Continuity of the density of the area integral in Hardy spaces (Q1611281)

Let \(u\) be a harmonic function on \(\mathbb R^{n+1}_{+}\). The area integral density \[ D_{\phi}u(\theta,a)=\int y\phi_{y}(\theta -x) \Delta(u-a)^{+}dxdy \] associated to \(u\) and the quantity \[ D^{\ast}_{\phi}u(\theta)=\sup\limits_{a\in \mathbb R} D_{\phi}u (\theta,a) \] were introduced by \textit{R. F. Gundy} [Harmonic Analysis, Conf. in Honor A. Zygmund, Chicago 1981, 138--149 (1983)]. \textit{R. F. Gundy} and \textit{M. L. Silverstein} [Ann. Inst. Fourier 35, 215--229 (1985; Zbl 0544.31012)] established that the \(L^p\)-norm of \(D^{\ast}_{\phi}(u)\) is equivalent to the \(H^p\)-norm of \(u\) for all \(0<p<+\infty\). This equivalence was improved by \textit{R. Banuelos} and \textit{Ch. N. Moore} [Ann. Inst. Fourier 41, 137--171 (1991; Zbl 0727.42016)]. It is proved in the present paper that, for all \(p>0\), there exists \(C_{p}>0\) such that, for all harmonic functions \(u,v\) and all \(a\in \mathbb R\), \[ \left\| \sup\limits_{h\in H_{\delta}} \left| D_{\phi}u(h,.,a)- D_{\phi}v(h,.,a)\right|\right\|_{p}\leq C_{p}\left(\left\| u\right\|_{H^p}+\left\| v\right\|_{H^p}\right)^{1/2} \left\| u-v\right\|_{H^p}^{1/2}, \] where \[ D_{\phi}u (h, \theta,a)=\int y\phi_{y}(\theta-x) h(x,y) \Delta(u-a)^{+}dxdy \] and \(h\in H_{\delta}\), which is a class of regularized versions of characteristic functions of Lipschitz sets. A good-\(\lambda\) inequality of exponential type on \(D_{\phi}u (h,\theta,a)\) is derived.