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

In this paper we extend classical Titchmarsh theorems on the Fourier transform of HölderLipschitz 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-Hölder-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-L boundedness of Fourier multipliers by extending a classical theorem of Hörmander 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-L boundedness of spectral multipliers of the Jacobi Laplacian.

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