Bounds on the Maximal functions for spectral multipliers
数学专题报告
报告题目(Title)：Bounds on the Maximal functions for spectral multipliers
报告人(Speaker)：陈鹏（中山大学）
地点(Place)：腾讯会议254359052
时间(Time)：12月1日15:30-16:30
邀请人(Inviter)：薛庆营 教授
报告摘要
Let $(X,d,\mu)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies Gaussian upper bound.Given H\"ormander type spectral multipliers $m_i,1\leq i\leq N,$ with uniform estimates, we prove an optimal $\sqrt{\log(1+N)}$ bound in $L^p$ for the maximal function $\sup_{1\leq i\leq N}|m_i(L)f|$ by making use of Doob transform and some techniques as in Grafakos-Honz\'ik-Seeger \cite{GHS2006} to use the ${\rm exp}(L^2)$ estimate by Chang-Wilson-Wolff \cite{CWW1985}. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function %$ M_{m,L}$ $M_{m,L}f(x) = \sup_{t>0} |m(tL)f(x)|$ is bounded on $L^p(X)$. The applications include Scattering operators, Schr\"odinger operators with inverse square potential, Dirichlet Laplacian with Dirichlet boundary, Bessel operators and Laplace-Beltrami operators.
主讲人简介
陈鹏，中山大学数学学院教授。主要从事调和分析的研究。主要学术成果发表在Math. Ann.，Adv. Math.，J. Math. Pures Appl.，Int. Math. Res. Not. IMRN，Trans. Amer. Math. Soc.，J. Funct. Anal.等国际重要数学期刊上。2021年入选国家高层次人才计划青年项目，先后主持多项国家自然科学基金面上项目、青年基金项目和广东省自然科学基金面上项目。
