Hardy spaces on the complex ball are isomorphic to Hardy spaces on the disc, \(1\leq p<\infty\)
Let B be the unit ball in \({\mathbb{C}}^ n\), \(n>1\) and let D be the unit disk in \({\mathbb{C}}\). Let \(H^ p(B)\) and \(H^ p(D)\) denote the usual Hardy spaces (1\(\leq p\leq\infty )\). The major result of this paper is the fact that as Banach spaces \(H^ p(B)\) and \(H^ p(D)\) are isomorphic if \(1\leq p<\infty\). This is in direct contrast to the polydisc case, where \textit{J. Bourgain} [J. Funct. Anal. 46, 45-57 (1983; Zbl 0492.46043)] has shown that \(H^ 1(D^ m)\) is isomorphic to \(H^ p(D^ n)\) if and only if \(n=m\). Also, it is known that for the ''disc algebra'' case A(B) is not isomorphic to A(D) and \(A(D^ n)\) is not isomorphic to A(D) if \(n>1\). One of the main tools of the proof is the solution of the inner function conjecture for the ball in \({\mathbb{C}}^ n\) [see \textit{A. B. Aleksandrov}, Math. USSR, Sb. 46, 143-159 (1983); translation from Mat. Sb., Nov. Ser. 118(160), 147-163 (1982; Zbl 0503.32001)]. This is used to show that \(H^ p(B)\) contains a complemented copy of \(H^ p(D)\). To show conversely that \(H^ p(D)\) contains a complemented copy of \(H^ p(B)\) the author uses an atomic \(H^ 1\) space and some results of \textit{B. Maurey} [Acta Math. 145, 79-120 (1980; Zbl 0506.46045)] on martingale \(H^ 1\) spaces.
scientific article; zbMATH DE number 3869715