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

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 $1<p\le 2\le q<\mathrm{\infty }\mathrm{.}$ Moreover, we obtain well-posedness of nonlinear heat and wave equations on the noncommutative Euclidean space.