Geodesic
Convergence of Fourier series almost everywhere and in the $L$-metric
Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 527-539 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following theorems are proved. Theorem 1. There exists a constant $C>0$ such that for any function $f\in L(0,2\pi)$ there is a measurable function $F$ for which $|F|=|f|$, and a) $\displaystyle\int_0^{2\pi}\sup_n|S_n(F)(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$, b) $\displaystyle\int_0^{2\pi}\sup_n|{\widetilde{S}}_n(F)(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$, c) $\displaystyle\int_0^{2\pi}|\widetilde{F}(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$, \noindent where $S_n(F)$ is a partial sum of the Fourier series of $F$, $\widetilde S_n(F)$ is a partial sum of the conjugate Fourier series, and $\widetilde F$ is the conjugate function to $F$. \medskip Theorem 2. {\it For any function $f\in L(0,2\pi)$ and $\varepsilon>0$ there exists a measurable function $F$ such that $|F|=|f|$, $\mu\{x\in[0,2\pi):F(x)\ne f(x)\}<\varepsilon$ ($\mu$ is Lebesgue measure), and both the Fourier series of $F$ and its conjugate series converge almost everywhere and in the metric of $L$.} Bibliography: 11 titles. 
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     author = {Sh. V. Kheladze},
     title = {Convergence of {Fourier} series almost everywhere and in the $L$-metric},
     journal = {Sbornik. Mathematics},
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     number = {4},
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}
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Sh. V. Kheladze. Convergence of Fourier series almost everywhere and in the $L$-metric. Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 527-539. http://geodesic.mathdoc.fr/item/SM_1979_35_4_a5/

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