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Hormander-Mikhlin type theorem on non-commutative spaces

Rauan Akylzhanov, Michael Ruzhansky, Kanat Tulenov

TL;DR

This work develops a non-commutative Fourier analysis framework for semifinite von Neumann algebras and locally compact Kac groups, introducing a Fourier structure that enables non-commutative versions of Hörmander-Mikhlin multiplier theorems. The authors prove general $L^{p}$-boundedness results for Fourier multipliers via a spectral operator $oldsymbol{\mathcal{D}}$ and a bounded operator $oldsymbol{\Psi}$, with bounds expressed in Lorentz spaces $L^{r,\, u}$ and involving $oldsymbol{}(t;oldsymbol{\mathcal{D}}^{eta}A)$; in the commutative case these reduce to the sharp classical criteria. Central to the approach are Paley-type and Hardy-Littlewood inequalities in the non-commutative setting, derived from the Fourier structure and generalized singular values. The paper also demonstrates applications to non-commutative evolution equations, establishing decay and regularity estimates for associated propagators and situating the results within the broader landscape of non-commutative harmonic analysis and quantum group theory.

Abstract

In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respectively. In the simplest case our result coincides with a sharp version of the classical Hormander Lp-multiplier theorem, which was obtained by Grafakos and Slavikova in [11]. Finally, we present some applications to the evolution equation in non-commutative setting.

Hormander-Mikhlin type theorem on non-commutative spaces

TL;DR

This work develops a non-commutative Fourier analysis framework for semifinite von Neumann algebras and locally compact Kac groups, introducing a Fourier structure that enables non-commutative versions of Hörmander-Mikhlin multiplier theorems. The authors prove general -boundedness results for Fourier multipliers via a spectral operator and a bounded operator , with bounds expressed in Lorentz spaces and involving ; in the commutative case these reduce to the sharp classical criteria. Central to the approach are Paley-type and Hardy-Littlewood inequalities in the non-commutative setting, derived from the Fourier structure and generalized singular values. The paper also demonstrates applications to non-commutative evolution equations, establishing decay and regularity estimates for associated propagators and situating the results within the broader landscape of non-commutative harmonic analysis and quantum group theory.

Abstract

In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respectively. In the simplest case our result coincides with a sharp version of the classical Hormander Lp-multiplier theorem, which was obtained by Grafakos and Slavikova in [11]. Finally, we present some applications to the evolution equation in non-commutative setting.

Paper Structure

This paper contains 11 sections, 16 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical $L^p$ and $L^{p,q}$ spaces
  4. Von Neumann algebras and $\tau$-measurable operators
  5. Noncommutative $L^p$ and $L^{p,q}$ spaces associated with a semi-finite von Neumann algebra
  6. Fourier structure on von Neumann algebras
  7. Paley and Hausdorff-Young-Paley type inequalities on von Neumann algebras endowed with Fourier structure
  8. Hardy-Littlewood-type inequalities associated with $\mathcal{D}$
  9. Hörmander-Mihlin type multiplier theorems
  10. Applications
  11. Acknowledgements

Key Result

Theorem 1.1

GS19. Let $\Psi$ be a Schwartz function on $\mathbb{R}^{n}$ whose Fourier transform is supported in the annulus $\frac{1}{2}<|\xi|<2$ and satisfies $\sum\limits_{j \in \mathbb{Z}} \widehat{\Psi}\left(2^{-j} \xi\right)=1, \quad \xi \neq 0$. Let $1<p<\infty$ and let $s \in(0, n), n \in \mathbb{N}$, sa Then for all functions $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$ we have the following priori

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 3.1
  • proof
  • ...and 25 more