Скачать или смотреть DOCTORAL SUMMER SCHOOL IN CONFORMAL GEOMETRY-RAFE MAZZEO (19 June, 2023)

TOWARD A GEOMETRIC THEORY OF FRACTIONAL LAPLACIANS
Abstract: In this series of lectures I will describe a method that provides,amongst other things, a fairly direct way to understand regularitytheory for solutions of fractional Laplacian equations on bounded domains.To start, I will briefly review the geometric theory of pseudodifferentialoperators (as opposed to the more standard oscillatory integral approach), and then introduce various classes of pseudodifferential operators associated to certain types of degenerate differential operators and describe the analytic properties of these new degenerate pseudodifferential operators.Applications include the analysis of linear and nonlinear problems onasymptotically hyperbolic spaces, the Graham-Zworski approach to GJMS operators, and finally, some new results, obtained jointlywith Gimperlein, about the boundary regularity theory for fractional Laplacians. This provides a new perspective to the comprehensive microlocal theory developed by Grubb.