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

transforms associated with Bessel operators

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.