BMO-strong means of Fourier series
Let f be an integrable function on the unit circle \({\mathbb{T}}\), let \(S_ k(f)\), resp. \(S_ k(\tilde f)\) denote the kth partial sum of the Fourier series of f, resp. its conjugate series, with respect to the trigonometric system. Consider the functions \(s_ n(f,x,t)\), \(s_ n(f,x,t)=s_ k(f,x)\) for \(k/n\leq t<(k+1)/n,\) \((k=0,1,...,n-1)\) on the interval \(<0,1)\). The Fourier series of f is called E-strongly summable at x if \(\lim_{n\to \infty}\| s_ n(f,x,t)-f(x)\|_ E=0.\) Here E is a Banach space of measurable functions \(<0,1)\). \(E=L^ p\) gives the classical notion of p-strong (or (H,p)-) summability. According to Marcinkiewicz-Zygmund theorem, both the series of f and its conjugate are p-strongly summable almost everywhere, for any \(f\in L^ 1({\mathbb{T}})\) and \(0<p<\infty\). The author proves a nice generalization of this result: the p-strong summability may be replaced by BMO-strong summability. Since BMO embeds into \(L_ M\), the Orlicz space with M-function \(M(u)=e^{| u|}-1\), this result also implies \(L_ M\)-strong summability, which is the Totikov conjecture. The crucial step in the proof is showing that, for \(E=BMO\), the corresponding majorization operators are of weak type (1,1). The strong summability with respect to the Walsh system on the group \(\prod^{\infty}_{1}{\mathbb{Z}}_ 2\) is also discussed.
scientific article; zbMATH DE number 4169349