报告人: 洪桂祥（武汉大学）

日  期: 2019年11月29日

时  间: 上午10:30

地  点: 理科楼 LA107

摘  要: In this talk, I shall talk about my recently finished joint work with Simeng Wang and Xumin Wang, which is the first progress on the pointwise convergence of noncommutative Fourier series, solving an open problem since Junge-Xu's remarkable ergodic maximal inequality in noncommutative analysis. Going back harmonic analysis on Euclidean space, one of our results suggests a new class of maximal inequalities which is quite interesting but challenging and deserves to be investigated. For more information, see the following abstract of the paper: This paper is devoted to the study of convergence of Fourier series for nonabelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. Based on this framework, this work studies the refined counterpart of pointwise convergence of these Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 pointwise, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence for all 1 < p < \infty. In a similar fashion, we also obtain results for a large subclass of groups (as well as discrete quantum groups) with the Haagerup property and weak amenability. We also consider the analogues of Fejer means and Bochner-Riesz means in the noncommutative setting. Our results in particular apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain as a byproduct the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies. As an ingredient, our proof also provides a refined version of Junge-Le Merdy-Xu’s square function estimates H_p(M) ≃ L_p(M) when p → 1.

报告人简介：洪桂祥，武汉大学数学与统计学院教授。本科毕业于南昌航空大学数学与信息科学学院，2007年考入北京师范大学攻读硕士，师从调和分析专家丁勇教授。2009年在国家公派项目的支持下，赴法国弗朗什-孔泰大学，跟随非交换鞅论奠基人之一许全华教授攻读博士，研究方向为“非交换调和分析及相关领域”，并于2012年6月获得博士学位。2012年9月，通过严格地筛选与激烈竞争（在全球范围内，从81位候选人中挑选3位），获得了Severo-Ochoa杰出项目的支持，在西班牙国家科学院数学科学研究所从事了3年的博士后研究。自2015年9月，洪桂祥加盟武汉大学数学与统计学院，至今在Communications in Mathematical Physics, Analysis and PDE, Journal of Functional Analysis，Revista Matematica Iberoamericana和International Mathematics Research Notices等国际重要数学期刊上发表了多篇论文。

学院联系人: 黄辉斥、蒋报捷

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