BMO functions and balayage of Carleson measures in the Bessel setting

Abstract

By \(BMO_{\text {o}}(\mathbb {R})\) we denote the space consisting of all those odd and bounded mean oscillation functions on \(\mathbb {R}\). In this paper we characterize the functions in \(BMO_{\text {o}}(\mathbb {R})\) with bounded support as those ones that can be written as a sum of a bounded function on \((0,\infty )\) plus the balayage of a Carleson measure on \((0,\infty )\times (0,\infty )\) with respect to the Poisson semigroup associated with the Bessel operator $$\begin{aligned} B_\lambda :=-x^{-\lambda }\frac{d}{dx}x^{2\lambda }\frac{d}{dx}x^{-\lambda },\quad \lambda >0. \end{aligned}$$ This result can be seen as an extension to Bessel setting of a classical result due to Carleson.