# Joris Roos

Van Vleck Visiting Assistant Professor

Department of Mathematics

University of Wisconsin-Madison

480 Lincoln Drive

Madison, WI 53706, USA

Office: VV 513

Email: jroos (at) math (dot) wisc (dot) edu

## Teaching

August 2018: Student workshop on Automated Theorem Proving and Artificial Intelligence (Leysin, Switzerland)

Fall 2018: Reading Seminar on Vapnik-Chervonenkis dimension and the independence property
## Research

### Research interests

- Harmonic analysis, in particular multilinear singular integrals, oscillatory integrals, singular and maximal Radon transforms, variants of Carleson's theorem, time-frequency analysis, decoupling inequalities
- Connections of harmonic analysis to related fields such as combinatorics, number theory and dispersive PDE

### Publications, preprints, theses

*A maximal function for families of Hilbert transforms along homogeneous curves (with S. Guo, A. Seeger, P.-L. Yung)*

Abstract
pdf
arXiv
arXiv:1902.00096, 2019, submitted.
Let $H^{(u)}$ be the Hilbert transform along the parabola $(t, ut^2)$ where
$u\in \mathbb R$. For a set $U$ of positive numbers consider the maximal
function $\mathcal{H}^U \!f= \sup\{|H^{(u)}\! f|: u\in U\}$. We obtain an
(essentially) optimal result for the $L^p$ operator norm of $\mathcal{H}^U$
when $2<p<\infty$. The results are proved for families of Hilbert transforms
along more general nonflat homogeneous curves.

*Averages of simplex Hilbert transforms (with P. Durcik)*

Abstract
pdf
arXiv
arXiv:1812.11701, 2018, submitted.
We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calderón commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two and three and give a conditional result in four and higher dimensions.

*Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem (with S. Guo, P.-L. Yung)*

Abstract
pdf
arXiv
arXiv:1710.10988, 2017, to appear in Analysis&PDE
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 Schr\"odinger-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 Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain--Guth iteration and the Bourgain--Demeter $\ell^2$ decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions $n\ge 4$.

*Bounds for anisotropic Carleson operators*

Abstract
pdf
arXiv
Journal
arXiv:1710.10962, 2017, J. Fourier Anal. Appl.
We prove weak $(2,2)$ bounds for maximally modulated anisotropically
homogeneous smooth multipliers on $\mathbb{R}^n$. These can be understood as
generalizing the classical one-dimensional Carleson operator. For the proof we
extend the time-frequency method by Lacey and Thiele to the anisotropic setting.
We also discuss a related open problem concerning Carleson operators along
monomial curves.

*A polynomial Roth theorem on the real line (with P. Durcik, S. Guo)*

Abstract
pdf
arXiv
Journal
Trans. Amer. Math. Soc. 371, no. 10, 6973-6993, 2019.
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.

*Maximal operators and Hilbert transforms along variable non-flat homogeneous curves (with S. Guo, J. Hickman, V. Lie)*

Abstract
pdf
arXiv
Journal
Proc. London Math. Soc. 115, no. 1, 177-219, 2017.
We prove that the maximal operator associated with variable homogeneous
planar curves $(t, u t^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive,
is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that
$u:\mathbb{R}^2 \to \mathbb{R}$ is a Lipschitz function. Furthermore, we prove
that the Hilbert transform associated with $(t, ut^{\alpha})_{t\in
\mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for
each $p>1$, under the assumption that $u:\mathbb{R}^2\to \mathbb{R}$ is a
measurable function and is constant in the second variable. Our proofs rely on
stationary phase methods, $TT^*$ arguments, local smoothing estimates and a
pointwise estimate for taking averages along curves.

*Polynomial Carleson operators along monomial curves in the plane (with S. Guo, L. B. Pierce, P.-L. Yung)*

Abstract
pdf
arXiv
Journal
J. Geom. Anal. 27, no. 4, 2977-3012, 2017.
We prove $L^p$ bounds for partial polynomial Carleson operators along
monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial
consisting of a single monomial. These operators are "partial" in the sense
that we consider linearizing stopping-time functions that depend on only one of
the two ambient variables. A motivation for studying these partial operators is
the curious feature that, despite their apparent limitations, for certain
combinations of curve and phase, $L^2$ bounds for partial operators along
curves imply the full strength of the $L^2$ bound for a one-dimensional
Carleson operator, and for a quadratic Carleson operator. Our methods, which
can at present only treat certain combinations of curves and phases, in some
cases adapt a $TT^*$ method to treat phases involving fractional monomials, and
in other cases use a known vector-valued variant of the Carleson-Hunt theorem.

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

Abstract
pdf
ULB
Doctoral dissertation, University of Bonn, 2017.
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.

*Inner-outer factorization of analytic matrix-valued functions*

Abstract
pdf
Master's thesis, University of Bonn, 2014.

**Local conferences**

Madison Lectures in Fourier Analysis 2019

AMS Sectional Meeting in Madison, September 14-15, 2019