  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.
