BMO functions and Balayage of Carleson measures in the Bessel setting

Abstract

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