BMO-strong means of Fourier series

Abstract

In the case when E = LD( p < =), we have the p-strong summability, considered for the first time by Hardy and Littlewood [2]. The following statement is due to Marcinkiewicz and Zygmund (see [2]): if f ~ L,(T) ,then the Fourier series and the conjugate series are p-strongly summable with any finite exponent p. Let L N be the Orlicz space (the principal term is N(u) = exp(u/in in u)). In [i], K. I. Oskolkov has proved the LN-Strong summability almost everwhere on T of the Fourier series and of the conjugate series. We denote by L M the Orlicz space with the M-function explu I i. In [I] one has formulated Totik's conjecture, whose equivalent formulation has the following form: if ! ~ L,(T) , then its Fourier series and its conjugate series are LM-strongly snmmable almost everywhere on T.