Estimates of Partial Derivatives for Harmonic Functions on the Unit Disc

Abstract

Let \(f = P[F]\) denote the Poisson integral of F in the unit disc \({{\mathbb {D}}}\) with F absolutely continuous on the unit circle \({{\mathbb {T}}}\) and \(\dot{F}\in L^p({{\mathbb {T}}})\) , where \(\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})\) . We show that for \(p\in (1,\infty )\) , the partial derivatives \(f_z\) and \(\overline{f_{\bar{z}}}\) belong to the holomorphic Hardy space \(H^p({{\mathbb {D}}})\) . In addition, for \(p=1\) or \(p=\infty \) , \(f_z\) and \(\overline{f_{\bar{z}}}\in H^p({{\mathbb {D}}})\) if and only if \(H(\dot{F})\in L^p({{\mathbb {T}}})\) , the Hilbert transform of \(\dot{F}\) and in that case, we have \(2izf_z=P[\dot{F}+iH(\dot{F})]\) . Our main tools are integral representations of \(f_z\) and \(f_{\overline{z}}\) in terms of \(\dot{F}\) 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).