Oscillation and variation for Riesz transform in setting of Bessel operators on H1 and BMO

Let λ > 0, and let the Bessel operator $${\Delta _\lambda }: = - {{{{\rm{d}}^2}} \over {{\rm{d}}{{{x}}^2}}} - {{2\lambda } \over x}{{\rm{d}} \over {{\rm{d}}x}}$$ defined on ℝ+:= (0, ∞). We show that the oscillation and ρ-variation operators of the Riesz transform $${R_{{\Delta _\lambda }}}$$ associated with Δλ are bounded on BMO(ℝ+, dmλ), where ρ > 2 and dmλ = x2λdx. Moreover, we construct a $${(1,\infty)_{{\Delta _\lambda }}}$$ -atom as a counterexample to show that the oscillation and ρ-variation operators of $${R_{{\Delta _\lambda }}}$$ are not bounded from H1 (ℝ+, dmλ) to L1 (ℝ+, dmλ). Finally, we prove that the oscillation and the ρ-variation operators for the smooth truncations associated with Bessel operators $${{\tilde R}_{{\Delta _\lambda }}}$$ are bounded from H1 (ℝ+, dmλ) to L1 (ℝ+, dmλ).