Wolff type estimates and the \(H^ p\) Corona problem in strictly pseudoconvex domains (Q1842620)

Let \(D = \{\rho < 0\}\) denote a strictly pseudoconvex domain in \(\mathbb{C}^n\) with smooth boundary (at least \(C^2)\). The main result is roughly as follows: If \(\varphi\) is a \(\overline \partial\)-closed \((0,1)\)-form such that (1) a certain measure built up from \(\varphi\) and \(\rho\) belongs to a space of measures between the space of finite measures in \(D\) and the space of Carleson measures (Wolff type estimate) and (2) \(\varphi\) satisfies a boundedness condition, then there is a solution \(u \in L^p(D)\) of \(\overline \partial_b u = \varphi\) for appropriate \(p\), \(1 \leq p \leq \infty\). Two proofs are given. The procedure involves blending and modifying various earlier estimates concerning diverse integral representations. The result is applied to solve the \(H^p\)- Corona problem for two generators.