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In this paper we will show that $| \\sup_{j} |S_{n_j}(f)| |_{1,\\infty}\\leq C |f|_{1} \\log\\log (10+\\frac{|f|_{\\infty}}{|f|_1})\\:.$ As a direct consequence we obtain that $S_{n_j}f \\rightarrow f $ a.e. for $f\\in L\\log\\log L\\log\\log\\log L$. 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