On the norms of generalized translation operators generated by the Jacobi-Dunkl operators (Q690580)

The author studies operators of generalized translation generated by the Dunkl type operators \(\Lambda\) of the form \[ T_\Lambda^y\,f(x)=\int_{-\pi}^\pi f(z)W(x,y,z)A_{\alpha,\beta}(z)\,dz \] in the spaces \(L^p_{\alpha,\beta}[-\pi,\pi]\) with the weight \[ A_{\alpha,\beta}(x)=(1-\cos x)^\alpha (1+\cos x)^\beta |\sin x|. \] Here, \(W\) is a specific kernel which involves only sine and cosine functions of \(x,y,z\). One of the main results provides a bound of the norm of such an operator for \(1\leq p\leq+\infty\) of the form \(\log \|T_\Lambda^y\|\leq |1-2/p|\log 2\).