On the behaviour of the partial sums of trigonometric Fourier series

In this paper we give a detailed exposition of Carleson's method of estimating partial sums of trigonometric Fourier series of functions that belong to the class$L(\ln^{+}L)^{1+\delta}$, $\delta>0$. We also improve slightly Carleson's relevant result by giving, instead of an estimate almost everywhere of the rate of growth of the partial sums of a Fourier series, an integral estimate of a majorant of the partial sums.