---
tag: NAX51M6025
kind: statement
role: theorem
name: Hunt's extension of Carleson's theorem
msc: [42A20, 42B25]
uses: [FBH6KG484Z, SASP14MA1Z, G12KXDHPPS, 5NN2YZZR3V, RARTMA0ZEC]
about: [G12KXDHPPS, V6Q24CHJG8]
sources:
  - url: https://en.wikipedia.org/wiki/Carleson%27s_theorem
---

## Statement

Let $1 < p < \infty$.
Let $f \colon \R \to \C$ be a $2\pi$-periodic function with $\int_{-\pi}^{\pi} \abs{f}^p < \infty$, that is, $f \in L^p(\mathbb{T})$ (see [[Lp space|SASP14MA1Z]]).
Let $S_N f(x) = \sum_{k=-N}^{N} \hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $\hat f(k)$ are its [[Fourier coefficients|5NN2YZZR3V]].
Then $S_N f(x) \to f(x)$ as $N \to \infty$ for [[almost every|RARTMA0ZEC]] $x \in \R$.

## Remarks

Hunt proved this in 1968 by extending the method of [[Carleson's theorem|FBH6KG484Z]], which is the case $p = 2$.
The key estimate is that the maximal operator $f \mapsto \sup_N \abs{S_N f}$ is bounded on $L^p(\mathbb{T})$ for $1 < p < \infty$.
The theorem is new only for $1 < p < 2$, since $L^p(\mathbb{T}) \subset L^2(\mathbb{T})$ for $p \ge 2$.
The case $p = 1$ fails: [[Kolmogorov's example|25V2MHPHZ7]] is an integrable function whose Fourier series diverges everywhere.

## Explanations

### L1 human

Carleson showed that Fourier series of square-integrable functions converge at almost every point.
Hunt showed the same for the larger classes $L^p$ with $p > 1$.
The boundary case $p = 1$ really is different, because there the convergence can fail everywhere.

### L2 human

The proof shows that the Carleson maximal operator is bounded on $L^p$ for $1 < p < \infty$.
With the maximal bound, a.e. convergence follows from convergence on the dense class of trigonometric polynomials.
The exact borderline between $L^1$ and the spaces $L^p$ with $p > 1$ for almost everywhere convergence is still studied.