BMO functions and Balayage of Carleson measures in the Bessel setting

TL;DR

This paper characterizes bounded support functions in the BMO space associated with the Bessel operator as sums of bounded functions and balayages of Carleson measures, extending classical results to the Bessel setting.

Contribution

It extends classical Carleson measure characterizations to the Bessel setting, specifically for BMO functions with bounded support.

Findings

Characterization of BMO_o(R) functions with bounded support as sums involving balayage of Carleson measures.

Extension of classical Carleson measure results to the Bessel operator context.

Provides a new perspective on BMO functions in the Bessel setting.

Abstract

By $BMO_o(R)$ we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in $BMO_o(R)$ 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 $B_\lambda =-x^{-\lambda }Dx^{2\lambda }Dx^{-\lambda}$, $\lambda >0$. This result can be seen as an extension to Bessel setting of a classical result due to Carleson.