researchseminars.org

Ciprian Demeter (Indiana University Bloomington)
2021-05-25 15:30 - 16:30 UTC | https://researchseminars.org/talk/HIMharmonicanalysis/1/
Title: Restriction of exponential sums to hypersurfaces (https://researchseminars.org/talk/HIMharmonicanalysis/1/)
by Ciprian Demeter (Indiana University Bloomington) as part of HIM Harmonic Analysis Seminar

Abstract
We discuss moment inequalities for exponential sums with respect to singular measures, whose Fourier decay matches those of curved hypersurfaces. Our emphasis will be on proving estimates that are sharp with respect to the scale parameter $N$, apart from $N^ϵ$ losses. Joint work with Bartosz Langowski.

Ruixiang Zhang (IAS)
2021-05-31 15:30 - 16:30 | https://researchseminars.org/talk/HIMharmonicanalysis/2/
Title: A stationary set method for estimating oscillatory integrals (https://researchseminars.org/talk/HIMharmonicanalysis/2/)
by Ruixiang Zhang (IAS) as part of HIM Harmonic Analysis Seminar

Abstract
Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

Mariusz Mirek
2021-06-21 15:30 - 16:30 | https://researchseminars.org/talk/HIMharmonicanalysis/3/
Title: Pointwise ergodic theorems for bilinear polynomial averages (https://researchseminars.org/talk/HIMharmonicanalysis/3/)
by Mariusz Mirek as part of HIM Harmonic Analysis Seminar

Abstract
We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg and Weiss) bilinear polynomial ergodic averages. This is joint work with Ben Krause and Terry Tao: arXiv:2008.00857. We will also talk about recent progress towards establishing Bergelson's conjecture.

Jonathan Hickman (University of Edinburgh)
2021-06-28 14:00 - 15:00 | https://researchseminars.org/talk/HIMharmonicanalysis/4/
Title: The helical maximal function (https://researchseminars.org/talk/HIMharmonicanalysis/4/)
by Jonathan Hickman (University of Edinburgh) as part of HIM Harmonic Analysis Seminar

Abstract
The circular maximal function is a singular variant of the familiar Hardy--Littlewood maximal function. Rather than take maximal averages over concentric balls, we take maximal averages over concentric circles in the plane. The study of this operator is closely related to certain GMT packing problems for circles, as well as the theory of the Euclidean wave propagator. A celebrated result of Bourgain from the mid 80s showed that the circular maximal function is bounded on $L^p$ if and only if $p > 2$. In this talk I will discuss a higher dimensional variant of Bourgain's theorem, in which the circles are replaced with space curves (such as helices) in $\mathbb{R}^3$. Our main theorem is that the resulting helical maximal operator is bounded on $L^p$ if and only if $p > 3$. The proof combines a number of recently developed Fourier analytic tools, and in particular a variant of the Littlewood--Paley theory for functions frequency supported in a neighbourhood of a cone. Joint work with David Beltran, Shaoming Guo and Andreas Seeger.

Michael Christ (UC Berkeley)
2021-07-05 15:30 - 16:30 | https://researchseminars.org/talk/HIMharmonicanalysis/5/
Title: On quadrilinear implicitly oscillatory integrals (https://researchseminars.org/talk/HIMharmonicanalysis/5/)
by Michael Christ (UC Berkeley) as part of HIM Harmonic Analysis Seminar

Abstract
The title refers to multilinear functionals
\[ \int_B \prod_{j\in J} (f_j\circ\varphi_j)\]
where $B\subset {\mathbb R}^D$ is a ball, $J$ is a finite index set, $\varphi_j:B\to {\mathbb R}^d$ are $C^\omega$ submersions,
$d$ $<$ $D$, and $f_j$ are measurable. The goal is majorization by a product of negative order Sobolev norms of $f_j$,
under appropriate hypotheses on the mappings $\varphi_j$.

Inequalities of this type are closely related to sublevel inequalities
\[ \big|\big\{x\in B: |\sum_{j\in J} a_j(x)\,(g_j\circ\varphi_j)(x)|<\varepsilon\big\}\big| = O(\varepsilon^c),\]
where the coefficients satisfy $a_j\in C^\omega$.

I will state results of this type with $(|J|,D,d) = (4,2,1)$ for the multiplicative inequality and $= (3,2,1)$
for the additive inequality, discuss connections between the two, and indicate some elements of proofs.

Po Lam Yung (Australian National University and the Chinese University of Hong Kong)
2021-07-12 14:00 - 15:00 | https://researchseminars.org/talk/HIMharmonicanalysis/6/
Title: A formula for Sobolev seminorms involving weak L^p (https://researchseminars.org/talk/HIMharmonicanalysis/6/)
by Po Lam Yung (Australian National University and the Chinese University of Hong Kong) as part of HIM Harmonic Analysis Seminar

Abstract
I will discuss some joint work with Haim Brezis and Jean Van Schaftingen, where a new formula was proved for the $W^{1,p}$ seminorm of any compactly supported smooth function on $\mathbb{R}^n$. The formula involves the weak $L^p$ norm of a modified difference quotient on the product space $\mathbb{R}^n \times \mathbb{R}^n$, and was partly inspired by the BBM formula by Bourgain, Brezis and Mironescu regarding fractional Sobolev seminorms. A similar formula for the $L^p$ norm of any $L^p$ function on $\mathbb{R}^n$ has been obtained in a recent paper with Qingsong Gu. The talk will conclude with some applications of this circle of ideas, that remedies the failures of certain critical Gagliardo-Nirenberg type embeddings.

Betsy Stovall (UW Madison)
2021-07-26 14:00 - 15:00 | https://researchseminars.org/talk/HIMharmonicanalysis/7/
Title: Fourier restriction to the sphere is extremizable more often than not (https://researchseminars.org/talk/HIMharmonicanalysis/7/)
by Betsy Stovall (UW Madison) as part of HIM Harmonic Analysis Seminar

Abstract
We will sketch a proof that the $L^p \to L^q$ Fourier extension inequality associated to the $d$-sphere possesses extremizers whenever $p < q < (d+2)p'/d$. This is joint work with Taryn Flock.

Jim Wright (University of Edinburgh)
2021-08-02 15:30 - 16:30 | https://researchseminars.org/talk/HIMharmonicanalysis/8/
Title: Exponential sums and oscillatory integrals: a unified approach (https://researchseminars.org/talk/HIMharmonicanalysis/8/)
by Jim Wright (University of Edinburgh) as part of HIM Harmonic Analysis Seminar

Abstract
In joint work with Gian Maria Dall'Ara, we have a simple argument which is powerful enough to effectively treat oscillatory integrals defined over general locally compact topological fields whose phase is a general polynomial of many variables. Our bounds have an interesting self-improving feature.

Shaoming Guo (UW Madison)
2021-08-09 14:00 - 15:00 | https://researchseminars.org/talk/HIMharmonicanalysis/9/
Title: Some recent progress on the Bochner-Riesz problem (https://researchseminars.org/talk/HIMharmonicanalysis/9/)
by Shaoming Guo (UW Madison) as part of HIM Harmonic Analysis Seminar

Abstract
I will report some recent progress on the Bochner-Riesz conjecture. We observe that recent tools developed to study the Fourier restriction conjecture, including wave packet decompositions, broad-narrow analysis, the polynomial methods, polynomial Wolff axioms, etc., work equally well for the Bochner-Riesz problem. This is joint work with Changkeun Oh, Hong Wang, Shukun Wu and Ruixiang Zhang.
