[Paper Review] A variation norm Carleson theorem

Richard Oberlin, Andreas Seeger+3 more|arXiv (Cornell University)|Oct 8, 2009

Advanced Harmonic Analysis Research18 references4 citations

TL;DR

This paper establishes $L^p$ bounds for the $r$-variation of the partial sum operators in Fourier analysis, proving that for $p > \max\{r', 2\}$, the $r$-variation of the Fourier partial sums is controlled in $L^p$-norm. The key result strengthens the Carleson-Hunt theorem by quantifying the rate of pointwise convergence via variation norms, which are sharper than the standard maximal function estimates.

ABSTRACT

We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

Motivation & Objective

• To strengthen the Carleson-Hunt theorem by replacing the maximal function control with $r$-variation norms for Fourier partial sums.
• To establish sharp $L^p$ estimates for the $r$-variation of the partial sum operators $S_n f$ on the torus $\mathbb{T}$, for $p > \max\{r', 2\}$.
• To extend the result to the Fourier integral operator on $\mathbb{R}$ via transference, proving boundedness from $L^p$ to $L^p(V^r)$.
• To demonstrate applications in ergodic theory, particularly to Wiener-Wintner type theorems, using the new variational estimates.

Proposed method

• Define the $r$-variation norm $\|a\|_{V^r} = \sup_K \sup_{n_0 < \cdots < n_K} \left( \sum_{\ell=1}^K |a_{n_\ell} - a_{n_{\ell-1}}|^r \right)^{1/r}$ for sequences.
• Apply the variation norm to the sequence of partial Fourier sums $S_n f(x)$, defining $\mathcal{V}^r S[f](x)$ as the $V^r$-norm of $\{S_n f(x)\}_{n \in \mathbb{N}_0}$.
• Prove $L^p$ boundedness of the operator $f \mapsto \mathcal{V}^r S[f]$ for $p > \max\{r', 2\}$, with operator norm depending on $p$ and $r$.
• Use transference techniques to lift the result from the circle $\mathbb{T}$ to the real line $\mathbb{R}$, proving boundedness of the integral operator $\mathcal{S}[f](\xi,x) = \int_{-\infty}^\xi \widehat{f}(\eta) e^{2\pi i \eta x} d\eta$ from $L^p$ to $L^p(V^r)$.
• Apply the main estimate to Wiener-Wintner-type ergodic theorems by showing that variation norm control implies existence of limits for oscillatory integrals.
• Use Fourier expansion of a mollifier $\phi$ and averaging over $\eta$ to reduce the variation norm estimate to the main theorem via $L^p$ bounds on oscillatory integrals.

Experimental results

Research questions

• RQ1 Can the pointwise a.e. convergence of Fourier series, guaranteed by the Carleson-Hunt theorem, be quantified via $r$-variation norms for $r > 2$?
• RQ2 What is the sharp range of $p$ for which the $r$-variation of the partial Fourier sum operators is bounded in $L^p$?
• RQ3 Can the variation norm estimate be extended from the circle to the real line via transference, and what are the corresponding $L^p$ bounds?
• RQ4 How can the $r$-variation estimate be applied to prove convergence in Wiener-Wintner type theorems in ergodic theory?
• RQ5 Is the endpoint Lorentz space estimate $L^{r',\infty}$ sharp for the $r$-variation operator at $p = r'$?

Key findings

• For $p > \max\{r', 2\}$, the $r$-variation of the Fourier partial sum operators satisfies $\| \mathcal{V}^r S[f] \|_{L^p(\mathbb{T})} \leq C_{p,r} \|f\|_{L^p(\mathbb{T})}$, establishing a quantitative refinement of the Carleson-Hunt theorem.
• At the endpoint $p = r'$, the $r$-variation operator maps $L^{r',1}(\mathbb{T})$ boundedly into $L^{r',\infty}(\mathbb{T})$, and this Lorentz space estimate is sharp.
• The $L^p(V^r)$ boundedness of the partial Fourier integral operator $\mathcal{S}$ on $\mathbb{R}$ holds for $r' < p < \infty$, with the same sharp range as on the torus.
• The main estimate implies convergence results in ergodic theory: for $p > \max\{r', 2\}$, the variation norm control of oscillatory integrals implies existence of limits a.e. for Wiener-Wintner type operators.
• The proof relies on decomposing the variation norm via Fourier expansion of a mollifier and applying the main $L^p$ estimate to each frequency component, with the $L^1$-norm of the Fourier transform of the mollifier ensuring summability.