Oscillation and variation for Riesz transform associated with Bessel operators

TL;DR

This paper proves boundedness properties of oscillation and variation operators related to the Riesz transform for Bessel operators on positive real line, extending harmonic analysis tools in this setting.

Contribution

It establishes boundedness of oscillation and variation operators for Riesz transforms associated with Bessel operators on various function spaces, a novel extension in harmonic analysis.

Findings

Boundedness on L^p spaces for p in (1, ∞)

Weak-type (1,1) boundedness

Boundedness from L^∞ to BMO

Abstract

Let $\lambda>0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ 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 $L^p(\mathbb R_+, dm_{\lambda})$ 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 $BMO(\mathbb{R}_{+},dm_{\lambda})$, where $\rho\in (2,\infty)$ and $dm_{\lambda}(x):=x^{2\lambda}dx$. As an application, we give the corresponding $L^p$-estimates for $\beta$-jump operators and the number of up-crossing.