Averages of simplex Hilbert transforms

We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calder\'on commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two and three and give a conditional result valid in all dimensions.Comment: 15 pages; final version to appear in Proc. AMS; fixed typos, reformulated main result

Singular integrals and maximal operators related to Carleson's theorem and curves in the plane

In this thesis we study several different operators that are related to Carleson's theorem and curves in the plane. An interesting open problem in harmonic analysis is the study of analogues of Carleson's operator that feature integration along curves. In that context it is natural to ask whether the established methods of time-frequency analysis carry over to an anisotropic setting. We answer that question and also provide certain partial bounds for the Carleson operator along monomial curves using entirely different methods. Another line of results in this thesis concerns maximal operators and Hilbert transforms along variable curves in the plane. These are related to Carleson-type operators via a partial Fourier transform in the second variable. A central motivation for studying these operators stems from Zygmund's conjecture on differentiation along Lipschitz vector fields. One of our results can be understood as proving a curved variant of this conjecture

Spherical maximal functions and fractal dimensions of dilation sets

For the spherical mean operators $\mathcal{A}_t$ in $\mathbb{R}^d$, $d\ge 2$, we consider the maximal functions $M_Ef =\sup_{t\in E} |\mathcal{A}_t f|$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.Comment: 30 pages, 3 figures. Slightly improved Theorem 1.

Discrete analogues of maximally modulated singular integrals of Stein-Wainger type: ℓp\ell^p bounds for p>1p>1

It is proved that certain discrete analogues of maximally modulated singular integrals of Stein-Wainger type are bounded on $\ell^p(\mathbb{Z}^n)$ for all $p\in (1,\infty)$. This extends earlier work of the authors concerning the case $p=2$. Some open problems for further investigation are briefly discussed.Comment: 15 pages; changed title, final version to appear in JF

Discrete analogues of maximally modulated singular integrals of Stein-Wainger type

Consider the maximal operator $$\mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\Big|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e(\lambda |y|^{2d}) K(y)\Big|,\quad (x\in\mathbb{Z}^n),$$ where $d$ is a positive integer, $K$ a Calder\'on-Zygmund kernel and $n\ge 1$. This is a discrete analogue of a real-variable operator studied by Stein and Wainger. The nonlinearity of the phase introduces a variety of new difficulties that are not present in the real-variable setting. We prove $\ell^2(\mathbb{Z}^n)$-bounds for $\mathscr{C}$, answering a question posed by Lillian Pierce.Comment: 24 pages. Additional details inserted; final version to appear in JEM

A new proof of an inequality of Bourgain

The purpose of this short note is to demonstrate how some techniques from additive combinatorics recently developed by Peluse and Peluse-Prendiville can be applied to give an alternative proof for a trilinear smoothing inequality originally due to Bourgain.Comment: 8 page

Spherical maximal operators on Heisenberg groups: Restricted dilation sets

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_Ef$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these maximal operators; the results depend on various notions of dimension of $E$.Comment: 27 pages, 1 figur

Endpoint sparse domination for classes of multiplier transformations

We prove endpoint results for sparse domination of translation invariant multiscale operators. The results are formulated in terms of dilation invariant classes of Fourier multipliers based on natural localized $M^{p\to q}$ norms which express appropriate endpoint regularity hypotheses. The applications include new and optimal sparse bounds for classical oscillatory multipliers and multi-scale versions of radial bump multipliers.Comment: 55 pages, 1 figur

A polynomial Roth theorem on the real line

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.Comment: 18 pages, to appear in Transactions of the AM