Abstract
By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash's inequality, Carlson's inequality and Sobolev's embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the L-infinity-norm. Finally, we obtain a logarithmic Sobolev inequality in L-p-spaces, from which we derive an L-p-Heisenberg-type uncertainty inequality and an L-p-Nash-type inequality for the Dunkl transform.