赵国焕，中国科学院数学与系统科学研究院

The regularity estimates have well-established for second-order elliptic and parabolic equations, as well as for equations with stable-like non-local operators. Whether similar results hold for non-local operators generated by Lévy processes with low-intensity small jumps (for instance $\log(1-\Delta)$) is unclear.

In this talk, I will first briefly introduce the Markov processes and operators satisfying the positive maximal principle, especially pure jump processes and non-local operators. After that, I plan to present some regularity results for non-local operators with critically low singularity kernels that do not allow standard scaling. For instance, Schauder-type estimates for log Laplacian operators and operators generated by geometric stable processes will be discussed. To obtain desired conclusions, a new notion of H?lder spaces (depending on the operators) will be introduced, and some probabilistic methods will be shown to overcome technical difficulties. In turn, I will also show how the analytic results can be applied to stochastic analysis.
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