Logarithm Sobolev and Shannon’s Inequalities Associated with the Deformed Fourier Transform and Applications

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∞-norm. Finally, we obtain a logarithmic Sobolev inequality in Lp-spaces, from which we derive an Lp-Heisenberg-type uncertainty inequality and an Lp-Nash-type inequality for the Dunkl transform.