Boundedness of Differential Transforms for Poisson Semigroups Generated by Bessel Operators

In this paper we analyze the convergence of the following type of series: TN \,\,f(x)=∑j=N₁N₂ vⱼ( P_aⱼ₊₁ \,\,f(x)- P_aⱼ \,\,f(x)), x∈ R₊, where \ Pₜ >0 is the Poisson semigroup associated with the Bessel operator Δ_λ:=-d² dx²-2λ xd dx, with λ being a positive constant, N=(N₁, N₂)∈ Z² with N₁ < N₂, ⱼ∈ Z is a bounded real sequence and ⱼ∈ Z is an increasing real sequence. Our analysis will consist in the boundedness, in Lᵖ(R₊) and in BMO(R₊), of the operators TN and its maximal operator T^*\,\,f(x)= N TN \,\,f(x). It is also shown that the local size of the maximal differential transform operators is the same with the order of a singular integral for functions f having local support.