On the norms of generalized translation operators generated by Dunkl type operators (Q690818)

An integral representation for the generalized translation operators generated by Dunkl type operators \[ \Lambda f(x) = f'(x) + \frac{A'(x)}{A(x)}\frac{f(x) - f( - x)}{2} \] in the spaces \( L_p(\mathbb{R})\) with weight \(A\) is established, and a known estimate for their norms is improved. It is proved that, under some natural assumptions on the functions \(A\), these norms do not exceed two.