Symmetry and functional inequalities for stable Lévy-type operators

Abstract

In this paper, we establish the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator $L$ on $R$: $L=a\left(x\right){\Delta }^{\alpha /2}+b\left(x\right)\frac{d}{dx},$where $a$ is a continuous and strictly positive function, and $b$ is a differentiable function. We then study the criteria for functional inequalities, such as logarithmic Sobolev inequalities, Nash inequalities and super-Poincaré inequalities under the assumption of symmetry. Our approach involves the Orlicz space theory and the estimates of the Green functions.