Carleson's Theorem: Proof, Complements, Variations

A variation norm Carleson theorem

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.

A variation norm Carleson theorem for vector-valued Walsh-Fourier series

We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hytönen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q<r \leq \infty and (r/(q-1))' < p < \infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type t …

Variational estimates for the bilinear iterated Fourier integral

We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.

高维下的变分范数Carleson定理

A variation norm Carleson theorem in higher dimensions

本文将Fefferman的多边形傅里叶部分和极大函数界推广到高维下的强$r$-变分范数界与2-振荡界，并针对嵌套矩形的部分傅里叶和建立了相关变分与振荡界，完善了高维Carleson定理的研究。

Restricted Carleson Variations at Endpoint and Discretized Hilbert Transforms in the Plane

We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to $\{ξ_1 + ξ_2 = 0\}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p \rightarrow L^p$ estimates when $2 < p <\infty$ for a smooth restricted variant of $V_2$ that is defined a priori on Schwartz functions by the formula \begin{eqnarray*} \mathcal{V}^{res}_2 : f \mapsto \sup_{R \in \mathbb{R}_+} ~~\sup_{0 \leq α< R} ~~\left(\sum_{j \in \mathbb{Z}} \left|f*\mathcal{F}^{-1} \left[ \tilde{1}_{[α+ j R, α+ (j+1)R]}\right] \right|^2 \right)^{1/2} \end{eqnarray*} where $…

Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeg…