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  "body": "\n## Statement\n\nLet $1 \u003c p \u003c \\infty$.\nLet $f \\colon \\R \\to \\C$ be a $2\\pi$-periodic function with $\\int_{-\\pi}^{\\pi} \\abs{f}^p \u003c \\infty$, that is, $f \\in L^p(\\mathbb{T})$ (see [[Lp space|SASP14MA1Z]]).\nLet $S_N f(x) = \\sum_{k=-N}^{N} \\hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $\\hat f(k)$ are its [[Fourier coefficients|5NN2YZZR3V]].\nThen $S_N f(x) \\to f(x)$ as $N \\to \\infty$ for [[almost every|RARTMA0ZEC]] $x \\in \\R$.\n\n## Remarks\n\nHunt proved this in 1968 by extending the method of [[Carleson's theorem|FBH6KG484Z]], which is the case $p = 2$.\nThe key estimate is that the maximal operator $f \\mapsto \\sup_N \\abs{S_N f}$ is bounded on $L^p(\\mathbb{T})$ for $1 \u003c p \u003c \\infty$.\nThe theorem is new only for $1 \u003c p \u003c 2$, since $L^p(\\mathbb{T}) \\subset L^2(\\mathbb{T})$ for $p \\ge 2$.\nThe case $p = 1$ fails: [[Kolmogorov's example|25V2MHPHZ7]] is an integrable function whose Fourier series diverges everywhere.\n\n## Explanations\n\n### L1 human\n\nCarleson showed that Fourier series of square-integrable functions converge at almost every point.\nHunt showed the same for the larger classes $L^p$ with $p \u003e 1$.\nThe boundary case $p = 1$ really is different, because there the convergence can fail everywhere.\n\n### L2 human\n\nThe proof shows that the Carleson maximal operator is bounded on $L^p$ for $1 \u003c p \u003c \\infty$.\nWith the maximal bound, a.e. convergence follows from convergence on the dense class of trigonometric polynomials.\nThe exact borderline between $L^1$ and the spaces $L^p$ with $p \u003e 1$ for almost everywhere convergence is still studied.\n",
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  "name": "Hunt's extension of Carleson's theorem",
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