BMO functions and balayage of Carleson measures in the Bessel setting

Abstract

By BMOₒ(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 BMOₒ(ℝ) with bounded support as those ones that can be written as a sum of a bounded function on (0, ∞) plus the balayage of a Carleson measure on (0, ∞)×(0, ∞) with respect to the Poisson semigroup associated with the Bessel operator Bₗ := –x⁻ˡ d dx x^{2ₗ} d dx x⁻ˡ, λ > 0. This result can be seen as an extension to Bessel setting of a classical result due to Carleson.