Oscillation and variation for semigroups associated with Bessel operators

TL;DR

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

Contribution

It establishes boundedness of oscillation and variation operators for Bessel-related semigroups on various function spaces, providing new characterizations of Hardy spaces in this context.

Findings

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

Boundedness from L^1 to weak-L^1 and from H^1 to L^1

Characterization of H^1 space via variation operators

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 ${\mathcal O(P^{[\lambda]}_\ast)}$ 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 $\rho\in(2, \infty)$ and $dm_\lambda(x):=x^{2\lambda}\,dx$. 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 $\{W^{[\lambda]}_t\}_{t>0}$. }