Asymptotic estimates for the growth of deformed Hankel transform by modulus of continuity

We derive asymptotic estimates for the growth of the norm of the deformed Hankel transform on the deformed Hankel--Lipschitz space defined via a generalised modulus of continuity. The established results are similar in nature to the well-known Titchmarsh theorem, which provide a characterization of the square integrable functions satisfying certain Cauchy--Lipschitz condition in terms of an asymptotic estimate for the growth of the norm of their Fourier transform. We also give some necessary conditions in terms of the generalised modulus of continuity for the boundedness of the Dunkl transform of functions in Dunkl-Lipschitz spaces, improving the Hausdorff-Young inequality for the Dunkl transform in this special scenario.