The space BMO and strong means of Fourier series (Q807911)

Let \(S_ k(f,x)\) be the partial sum of the Fourier (trigonometric) series of the function f and \(\tilde S_ k(f,x)\) the partial sum of the conjugate series. Let \(\Delta^ n_ k[k/n,(k+1)/n)\quad (k=0,1,...,n- 1)\) and define \(\kappa_{\Delta^ n_ k}(t)=1\) if \(t\in \Delta^ n_ k\) and \(\kappa_{\Delta^ n_ k}(t)=0\) if \(t\in [0,1)\setminus \Delta^ n_ k.\) The author proves that the operator \[ (Bf)(x)=\sup_{0\leq n<\infty}\| \sum^{2^ n-1}_{k=0}S_ k(f,x)\kappa_{\Delta_ k^{2^ n}}(t)\|_{BMO[0,1)} \] is of weak ype (1,1) and similarly for the operator corresponding to \(\tilde S_ k(f,x)\).