Rearrangement estimates of the area integrals (Q2773273)

The author considers a general second order differential operator \(L\) and \(v\in L^{1}_{\text{loc}}({\mathbb{R}}_+^{n+1})\) for which \(Lv\) is a positive Borel measure \(\mu_{Lv}\) on \(\mathbb{R}_+^{n+1}\). The area integral is defined by \((S_{\alpha}v)(x)=\int_{\Gamma_{\alpha}(x)}t^{1-n} d\mu_{L_{v}} (y,t).\) The main result is a relation between \(((S_{\alpha}v)^{\delta})^{*}_{\omega}\), the non-increasing rearrangement with respect to \(\omega\in A_{\infty}\), and the nontangential maximal function \(((N_{\beta}v)^{\delta})^{*}_{\omega}\) (\(0<\delta\leq 1\)).