Boundedness of Differential Transforms for Poisson Semigroups Generated by Bessel Operators

Abstract In this paper we analyze the convergence of the following type of series: TN \, \, f (x) =₉=₍䃑^N₂ vⱼ (P₀_₉+₁ \, \, f (x) - P₀_₉ \, \, f (x) ), x R_+, where \{ Pₓ \}ₓ₀ is the Poisson semigroup associated with the Bessel operator _: =-d² dx²-2 xd dx, with λ being a positive constant, N= (N₁, N₂) Z² with N₁ N₂, \vⱼ\₉ ₙ is a bounded real sequence and \aⱼ\₉ ₙ 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.