Estimates of solutions of the \(H^p\) and \(BMOA\) corona problem
For a bounded domain \(D\) in \(\mathbb C^n\), the classical corona problem may be stated as follows: Suppose \(g_1,\dots{},g_m\) are holomorphic functions on \(D\) satisfying \(0<\delta \leq |g(z)|\leq 1, z\in D\), where \(|g|^2 =\sum |g_j|^2\). Then there exist bounded holomorphic functions \(u_1,\dots{},u_m\) on \(D\) such that \(\sum g_ju_j = 1\). When \(n=1\), this is true for all finitely connected domains. However, when \(n>1\), the corona problem is known to fail, even for smoothly bounded pseudoconvex domains [see \textit{N. Sibony}, Ann. Math., II. Ser. 126, 675-682 (1987; Zbl 0636.32007)]. The question is still unsettled in strictly pseudoconvex domains, even in the ball. In the main result of the paper, the authors prove the following \(H^p\) version of the corona problem. Let \(D\) be a strictly pseudoconvex domain in \(\mathbb C^n\) with \(C^3\) boundary, assume that \(g_1,\dots{},g_m \in H^{\infty}\) satisfy \(0<\delta<|g|\leq 1\), and let \(p<\infty\). For any \(\varphi\in H^p\) there are \(u_j\in H^p\) such that \(\sum g_ju_j =\varphi\) and \(\|u\|_{H^p}\leq C_{\delta}\|\varphi\|_{H^p}\), where \(C_\delta\leq C(\log(1/\delta))^{r/2}\big/\delta^{1+r}\) if \(r=\min(n,m-1)\neq 1\), and where \(C_{\delta}\leq C\log(1/\delta)\big/\delta^2\) if \(r=\min(n,m-1)=1\), and \(C\) is independent of \(m\), but depends on \(p\). If \(p=\infty\), the proof yields a solution \(u\) in \(H^{\infty}\cdot BMO\). The result for \(p\leq 2\) was previously proved by the first author [Trans. Am. Math. Soc. 342, No. 1, 241-255 (1994; Zbl 0801.32001) and Bull. Sci. Math., II. Ser. 118, No. 3, 287-306 (1994; Zbl 0810.32002)]. The proofs also lead to new results about the Taylor spectrum of analytic Toeplitz operators on \(H^p\) and \(BMOA\).
strictly pseudoconvex domain in \(\mathbb{C}^n\)
scientific article; zbMATH DE number 1406572