Fourier multipliers and their applications to PDE on the quantum Euclidean space

Abstract

In this work, we present some applications of the $L^p$-$L^q$ boundedness of Fourier multipliers to PDEs on the noncommutative (or quantum) Euclidean space. More precisely, we establish $L^p$-$L^q$ norm estimates for solutions of heat, wave, and Schrödinger type equations with Caputo fractional derivative in the case $1<p\le 2\le q<\infty .$ Moreover, we obtain well-posedness of nonlinear heat and wave equations on the noncommutative Euclidean space.

arXiv comment21 pages. Accepted to NoDEA. Some inaccuracies regarding the symbol $\sigma$ are corrected

References in corpus (7)

• Moyal Planes are Spectral Triples
• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Quantum differentiability on noncommutative Euclidean spaces
• Quantum-Classical Hybrid Systems and their Quasifree Transformations
• $L^p$-$L^q$ boundedness of $(k,a)$-Fourier multipliers with applications to Nonlinear equations
• Hybrid quantum-classical systems: Quasi-free Markovian dynamics
• On Quantum Sobolev Inequalities

Cited by in corpus (3)

• Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces
• $L^p-L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces
• Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications