Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	赵国焕，中国科学院数学与系统科学研究院
Inviter:	中科院数学院学术报告会组委
Title:	Non-local operators with low-order singular kernels
Language:	Chinese
Time & Venue:	2023.01.13 11:20-12:00 腾讯会议：671-4985-6447
Abstract:	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. ng)