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The impact of Schur multipliers in harmonic analysis and operator algebras

Javier Parcet

TL;DR

This survey highlights how Schur multipliers act as a bridge between harmonic analysis and operator algebras, emphasizing their impact on Schatten classes, group von Neumann algebras, and rigidity phenomena in higher-rank lattices. It develops and collects new inequalities for Schur multipliers, extends Fourier-Schur transference to nonToeplitz settings, and derives Hörmander–Mikhlin-type criteria in matrix-valued contexts, including both cocycle and Lie-derivative approaches. A key contribution is the local geometry of $S_p$-idempotents and a classification of Hilbert transforms on Lie groups, revealing deep links between Euclidean harmonic analysis and noncommutative analysis. The work applies these tools to operator rigidity results for higher-rank lattices, notably the Lafforgue–de Laat–de la Salle rigidity theorem, showing failure of Fourier $L_p$-approximation in these groups and connecting to Connes' rigidity program. It also explores weaker decay regimes and conjectural connections to Bochner–Riesz and Kakeya phenomena, outlining new directions at the intersection of harmonic analysis and operator algebras.

Abstract

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.

The impact of Schur multipliers in harmonic analysis and operator algebras

TL;DR

This survey highlights how Schur multipliers act as a bridge between harmonic analysis and operator algebras, emphasizing their impact on Schatten classes, group von Neumann algebras, and rigidity phenomena in higher-rank lattices. It develops and collects new inequalities for Schur multipliers, extends Fourier-Schur transference to nonToeplitz settings, and derives Hörmander–Mikhlin-type criteria in matrix-valued contexts, including both cocycle and Lie-derivative approaches. A key contribution is the local geometry of -idempotents and a classification of Hilbert transforms on Lie groups, revealing deep links between Euclidean harmonic analysis and noncommutative analysis. The work applies these tools to operator rigidity results for higher-rank lattices, notably the Lafforgue–de Laat–de la Salle rigidity theorem, showing failure of Fourier -approximation in these groups and connecting to Connes' rigidity program. It also explores weaker decay regimes and conjectural connections to Bochner–Riesz and Kakeya phenomena, outlining new directions at the intersection of harmonic analysis and operator algebras.

Abstract

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 -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.
Paper Structure (14 sections, 11 theorems, 44 equations)

This paper contains 14 sections, 11 theorems, 44 equations.

Table of Contents

  1. New inequalities for Schur multipliers
  2. Fourier-Schur transference
  3. Hörmander-Mikhlin-Schur multipliers
  4. The local geometry of idempotent Schur multipliers
  5. Harmonic analysis in group von Neumann algebras
  6. More on Fourier-Schur transference
  7. Fourier multipliers with one-point singularities
  8. Hörmander-Mikhlin criteria via cocycles
  9. Hörmander-Mikhlin criteria via Lie derivatives
  10. Characterizing Hilbert transforms on Lie groups
  11. Operator rigidity phenomena in higher-rank lattices
  12. Lafforgue/de Laat/de la Salle's rigidity theorem
  13. On Fourier and Schur $L_p$-approximations
  14. Weaker changes of regime -- Polynomial decay

Key Result

Theorem 1.1

(Fourier-Schur transference I). Let $1 \le p \le \infty$ and consider a locally compact abelian group $\mathrm{G}$. Assume that $m \colon \mathrm{G} \to \mathbf{C}$ defines a completely $L_p$-bounded Fourier multiplier and set $M(g,h) = m(gh^{-1})$. Then

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 2.1
  • Theorem 2.2
  • ...and 18 more