The Hardy and Bellman transforms of the spaces \(H^1\) and BMO
Hardy transform associates to a function a trigonometric series in which the \(n\)-th coefficients are means of the corresponding coefficients of the function. Bellman transform is the adjoint to Hardy transform. The author proves that the Hardy transform is bounded on \(H^1(T)\) and unbounded on \(\text{BMO}(T)\), and so the Bellman transform is bounded on \(\text{BMO}(T)\) and unbounded on \(H^1(T)\). This continues the old results about boundedness of both Hardy and Bellman transforms on \(L^p\), \(1<p<\infty\).
scientific article; zbMATH DE number 1262574
scientific article