Some generalizations of the Hardy-Littlewood-Paley theorem
The Hardy-Littlewood-Paley theorem on an estimate for the norm of a function in terms of its Fourier coefficients in an orthonormal system is generalized in three directions. Firstly, an estimate of the \(L^ q\)- norm of a function or the majorant of partial sums of its transposed series in terms of its Fourier coefficients and the \(L^ r\)-norms \((2< q<r\leq\infty)\) of operators of partial sums in an orthonormal system is given. Secondly, several analogous bounds for Orlicz classes \(\varphi(L)\) in the case of orthonormal systems of bounded functions are proved. Thirdly, necessary and sufficient conditions for imbedding into \(\varphi(L)\) of classes of functions \(L^ p(\varepsilon)\) with a given majorant of best approximations in \(L^ p[0;2\pi]\) in the trigonometric system.
scientific article; zbMATH DE number 484258
scientific article