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In this paper we extend classical Titchmarsh theorems on the Fouriertransform of H$\ddot{\text{o}}$lder-Lipschitz functions to the setting ofharmonic $NA$ groups, which relate smoothness properties of functions to thegrowth and integrability of their Fourier transform. We prove a Fouriermultiplier theorem for $L^2$-H$\ddot{\text{o}}$lder-Lipschitz spaces onHarmonic $NA$ groups. We also derive conditions and a characterisation ofDini-Lipschitz classes on Harmonic $NA$ groups in terms of the behaviour oftheir Fourier transform. Then, we shift our attention to the spherical analysison Harmonic $NA$ group. Since the spherical analysis on these groups fits wellin the setting of Jacobi analysis we prefer to work in the Jacobi setting. Weprove $L^p$-$L^q$ boundedness of Fourier multipliers by extending a classicaltheorem of H$\ddot{\text{o}}$rmander to the Jacobi analysis setting. On the wayto accomplish this classical result we prove Paley-type inequality andHausdorff-Young-Paley inequality. We also establish $L^p$-$L^q$ boundedness ofspectral multipliers of the Jacobi Laplacian.

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