Hardy spaces for Dunkl-Gegenbauer expansions

摘要

We develop the harmonic analysis associated with the Dunkl-Gegenbauer expansions, which is in terms of the system {φ_n ~λ(e~(iθ)), e~(-iθ)φ_(n-1) ~λ(e~(-iθ))}, orthogonal with respect to the weight |sinθ|~(2λ) on the circumference S~1 = ?D, where φ_n ~λ(e~(iθ))=P_n ~λ(cosθ)+i2λ+2λP_(n-1) ~(λ+1)(cosθ)sinθ, and Pnλ's are the Gegenbauer polynomials. This system is connected with the operators T_zf(z)=?zf+λ[f(z)-f(z)]/(z-z) and T_zf(z)=?zf-λ[f(z)-f(z)]/(z-z). The theory of the Hardy spaces H_λ p(D) for p≥p_0:=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 Riesz's theorem is obtained. Under certain sharp estimates on the mean growth of H_λ ~p(D) functions, a variety of inequalities is proved for all p>p_0. The paper concludes with the L~p-L~q 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 H_λ ~p(D)-H_λ ~q(D) boundedness of I_λ ~δf for p_0<(2λ+1)/δ and q~(-1)=p~(-1)-δ/(2λ+1).