Carleson inequalities in classes of derivatives of harmonic Bergman functions with \(0<p\leq 1\) (Q1293790)

The author studies conditions on a \(\sigma\)-finite positive Borel measure \(\mu\), for the inequality \[ \Biggl(\int|D^\alpha u|^q d\mu\Biggr)^{1/q}\leq C\Biggl(\int|D^m_y u|^p y^r dV\Biggr)^{1/p} \] to be true (and in particularly \(\int|u|^p d\mu\leq C\int|D_y u|^p y^r dV\), where \(p\leq 1\), \(u\) belongs to the Bergman space \(b^p\)). Part 3 contains a necessary and sufficient condition for \(\mu\) on the upper half space to satisfy the considered inequality. In Part 4, the author shows that when \(p\leq 1\) the Bergman norm is comparable to several derivative norms. The paper is very interesting and contains several very important lemmas and theorems with long and full proofs.