Symmetry and functional inequalities for stable L\'evy-type operators

In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable L\'evy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$\mathcal{L}=a(x){\Delta^{\alpha/2}}+b(x)\frac{\d}{\d x},$$ where $a,b$ are the continuous positive and differentiable functions, respectively. Under the assumption of symmetry, we further study the criteria for functional inequalities, including Poincar\'e inequalities, logarithmic Sobolev inequalities and Nash inequalities. Our proofs rely on the Orlicz space theory and the estimates of the Green functions.