Calderón–Zygmund Operators Associated to Ultraspherical Expansions

We define the higher order Riesz transforms and the Littlewood–Paley g-function associated to the differential operator Lλf(θ) = –f′′(θ)–2λ cot θ f′(θ) + λ2f(θ). We prove that these operators are Calderón–Zygmund operators in the homogeneous type space ((0, π), (sin t)2λdt). Consequently, Lp weighted, H1 – L1 and L∞ – BMO inequalities are obtained.