Holomorphic martingales and interpolation between Hardy spaces
The concepts of holomorphic functions and holomorphic martingales introduced by Varopoulos are used to give simple probabilistic proofs of the following interpolation results. If a linear operator \(T\) is bounded both on \(H^ 1(T)\) and on \(H^ \infty(T)\), then it is bounded on \(H^ p(T)\) \((1<p<\infty)\) (P. W. Jones); if it is bounded both on \(H^ 1(T\times T)\) and on \(H^ 2(T\times T)\), then it is bounded on \(H^ p(T\times T)\) \((1<p<2)\) (S. Y. Chang, R. Fefferman) where \(T\) denotes the unit circle in the complex plane; and, moreover, if it is bounded both on \(H^ 1(R_ +^{n+1})\) and on \(L^ \infty(R_ +^{n+1})\), then it is bounded on \(H^ p(R_ +^{n+1})\) \((1<p<\infty)\) (N. M. Riviere, Y. Sagher), too. As the first step these theorems are verified for holomorphic martingale spaces and then they follow easily by a principle of transference.
scientific article; zbMATH DE number 528679
scientific article