Estimates of the first partial derivatives of (α,β)-harmonic functions on the unit disc

Suppose α,β∈R∖Z− such that α+β>−1 and 1≤p≤∞. Let u=Pα,β[f] be an (α,β)-harmonic function on D, the unit disc of C, with the boundary f being absolutely continuous and f˙∈Lp(0,2π), where f˙(eiθ):=ddθf(eiθ). In this paper, we investigate the membership of the partial derivatives ∂zu and ∂z‾u in the space HGp(D), the generalized Hardy space. We prove, if α+β>0, then both ∂zu and ∂z‾u are in HGp(D). For α+β<0, we show if ∂zu or ∂z‾u∈HG1(D) then u=0 or u is a polyharmonic function.