Estimates of Partial Derivatives for Harmonic Functions on the Unit Disc

Let f=P[F] denote the Poisson integral of F in the unit disc D with F absolutely continuous on the unit circle T and F˙∈L^p(T), where F˙(e^it)=ddtF(e^it). We show that for p∈(1,∞), the partial derivatives f_z and f_z¯¯ belong to the holomorphic Hardy space H^p(D). In addition, for p=1 or p=∞, f_z and f_z¯¯∈H^p(D) if and only if H(F˙)∈L^p(T), the Hilbert transform of F˙ and in that case, we have 2izf_z=P[F˙+iH(F˙)]. Our main tools are integral representations of f_z and f_z¯ in terms of F˙ and M. Riesz’s theorem on conjugate functions. This simplifies and extends the recent results of Chen et al. [1] (J. Geom. Anal., 2021) and Zhu [17] (J. Geom. Anal., 2021).