| Abstract: |
| In this talk, we establish the sufficient and necessary conditions for the symmetry of the following stable L\`evy-type operator $\mathcal{L}$ on $\mathbb{R}$:
$$
\mathcal{L}=a(x){\Delta^{\alpha/2}}+b(x)\frac{\mathrm{d}}{\mathrm{d} x},
$$
where $a,b$ are the continuous positive and differentiable functions, respectively. We then study the criteria for functional inequalities, such as logarithmic Sobolev inequalities, Nash inequalities and super-Poincar\`e inequalities
under the assumption of symmetry. Our approach involves the Orlicz space theory and the estimates of the Green functions. This is based on a joint work with Tao Wang. |
|