Oscillation and variation for semigroups associated with Bessel operators

Let λ>0λ>0 and △λ:=−d2dx2−2λxddx be the Bessel operator on R+:=(0,∞)R+:=(0,∞). The authors show that the oscillation operator O(P⁎[λ]) and variation operator Vρ(P⁎[λ]) of the Poisson semigroup {Pt[λ]}t>0 associated with ΔλΔλ are both bounded on Lp(R+,dmλ)Lp(R+,dmλ) for p∈(1,∞)p∈(1,∞), BMO(R+,dmλ)BMO(R+,dmλ), from L1(R+,dmλ)L1(R+,dmλ) to L1,∞(R+,dmλ), and from H1(R+,dmλ)H1(R+,dmλ) to L1(R+,dmλ)L1(R+,dmλ), where ρ∈(2,∞)ρ∈(2,∞) and dmλ(x):=x2λdx. As an application, an equivalent characterization of H1(R+,dmλ)H1(R+,dmλ) in terms of Vρ(P⁎[λ]) is also established. All these results hold if {Pt[λ]}t>0 is replaced by the heat semigroup {Wt[λ]}t>0.