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The nonlinear estimates on quantum Besov space

Deyu Chen, Guixiang Hong

TL;DR

The paper addresses nonlinear PDEs on quantum Euclidean spaces by proving boundedness and Lipschitz-type estimates for nonlinear superposition operators with non-smooth symbols on quantum Besov spaces. It introduces a quantum chain rule and leverages multiple operator integrals together with difference characterizations to obtain sharp nonlinear estimates for $T_F$ on $B_{p,q}^s(R^d_ heta)$, covering cases $s> frac{d}{p}$ and $F$ in various regularity classes. A key achievement is resolving McDonald’s conjecture on the equivalence of two descriptions of quantum Besov spaces and deriving a quantum chain rule that underpins a systematic approach to nonlinear estimates, including perturbation formulas. The results yield local and, under Lipschitz assumptions on $F$, global well-posedness results for noncommutative Allen-Cahn equations, highlighting a robust framework for nonlinear analysis in noncommutative PDEs with non-smooth nonlinearities.

Abstract

The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in \cite[Remark 3.16]{McNLE}. The new ingredients in the proof also involve quantum chain rule and nonlinear interpolation.

The nonlinear estimates on quantum Besov space

TL;DR

The paper addresses nonlinear PDEs on quantum Euclidean spaces by proving boundedness and Lipschitz-type estimates for nonlinear superposition operators with non-smooth symbols on quantum Besov spaces. It introduces a quantum chain rule and leverages multiple operator integrals together with difference characterizations to obtain sharp nonlinear estimates for on , covering cases and in various regularity classes. A key achievement is resolving McDonald’s conjecture on the equivalence of two descriptions of quantum Besov spaces and deriving a quantum chain rule that underpins a systematic approach to nonlinear estimates, including perturbation formulas. The results yield local and, under Lipschitz assumptions on , global well-posedness results for noncommutative Allen-Cahn equations, highlighting a robust framework for nonlinear analysis in noncommutative PDEs with non-smooth nonlinearities.

Abstract

The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in \cite[Remark 3.16]{McNLE}. The new ingredients in the proof also involve quantum chain rule and nonlinear interpolation.
Paper Structure (24 sections, 33 theorems, 260 equations)

This paper contains 24 sections, 33 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Background
  3. PDE theory on quantum Euclidean spaces
  4. Nonlinear superposition operators
  5. Main results
  6. Preliminaries
  7. Quantum Euclidean spaces
  8. Differential calculus on quantum Euclidean spaces
  9. Fourier multiplier on quantum Euclidean spaces
  10. Quantum Besov spaces
  11. Characterizations of quantum Besov spaces
  12. Multiplier characterization
  13. Difference characterization
  14. Heat semigroup on quantum Besov spaces
  15. Multiple operator integral
  16. ...and 9 more sections

Key Result

Theorem 1.1

For $p,q\in [1,\infty],s>\frac{d}{p}$ and $u\in B_{p,q}^s(\mathbb{R}^d_{\theta})$ self-adjoint, if $F\in C^{\left \lceil s \right \rceil}({\mathbb R})$ and $F(0)=0,$ then there exists a positive non-decreasing continuous function $h$ such that for $p\in (1,\infty),$ we have If $F \in_{\mathrm{loc}} \tilde{B}_{\infty,1}^{1}(\mathbb{R})\cap \tilde{B}_{\infty,1}^{\left \lceil s \right \rceil}(\mathb

Theorems & Definitions (74)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 64 more