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Oscillations and variations for semigroups and Riesz transforms associated with Bessel operators
发布时间: 2017-09-05     12:16   【返回上一页】 发布人:杨东勇


 北京师范大学数学科学学院

 

调和分析学术报告

 

 

 

报告题目Oscillations and variations for semigroups and Riesz

transforms associated with Bessel operators

 


报告人:杨东勇(厦门大学)

 

 

时间地点:20179816:0017:00, 后主楼1220

 

 

邀请人: 杨大春

 

 

摘要: Consider the space $/rrp=(0,/infty)$ equipped with the Euclidean distance and the measure $dm_/lz(x)=x^{2/lz}dx$, where $/lz /in (0,/infty )$ is a fixed constant and $dx$ is the Lebesgue measure. Consider the Bessel operator $/Delta_/lz=-/frac{d^2}{dx^2}-/frac{2/lz}{x}/frac{d}{dx}$ on $/rrp$.

 

In this talk, we do the following:

 (i) show that the oscillation operator ${/mathcal O(P^{[/lambda]}_/ast)}$ and variation operator

${/mathcal V}_/rho(P^{[/lambda]}_/ast)$ of the Poisson semigroup $/{P^{[/lambda]}_t/}_{t>0}$ associated with $/Delta_/lambda$ are both bounded on $L^p(/mathbb R_+, dm_/lambda)$ for $p/in(1, /infty)$, $BMO({{/mathbb R}_+},dm_/lambda)$, from $L^1({{/mathbb R}_+},dm_/lambda)$ to $L^{1,/,/infty}({{/mathbb R}_+},dm_/lambda)$, and from $H^1({{/mathbb R}_+},dm_/lambda)$ to $L^1({{/mathbb R}_+},dm_/lambda)$, where  $/rho/in(2, /infty)$.

As an application, an equivalent characterization of

$H^1({{/mathbb R}_+},dm_/lambda)$ in terms of ${/mathcal V}_/rho(P^{[/lambda]}_/ast)$ is also established.

All these results hold if $/{P^{[/lambda]}_t/}_{t>0}$ is replaced by the heat semigroup $/{W^{[/lambda]}_t/}_{t>0}$.

 

  (ii) show that the oscillation operator $/mathcal{O}(R_{/Delta_{/lambda},/ast})$ and variation operator $/mathcal{V}_{/rho}(R_{/Delta_{/lambda},/ast})$ of the Riesz transform $R_{/Delta_{/lambda}}$ are both bounded on $L^p(/mathbb R_+, dm_{/lambda})$ for $p/in(1,/,/infty)$, from $L^1(/mathbb{R}_{+},dm_{/lambda})$ to $L^{1,/,/infty}(/mathbb{R}_{+},dm_{/lambda})$, and from $L^{/infty}(/mathbb{R}_{+},dm_{/lambda})$ to $BMO(/mathbb{R}_{+},dm_{/lambda})$.  As an application, we give the corresponding $L^p$-estimates for $/beta$-jump operators and the number of up-crossing

 

This is joint work with Jing Zhang and Huoxiong Wu.