Boundedness of Fourier Multipliers and Applications to Nonlinear PDEs for the Strichartz Fourier Transform on the Heisenberg Group

Abstract

We investigate Fourier multipliers associated with the Strichartz Fourier transform on the Heisenberg group. In particular, we establish Hörmander-type $L^p-L^q$ boundedness results for the range $1<p\le 2\le q<\infty$. The analysis is based on deriving suitable analogues of the Hausdorff-Young and Paley inequalities for the Strichartz Fourier transform, followed by interpolation arguments to obtain the desired multiplier estimates. As an application, we study the local well-posedness of certain nonlinear partial differential equations. Furthermore, we establish an $L^p$-boundedness theorem for Fourier multipliers associated with the Strichartz Fourier transform for the full range $1<p<\infty$.