  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.
