Monday, 26 June 2023
Sharp inequalities have a rich tradition in harmonic analysis, going back to the epoch-making works of Beckner and Lieb for the sharp Hausdorff-Young and Hardy-Littlewood-Sobolev inequalities. Even though the history of sharp restriction theory is considerably shorter, it is moving at an incredible pace. In this talk, we survey selected highlights from the past decade, describe some of our own contributions, and pose a few open problems which lie at the interface of euclidean harmonic analysis and dispersive PDE. The talk also serves as an introduction to the talk of Giuseppe Negro in this program.
References:
- G. Negro and D. Oliveira e Silva, "Intermittent symmetry breaking and stability of the sharp Agmon-Hörmander estimate on the sphere". Proc. Amer. Math. Soc. 151 (2023), no. 1, 87–99. arXiv:2107.14273.
- E. Carneiro, D. Oliveira e Silva and M. Sousa, "Sharp mixed norm spherical restriction". Adv. Math. 341(2019), 583-608. arXiv:1710.10365.
- D. Oliveira e Silva and R. Quilodrán, "On extremizers for Strichartz estimates for higher order Schrödinger equations". Trans. Amer. Math. Soc. 370 (2018), no. 10, 6871-6907. arXiv:1606.02623.
- D. Oliveira e Silva and R. Quilodrán, "A comparison principle for convolution measures with applications".Math. Proc. Cambridge Philos. Soc. 169 (2020), no. 2, 307-322. arXiv:1804.10463.
- G. Brocchi, D. Oliveira e Silva and R. Quilodrán, "Sharp Strichartz inequalities for fractional and higher order Schrödinger equations". Anal. PDE 13 (2020), no. 2, 477-526. arXiv:1804.11291.
We consider the general Fourier extension estimate on the $d$-dimensional cone in $\mathbb{R}^{1+d}$: \begin{equation*} \lVert\widehat{f\mu}\rVert_{L^q(\mathbb R^{1+d})}\le C_{d, p}\lVert f\rVert_{L^p(d\mu)}.\end{equation*} We prove that it admits \emph{maximizers}, i.e.~functions $f$ that attain the best possible constant $C_{d,p}$. For $p=2$, this estimate coincides with the \emph{Strichartz estimate} for the wave equation; in this case, Foschi found that exponentials $f(\tau, \xi)=\exp(-A\tau +b\cdot \xi + c)$ are the maximizers for $d\in\{2, 3\}$. We further provethat, for arbitrary $d$, these exponentials have a chance of being maximizers if and only if $p=2$. This parallels Christ and Quilodr\'an result that gaussian functions rarely maximize Fourier extension estimates on the paraboloid. Our proof is entirely different, and it is based on the conformal compactification of $\mathbb{R}^{1+d}$ given by the Penrose transform. Joint work with Diogo Oliveira e Silva, Betsy Stovall and James Tautges.
We will discuss the question of existence of maximizers for Fourier restriction onto the parabola $(t, t^2)$ and the cubic monomial $(t, t^3)$ in $\R^2$.
We present new weighted and sparse bounds for Bochner--Riesz means at the critical index \lambda(p)=d(1/2-1/p)-1/2. In particular, for every A_1 weight, we prove an extension of Vargas' weak type (1,1) weighted bounds for some p>1. This is achieved by proving new endpoint sparse bounds which extend those previously obtained by Kesler and Lacey in two dimensions and by Conde-Alonso, Culiuc, Di Plinio and Ou for p=1. A key novelty in our approach is an improved decomposition of the Bochner-Riesz multipliers which satisfies both orthogonality and kernel localization properties. In particular, this allows us to prove off-diagonal versions of certain vector-valued L^p estimates featuring in the works of Tao and Seeger on weak-type estimates.
This is joint work with Joris Roos and Andreas Seeger.
Tuesday, 27 June 2023
I will discuss a class of maximal operators that arise from averaging functions over thin neighborhoods of curves in the plane. Examples of such operators are the Kakeya maximal function and the Wolff and Bourgain circular maximal functions. To understand the behavior of these operators, we need to study the possible intersection patterns for collections of curves in the plane: how often can these curves intersect, how often can they be tangent, and how often can they be tangent to higher order?
The Stein-Tomas inequality dates back to 1975 and is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. The goal of this talk is three-fold: we present a recent proof of sharp non-endpoint Stein-Tomas in low space dimensions whenever the exponent is an even integer; we present a variational refinement and withdraw some consequences to restriction theory; and we discuss how to improve the Stein-Tomasinequality in the presence of block radial symmetries.
References:
- V. Kovač and D. Oliveira e Silva, "A variational restriction theorem". Arch. Math. (Basel) 117 (2021), 65-78. arXiv:1809.09611.
- G. Negro, D. Oliveira e Silva and C. Thiele, "When does exp(-τ) maximize Fourier extension for a conic section?". Preprint at arXiv:2209.03916.
- C. Bilz, "Large sets without Fourier restriction theorems." Trans. Amer. Math. Soc. (2022). https://doi.org/10.1090/tran/8714. arXiv: 2001.10016.
- V. Kovač, "Fourier restriction implies maximal and variational Fourier restriction." J. Funct. Anal. 277 (2019), issue 10, 3355-3372.
We will discuss the question of existence of maximizers for Fourier restriction onto the parabola $(t, t^2)$ and the cubic monomial $(t, t^3)$ in $\R^2$.
The Brascamp-Lieb inequality has played a significant role in a number of developments in harmonic analysis in recent years, including Fourier restriction and Kakeya estimates. One of the pillars in the theory of the Brascamp-Lieb inequality is Lieb's theorem on the existence of gaussian near-maximizers. This talk is concerned with the inverse version of the Brascamp-Lieb inequality. This includes a discussion of the analogue of Lieb's theorem due to Barthe and Wolff, and a certain regularized version of the inverse Brascamp-Lieb inequality. The talk is mostly based on joint work with Shohei Nakamura, but I also plan to discuss subsequent work with Shohei Nakamura and Hiroshi Tsuji.
Wednesday, 28 June 2023
In this talk, we will describe some new ways of characterising Sobolev and BV functions, using sizes of superlevel sets of suitable difference quotients. They provide remedy in certain cases where some critical Gagliardo-Nirenberg interpolation inequalities fail, and lead us to investigate real interpolations of certain fractional Besov spaces. Some connections will be drawn to earlier work by Bourgain, Brezis and Mironescu, and an image processing application will be given. Joint work with Haim Brezis, Jean Van Schaftingen, Qingsong Gu, Andreas Seeger, Brian Street and Oscar Dominguez.
References:
- D. Oliveira e Silva and R. Quilodrán, "Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres". J. Funct. Anal. 280 (2021), no. 7, 108825. arXiv:1909.10230.
- D. Oliveira e Silva and R. Quilodrán, "Smoothness of solutions of a convolution equation of restricted-type on the sphere". Forum Math. Sigma 9 (2021), Paper No. e12, 40 pp. arXiv:1909.10220.
- D. Foschi and D. Oliveira e Silva, "Some recent progress on sharp Fourier restriction theory". Anal. Math.43 (2017), no. 2, 241-265. arXiv:1701.06895.
Any set containing a sphere centered at every point cannot have 0 Lebesgue measure. This is a consequence of the L^p boundedness of the spherical maximal function. On the other hand, there are sets of 0 Lebesgue measure which contain a large family of spheres, which may be considered as Kakeya/Nikodym sets for spheres. This talk will be a survey of such sets and their Hausdorff dimension, and mapping properties of related maximal functions.
In this talk, we prove the optimal eigenfunction bound for the Hermite operator in every dimension bigger than and equal to 3. Our result is based on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere.
Friday, 30 June 2023
The ancient Pythagorean theorem gives a formula for computing the Euclidean distance between two points. It is simply astounding that a concept so simple and classical has continued to fascinate mathematicians over the ages, and remains a tantalizing source of open problems to this day.
Given a set E, its distance set consists of numbers representing distances between points of E. If E is large, how large is its distance set? How does the structure of a set influence the structure of distances in the set? Such questions play an important role in many areas of mathematics and beyond. The talk will survey a few research problems associated with Euclidean distances between points and discuss a few landmark results in some of them.
The presentation is intended to be an introduction to a vibrant research area; no advanced mathematical background will be assumed.
References:
- R. Mandel and D. Oliveira e Silva, "The Stein-Tomas inequality under the effect of symmetries". Preprint at arXiv:2106.08255. To appear in Journal d'Analyse Mathématique.
- R. Mandel and D. Oliveira e Silva, "Block-radial symmetry breaking for ground states of biharmonic NLS". Preprint at arXiv:2306.03720.
Any set containing a sphere centered at every point cannot have 0 Lebesgue measure. This is a consequence of the L^p boundedness of the spherical maximal function. On the other hand, there are sets of 0 Lebesgue measure which contain a large family of spheres, which may be considered as Kakeya/Nikodym sets for spheres. This talk will be a survey of such sets and their Hausdorff dimension, and mapping properties of related maximal functions.
We consider the \emph{focusing cubic wave equation} $\partial_t^2 u - \Delta u = u^3$ on $\mathbb{R}^{1+3}$. By means of the Penrose transform, we construct a two-parameter family of solutions. Depending on the values of the parameters, these solutions either scatter to linear ones, blow-up in finite time, or exhibit a new type of unstable behaviour that acts as a threshold between the other two. We characterize this threshold behaviour precisely, and relate it to an existing conjecture of Bizoń--Zenginoğlu. Joint work with Thomas Duyckaerts.
Monday, 03 July 2023
In this series of talks I will discuss numerical integration on Riemannian manifolds, and perhaps more general spaces. The attention will be focused on estimates of the worst-case error for functions in Sobolev spaces, that is we look for point configurations (and weights) that give small errors for the integration of all functions on some Sobolev space at the same time. In this first talk, after a quick introduction to the problem and the context, I will discuss results based directly on the possibility to partition the manifold into regions of equal volume and small diameter.
We will survey the role of improving estimates and sparse bounds in the setting of discrete Harmonic Analysis. The first talk will highlight the role of the continuous setting. We will then move to the discrete setting, starting with a illustrative example of the square integers. Especially important here is the interplay between the continuous and periodic versions of these averages. We then move to other examples, including the discrete sphere, and the primes. A highlight of the series of talks will be a discussion of a new variant of the Vinogradov Three Primes theorem in the setting of the Gaussian Integers.
Reference: https://bookstore.ams.org/browse?simplesearch=Krause%20discrete
Two classic questions - the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem - both focus on distance. Here, distance can be thought of as a simple two point configuration. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as three point configurations. In this pair of talks I will go through some of the history of such point configuration questions and end with some recent results, for instance on triangles.
Tuesday, 04 July 2023
In the second talk I will show to what degree a design (that is a quadrature rule which is exact on all eigenfunctions of the Laplace-Beltrami operator, up to a certain eigenvalue) can be used to approximate the integrals of functions in Sobolev spaces. Then I will talk about the existence of optimal designs on manifolds: in the case of the sphere this was the conjecture of Korevaar and Meyers, proved in 2013 by Bondarenko, Radchenko and Viazovska.
We will survey the role of improving estimates and sparse bounds in the setting of discrete Harmonic Analysis. The first talk will highlight the role of the continuous setting. We will then move to the discrete setting, starting with a illustrative example of the square integers. Especially important here is the interplay between the continuous and periodic versions of these averages. We then move to other examples, including the discrete sphere, and the primes. A highlight of the series of talks will be a discussion of a new variant of the Vinogradov Three Primes theorem in the setting of the Gaussian Integers.
Reference: https://bookstore.ams.org/browse?simplesearch=Krause%20discrete
Two classic questions - the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem - both focus on distance. Here, distance can be thought of as a simple two point configuration. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as three point configurations. In this pair of talks I will go through some of the history of such point configuration questions and end with some recent results, for instance on triangles.
Wednesday, 05 July 2023
In the last talk, I will discuss certain results that are needed to prove the estimates in the previous talks. Among others, Cassels inequality for manifolds, almost positive kernels on manifolds, and partitions of manifolds into regions of equal volume and small diameter.
We will survey the role of improving estimates and sparse bounds in the setting of discrete Harmonic Analysis. The first talk will highlight the role of the continuous setting. We will then move to the discrete setting, starting with a illustrative example of the square integers. Especially important here is the interplay between the continuous and periodic versions of these averages. We then move to other examples, including the discrete sphere, and the primes. A highlight of the series of talks will be a discussion of a new variant of the Vinogradov Three Primes theorem in the setting of the Gaussian Integers.
Reference: https://bookstore.ams.org/browse?simplesearch=Krause%20discrete
Several years ago, in collaboration with J. Bourgain, M. Mirek, and E.M. Stein, we initiated systematic studies of dimension-free estimates for maximal functions of discrete averaging operators on $\mathbb{Z}^d$. During my talk I would like to overview existing results, state interesting open questions, and present directions for potential progress.
Dimension-free estimates for continuous maximal functions defined over symmetric convex sets in $\mathbb{R}^d$ has been an important theme in harmonic analysis for about 40 years. Yet, it turns out that discrete dimension-free estimates are much harder to achieve and there is no general result that can hold for all symmetric convex sets. Indeed, for a certain family of ellipsoids the $\ell^p(\mathbb{Z}^d}$ norms of the corresponding discrete maximal functions grows to infinity with the dimension. On the other hand the comparison between discerete and continuous maximal functions is possible for suprema restricted to large radii. Moreover, for specific families of sets: cubes, balls, and spheres, one can prove dimension-free estimates for their discrete maximal functions. In the case of maximal functions for the discrete spheres the question is closely related to Waring's problem in analytic number theory.
Curvature adds exciting number theoretic interactions to discrete versions of continuous operators in harmonic analysis. Exploring the ways spheres can interact in this setting is a rich area of study with many unanswered questions and developing tools. We will begin by counting triangles in Euclidean space, and connect this to a web of both analysis and number theory.
Thursday, 06 July 2023
We will survey the role of improving estimates and sparse bounds in the setting of discrete Harmonic Analysis. The first talk will highlight the role of the continuous setting. We will then move to the discrete setting, starting with a illustrative example of the square integers. Especially important here is the interplay between the continuous and periodic versions of these averages. We then move to other examples, including the discrete sphere, and the primes. A highlight of the series of talks will be a discussion of a new variant of the Vinogradov Three Primes theorem in the setting of the Gaussian Integers.
Reference: https://bookstore.ams.org/browse?simplesearch=Krause%20discrete
In this talk I will discuss how certain Fourier optimization problems arise in connection to two number theory problems: (i) bounds for the least quadratic non-residue and (ii) bounds for the least prime in a given arithmetic progression.
In the early 1970s, Montgomery proved some interesting estimates about Dirichlet polynomials, which are motivated by applications to the Riemann zeta function. This problem is related to the restriction problem for the sphere raised by Stein at about the same time, and Montgomery's estimates use techniques similar to techniques used on the restriction problem during the 1970s, such as the TT* method. Since Montgomery's work, there has been hardly any progress. In this talk, we will describe Montgomery's work and discuss some of the issues that make further progress difficult.
Friday, 07 July 2023
We will survey the role of improving estimates and sparse bounds in the setting of discrete Harmonic Analysis. The first talk will highlight the role of the continuous setting. We will then move to the discrete setting, starting with a illustrative example of the square integers. Especially important here is the interplay between the continuous and periodic versions of these averages. We then move to other examples, including the discrete sphere, and the primes. A highlight of the series of talks will be a discussion of a new variant of the Vinogradov Three Primes theorem in the setting of the Gaussian Integers.
Reference: https://bookstore.ams.org/browse?simplesearch=Krause%20discrete