbert-transform-like examples: the classical Hilbert transform and its affine and projective variants. This yields a comprehensive local classification of Fourier $L_p$-idempotents on broad classes of Lie groups, including a precise description for $SL_2(\mathbf{R})$ and stratified groups, and connects noncommutative Schur multiplier theory with Fefferman’s ball multiplier paradigm. The results illuminate deep structural similarities between Schur and Fourier multipliers beyond the Fourier-analytic setting and have implications for understanding $L_p$-boundedness of idempotent multipliers in noncommutative harmonic analysis on groups.

Abstract

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.

The local geometry of idempotent Schur multipliers

TL;DR

The paper analyzes the local geometry of idempotent Schur multipliers, proving that -boundedness is equivalent to a zero-curvature condition and to a triangular truncation representation, thereby showing that all Schur idempotents are locally modeled by the triangular projection. It then extends these local models to idempotent Fourier multipliers on Lie groups, establishing a transference framework that reduces local cb--multipliers to three fundamental Hilbert-transform-like examples: the classical Hilbert transform and its affine and projective variants. This yields a comprehensive local classification of Fourier -idempotents on broad classes of Lie groups, including a precise description for and stratified groups, and connects noncommutative Schur multiplier theory with Fefferman’s ball multiplier paradigm. The results illuminate deep structural similarities between Schur and Fourier multipliers beyond the Fourier-analytic setting and have implications for understanding -boundedness of idempotent multipliers in noncommutative harmonic analysis on groups.

Abstract

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 , we provide a local characterization (under some mild transversality condition) for the boundedness on Schatten -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 -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 -idempotents.
Paper Structure (14 sections, 10 theorems, 53 equations, 1 figure)

This paper contains 14 sections, 10 theorems, 53 equations, 1 figure.

Table of Contents

  1. Idempotent Schur multipliers
  2. Schur multipliers
  3. Proof of Theorem A: Boundedness implies zero-curvature
  4. Proof of Theorem A: Zero-curvature implies triangular truncations
  5. Proof of Theorem A: Transference on triangular truncations
  6. Relatively compact domains.
  7. Curvature on smoother domains.
  8. On the transversality condition.
  9. Idempotent Fourier multipliers on Lie groups
  10. Idempotent multipliers
  11. Local Fourier-Schur transference
  12. Nonunimodular groups
  13. The group $\mathrm{SL}_2(\mathbf{R})$
  14. Stratified Lie groups

Key Result

Lemma 1.1

Let $(X,\mu),(X',\mu'),(Y,\nu),(Y',\nu')$ be atomless $\sigma$-finite measure spaces and $f\colon X \to X'$ and $g\colon Y \to Y'$ be measurable maps. Assume the pushforward measures $f_* \mu$ and $g_* \nu$ are absolutely continuous with respect to the measures $\mu'$ and $\nu'$ respectively. Then,

Figures (1)

  • Figure 1: Failure of \ref{['item:SteinCurvature']} for spherical Hilbert transforms $H_{{\mathbf{S},\delta}}$

Theorems & Definitions (31)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof : Proof of Lemma \ref{['prop:Meyer_Lemma_NC']}.
  • Remark 1.3
  • Theorem 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Corollary 1.7
  • ...and 21 more