Hardy spaces for Dunkl–Gegenbauer expansions

We develop the harmonic analysis associated with the Dunkl–Gegenbauer expansions, which is in terms of the system {ϕnλ(eiθ),e−iθϕn−1λ(e−iθ)}, orthogonal with respect to the weight |sinθ|2λ|sinθ|2λ on the circumference S1=∂DS1=∂D, where ϕnλ(eiθ)=Pnλ(cosθ)+i2λn+2λPn−1λ+1(cosθ)sinθ, and Pnλʼs are the Gegenbauer polynomials. This system is connected with the operators Tzf(z)=∂zf+λ[f(z)−f(z¯)]/(z−z¯) and Tz¯f(z)=∂z¯f−λ[f(z)−f(z¯)]/(z−z¯). The theory of the Hardy spaces Hλp(D) for p⩾p0:=2λ/(2λ+1)p⩾p0:=2λ/(2λ+1) is studied, which extends the theory of Muckenhoupt and Stein from the upper half-disk to the whole disk. As a corollary, a remarkable generalization of Rieszʼs theorem is obtained. Under certain sharp estimates on the mean growth of Hλp(D) functions, a variety of inequalities is proved for all p>p0p>p0. The paper concludes with the LpLp–LqLq boundedness and the boundedness on weighted Morrey spaces of the associated Riesz potential Iλδf, by means of two different fractional maximal functions, and also the Hλp(D)–Hλq(D) boundedness of Iλδf for p0