  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.
