Titchmarsh theorems, Hausdorff-Young-Paley inequality and $L^p$-$L^q$ boundedness of Fourier multipliers on harmonic $NA$ groups

TL;DR

This paper extends classical Fourier analysis theorems to harmonic $NA$ groups, establishing new multiplier theorems, inequalities, and boundedness results in the context of Jacobi analysis and harmonic analysis on these groups.

Contribution

It introduces Fourier multiplier theorems and boundedness results for harmonic $NA$ groups, extending classical analysis to a non-commutative setting with Jacobi analysis techniques.

Findings

Proved Fourier multiplier theorem for $L^2$-Hölder-Lipschitz spaces on harmonic $NA$ groups.

Derived conditions for Dini-Lipschitz classes via Fourier transform behavior.

Established $L^p$-$L^q$ boundedness of spectral multipliers of the Jacobi Laplacian.

Abstract

In this paper we extend classical Titchmarsh theorems on the Fourier transform of H$\ddot{\text{o}}$lder-Lipschitz functions to the setting of harmonic $NA$ groups, which relate smoothness properties of functions to the growth and integrability of their Fourier transform. We prove a Fourier multiplier theorem for $L^2$-H$\ddot{\text{o}}$lder-Lipschitz spaces on Harmonic $NA$ groups. We also derive conditions and a characterisation of Dini-Lipschitz classes on Harmonic $NA$ groups in terms of the behaviour of their Fourier transform. Then, we shift our attention to the spherical analysis on Harmonic $NA$ group. Since the spherical analysis on these groups fits well in the setting of Jacobi analysis we prefer to work in the Jacobi setting. We prove $L^p$-$L^q$ boundedness of Fourier multipliers by extending a classical theorem of H$\ddot{\text{o}}$rmander to the Jacobi analysis setting. On the way to accomplish this classical result we prove Paley-type inequality and Hausdorff-Young-Paley inequality. We also establish $L^p$-$L^q$ boundedness of spectral multipliers of the Jacobi Laplacian.