澳大利亚科学院院士Fedor Sukochev系列学术报告

发布时间:2018年05月11日 作者:唐颖   消息来源:业务办    阅读次数:[]

澳大利亚科学院院士Fedor Sukochev系列学术报告

系列报告题目1: Cwikel estimates revisited

报告人: Fedor Sukochev,University of New South Wales

报告时间:2018年5月28日下午4:00-6:00,数理楼小报告厅

摘要:The so-called Cwikel estimates provide an estimate on the decay the singular values of the operator $M_f g(-i\nabla)$, where $M_f$ denotes the multiplication operator on $L_2(\mathbb{R}^d)$ by the function $f$ and $\nabla$ denotes the gradient, in terms of decay of functions $f$ and $g$. We prove a stronger version of this estimates by showing that under the condition $f\otimes g\in (L_2+L_\infty)(\mathbb{R}^{2d})$ we have

$$\mu^2(M_fg(-i\nabla))\prec\prec 532 \mu^2(f\otimes g),$$

in the sense of Hardy-Littlewood majorisation, where $\mu(M_fg(-i\nabla))$ denotes the sequence of singular values of the operator $M_fg(-i\nabla)$ and $\mu(f\otimes g)$ denotes the decreasing rearrangement of $f\otimes g$. As a corollary of this estimate and the Lorentz-Shimogaki interpolation theorem we obtain Cwikel estimates in any arbitrary symmetric space $E$, which is an interpolation space for the Banach couple $(L_2,L_\infty)$, that is

if $f\otimes g\in E(\mathbb{R}^{2d})$, then $M_f g(-i\nabla)\in E(L_2(\mathbb{R}^d))$ and

$$\|M_f g(-i\nabla)\|_E\leq {\rm const}(E)\, \|f\otimes g\|_E.$$

Furthermore, we show that our assumption on the symmetric function space $E$ is optimal, that is, if we omit the assumption that $E$ is an $(L_2,L_\infty)$-interpolation space, then the corresponding Cwikel estimates fail.

We also provide a version of these estimates for a class of interpolation spaces in the pair $(L_q, L_2)$ with $0<q<2.$

Joint work with G. Levitina and D. Zanin.

报告人简介:Fedor Sukochev是澳大利亚新兰威尔士大学教授,澳大利亚科学院院士,主要研究领域是非交换几何,非交换分析和概率;目前在《Acta Math》、《Invent Math》、《Advance in Math》、《Amer. J. Math》、《Proc. London Maht Society》、《J. London Maht Society》《Trans. AMS》、《JFA》等国际主流数学期刊上发表学术论文200多篇。主持澳大利亚ARC基金10多项,包括ARC Laureate Fellowships和ARC Discovery Outstanding Researcher Award。

系列报告研讨2:Noncommutative maximal inequalities for martingale trasformation

报告人:Fedor Sukochev,University of New South Wales

报告讨论时间:2018年5月30日下午4:00-6:00

摘要:We investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on the noncommutative Gundy decomposition and some recent advance on algebraic atomic decomposition. We also prove corresponding maximal inequalities for fractional Doob operator.

地点: 数理楼547



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