Bochner-Riesz平均及Stein球面平均相关问题研究

本项目研究Bochner-Riesz平均、Stein球面平均算子及其相关算子的若干问题.研究内容主要有以下两部分:在Bochner-Riesz平均算子方面,探寻Bochner-Riesz平均算子在紧流形上依范数收敛速度的K-泛函刻画;在Stein球面平均算子及其相关算子方面,讨论广义球平均算子在齐次Triebel-Lizorkin空间上的依范数收敛及收敛速度,探究迭代球面平均算子(A_t)^N(f)的导算子在H^p(M)上有界的充要条件,研究与S_t^γ(f)相关的分数次波算子的收敛性等若干调和分析问题.本项目研究的问题源于调和分析,同时又与逼近论、偏微分方程和概率论有着紧密联系,其研究结果对上述领域有深刻的理论意义和广泛的应用价值.

This project is focused on the convergence of Bochner-Riesz mean and Stein's spherical average operators, and related topics. The project contains two parts. In part I, we will explore the L^p equivalence relation between K-functional and approximation of Bochner-Riesz multiplier operators on general compact manifolds, under the sharp condition on its index. In part II, the boundedness properties and the norm convergence rate of the averaging operator S_t^γ(f) on Triebel-Lizorkin Spaces are studied. Furthermore, the derivative estimates of the iterated spherical averages (A_t)^N(f) to determine the optimal range of exponents (α,N,p) which ensures H^p(M) boundedness of P(∂/∂x)(A_t)^N(f) are addressed, where P is a general polynomial of degree α. The convergence of fractional wave operators on R^n and general compact manifolds is also considered. Even though the problems considered in this project mainly originate from Harmonic Analysis, they are also closely related to Approximation Theory, Analysis of PDEs, and Probability Theory. The established theoretical results in this project will be of profound significance and widespread application prospects with respect to the above-mentioned fields.