A Littlewood-Paley type inequality (Q1416078)

Let \(n\geq 3\) and \(B\) be the unit ball in \(\mathbb{R}^{n}\) and \(S=\partial B\). The following theorem is proved: Let \(u\) be a harmonic function in the unit ball \(B\) and \(p\in [(n-2)/(n-1),1]\). Then there is a constant \(C=C(p,n)\) such that \[ \sup_{0\leq r<1}\int_{S} | u(r\zeta)| ^{p} \,d\sigma(\zeta)\leq C\left( | u(0)| ^{p}+\int_{B}| \nabla u(x)| ^{p}(1-| x| )^{p} \,dV(x)\right). \]