Oscillation and variation for Riesz transform associated with Bessel operators

Abstract

Let $\lambda >0$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${\mathbb{R}}_{+}:=(0,\infty )$. We 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 $R_{{\Delta }_{\lambda }}$ associated with ${\Delta }_{\lambda }$ are both bounded on $L^p({\mathbb{R}}_{+},d{m}_{\lambda })$ for $p\in (1,\infty )$, from $L^1({\mathbb{R}}_{+},d{m}_{\lambda })$ to $L^{1,\infty }({\mathbb{R}}_{+},d{m}_{\lambda })$, and from $L^{\infty }({\mathbb{R}}_{+},d{m}_{\lambda })$ to $BMO({\mathbb{R}}_{+},d{m}_{\lambda })$, where $\rho \in (2,\infty )$ and $d{m}_{\lambda }(x):=x^{2\lambda }dx$. As an application, we give the corresponding $L^p$-estimates for $\beta$-jump operators and the number of up-crossing.

20 pages