Carleson measures via BMO (Q2380358)

A finite Borel measure \(\mu\) on the unit disk \(\Delta\) is an \(\alpha\)-Carleson measure if \[ \sup_I\frac{\mu(SI)}{|I|^\alpha}<\infty. \] Here \(I\) is an arc on \(\partial\Delta\), and \(SI\) is the corresponding Carleson square. It is well-known that \(\mu\) is an \(1\)-Carleson square if and only if \(H^p\subset L^p(\mu)\) for all \(0<p<\infty\). The authors define \[ F_\mu^\alpha(t)=\int\!\!\int_{|z|<1-t}\frac{d\mu(z)}{1-|z|^2},\qquad G_\mu^\alpha(t)=\frac{\mu(\{1-t\leq |z|<1\})}{t^\alpha},\qquad 0<t\leq 1, \] and \(\Phi_\mu^\alpha=F_\mu^\alpha+G_\mu^\alpha\), and prove that \[ \sup_t G_\mu^\alpha(t)\approx\mu(\Delta)+\|F_\mu^\alpha\|_*\approx \mu(\Delta)+\|\Phi_\mu^\alpha\|_*, \] where \[ \|f\|_*=\sup_{J\subset (0,1]}\frac{1}{|J|}\int_J|f(x)-f_J|\,dx,\qquad f\in L^1_{\text{loc}}. \] This way, the authors provide characterizations of Carleson measures via uniform bounds for BMO norms of the functions \(F_\mu^\alpha\) and \(\Phi_\mu^\alpha\), which are naturally associated with \(\mu\). They also obtain more quantitative versions of these characterizations involving the restriction of \(\mu\) on non-tangential approach regions.