The area integral and its density for BMO and VMO functions (Q1320526)

Let \(f\in L_ 1(\mathbb{R}^ \nu)\) and let \(u\) be its harmonic extension to \(\mathbb{R}^{\nu+1}_ +\). The conic area integral of \(f\) at \(\theta\in\mathbb{R}^ \nu\) is defined by \[ A_ a^ 2(\theta)= \int_{\Gamma_ a(\theta)} y^{1-\nu}|\nabla u(x,y)|^ 2 dx dy, \] where \(\Gamma_ a(\theta)\) is the cone with vertex at \(\theta\) and aperture \(a\). The Littlewood-Paley square function of \(f\) is given by \[ g_ *^ 2(\theta)= \int_{\mathbb{R}_ +^{\nu+1}} y p_ \theta(x,y)|\nabla u(x,y)|^ 2 dx dy, \] where \(p_ \theta\) is the Poisson kernel on the half space. It is known that \(A_ a\) is bounded on \(L_ p\), \(1< p<\infty\), and \(g_ *\) is bounded on \(L_ p\), \(2\leq p<\infty\), and both are bounded on BMO. In the paper under review these results are generalized in several ways. It is shown that \(A_ a^ 2\) and \(g_ *^ 2\), together with certain of their modifications, are in VMO if \(f\) is in VMO (VMO is a closure in BMO of the space of uniformly continuous functions). The results are further extended to the maximal density of the area integral. Some results on the functional \[ D_ a^ r(\theta)= \int_{\Gamma_ a(\theta)} y^{1-\nu}\Delta| u- r|(dz) \] in BMO and VMO are obtained as well.