The nonlinear estimates on quantum Besov spaces

TL;DR

This paper establishes boundedness estimates for superposition operators with non-smooth symbols on quantum Besov spaces, advancing the understanding of noncommutative PDEs and resolving a key conjecture.

Contribution

It generalizes previous results to non-smooth symbols and proves the equivalence of two quantum Besov space descriptions, using a novel quantum chain rule.

Findings

Boundedness estimates for superposition operators on quantum Besov spaces.

Resolution of the conjecture on the equivalence of quantum Besov space descriptions.

Introduction of a quantum chain rule and nonlinear interpolation techniques.

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. The ingredients in the proof involve a novel quantum chain rule and nonlinear interpolation. 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}.