# Harmonic Analysis ICM Satellite Conference 2018 Porto Alegre Abstract: We show that the classical...

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Harmonic Analysis ICM Satellite Conference 2018

Porto Alegre

Plenary Speakers

The nonlinear Brascamp—Lieb inequality and applications to oscillatory integrals

Jonathan Bennett (University of Birmingham) J.Bennett@bham.ac.uk

Abstract: We discuss a certain nonlinear variant of the classical Brascamp—Lieb inequality, and describe a recent application in the theory of multilinear oscillatory integrals. This is joint work with Stefan Buschenhenke.

Recent advances in Dynamical Sampling

Carlos Cabrelli (Universidad de Buenos Aires) cabrelli@dm.uba.ar

Abstract: In this talk we will describe this novel theory and review some of the more relevant results.

Recent developments in sharp Fourier restriction

Emanuel Carneiro (Instituto de Matemática Pura e Aplicada) carneiro@impa.br

Abstract: This talk will be a brief survey on the problem of finding the sharp forms and classifying the extremizers of some Fourier restriction inequalities. At the end, we aim to present some recent developments for the hyperboloid restriction, related to the Klein-Gordon equation, and for a mixed-nom spherical restriction inequality of L. Vega of 1988.

1

The Kakeya needle problem for rectifiable sets

Marianna Csörnyei (University of Chicago) csornyei@math.uchicago.edu

Abstract: We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is a joint work with Alan Chang.

Some recent progress on harmonic analysis on certain non- doubling spaces

Xuan Thinh Duong (Macquarie University) xuan.duong@mq.edu.au

Abstract: Consider the Schrödinger operator L = ∆ + V where ∆ is the Laplace Beltrami operator on a non-doubling manifold with ends Rm]Rn (with m > n ≥ 3) and the potential V is non-negative. Let T = m(L) be the holomorphic functional calculus of Laplace transform type of L which is a singular integral with rough kernel. Our results have 2 parts: (a) We prove that T is of weak type (1, 1), hence by interpolation is bounded on Lp(Rm]Rn) for (1 < p < ∞) and (b) We study BMOL(Rm]Rn) which is the BMO space associated to L on Rm]Rn and show that m(L) is bounded from L∞ to BMOL which also implies that m(L) is bounded on Lp(Rm]Rn) for (1 < p

Riesz Transforms: Laplace versus Schrödinger

Eleonor Harboure (Univesidad Nacional del Litoral) harbour@santafe-conicet.gov.ar

Abstract: As it is well known, the behaviour of Riesz Transforms is essential to obtain regularity properties for solutions of the equation ∆u = f .

For the Schrödinger operator L = −∆ + V with V a non-negative function sat- isfying a very mild condition, we consider the corresponding Riesz operators. The presence of the potential V has two side effects: on one hand their kernels loose reg- ularity but, on the other hand, their size, in some sense, is much better at infinity.

I will present several results on the behaviour of Schrödinger Riesz Transforms that show their similarities and differences with the standardad Riesz Transforms.

In particular we will review the fundamental work of Shen about continuity on Lebesgue spaces. Also, we will go over on some more recent results: weighted Lp

theory, boundedness on suitable versions of H1, BMO and regularity spaces. In this context, second order Riesz Transforms are not compositions of those of

first order. In particular, to obtain regularity results for that operator will eventually require adding more assumptions on the potential.

Of commutators and Jacobians

Tuomas Hytönen (University of Helsinki) tuomas.hytonen@helsinki.fi

Abstract: The Lp boundedness of commutators [b, T ] = bT − Tb of pointwise multiplication b and singular integral operators T has been well studied for a long time. There are also many results about Lp to Lq boundedness for p < q, but (until recently) almost nothing for p > q. I will supply the missing pieces to present a complete picture of the Lp to Lq boundedness for all p, q ∈ (1,∞), and relate the regime of exponents p > q to the mapping properties of the Jacobian on first order Sobolev spaces.

3

Strichartz estimates for orthonormal systems in Sobolev spaces

Sanghyuk Lee (Seoul National University) shklee@snu.ac.kr

Abstract: This talk concerns the Strichartz estimates for the Schrödinger equation with orthonormal systems of initial data which have extra regularity. The estimates can be thought of a vector valued generalization of the classical Strichartz estimates. We discuss the optimal range of these estimates and connection to the Strichartz estimates for the kinetic transport equation.

On the sharp upper bound related to the weak Muckenhoupt- Wheeden conjecture

Andrei Lerner (Bar-Ilan University) lernera@math.biu.ac.il

Abstract: We show that the upper bound [w]A1 log(e + [w]A1) for the L 1(w) →

L1,∞(w) norm of the Hilbert transform cannot be improved in general. This is joint work with Fedor Nazarov and Sheldy Ombrosi.

The Hilbert Transform and the maximal (Hardy-Littlewood) operator along variable families of non-flat curves

Victor Lie (Purdue University) vlie@purdue.edu

Abstract: In this talk we will investigate the following

Main Problem: Let Γ ≡ Γ(x,y) = (t, γ(x, y, t)) be a variable curve in the plane, where here t ∈ R and (x, y) ∈ R2 while

γ(x,y)(·) := γ(x, y, ·) : R → R

is a “suitable” real function. Under what conditions on the curve Γ ≡ Γ(x,y) - (our main target: minimal

regularity in x and y) - do we have that

4

• the Hilbert transform along curve Γ defined by

HΓf(x, y) := p.v.

∫ R f(x− t, y + γ(x, y, t)) dt

t ,

• the maximal (Hardy-Littlewood) operator along curve Γ defined by

MΓf(x, y) := sup �>0

1

2�

∫ � −� |f(x− t, y + γ(x, y, t))| dt ,

are bounded operators from Lp(R2) to Lp(R2) for 1 < p

Crystal Groups in Harmonic Analysis

Ursula Molter (Universidad de Buenos Aires) umolter@dm.uba.ar

Abstract: In this talk we will look at some properties of spaces invariant under the action of a crystal group and at crystal wavelets or frames. The main tool will be the relation between the action of a crystal group and translations of multiply generated spaces.

The Helicoidal Method

Camil Muscalu (Cornell University) camil@math.cornell.edu

Abstract: The goal of the lecture is to describe some of our recent work with Cristina Benea, on what we called the helicoidal method. One possible way to think of this new, iterative, method, is to view it as a modern and more powerful analogue of Rubio de Francia extrapolation theory. Just as the technique of Rubio de Francia allows one to obtain weighted norm inequalities for the operator in question and its multiple vector valued extensions, so does the helicoidal method allow one to prove sparse domination for the corresponding operator and its multiple vector valued extensions. Using these ideas, in the last few years, we have been able to give complete, positive answers, to a number of natural open questions, that have been circulating for some time. It is also interesting to mention that the fundamental localized estimates that lie at the hart of the method, have nontrivial consequences even in the scalar setting. When applied to the case of the bilinear Hilbert transform, for instance, they imply the known Lp estimates form the Lacey and Thiele theorem directly, without the use of interpolation of trilinear forms, while when applied to the case of the variational Carleson operator, they provide a significant simplification of the proof of Oberlin et al. theorem.

6

Extrapolation for linear and multilinear Muckenhoupt classes

Sheldy Ombrosi (Universidad Nacional del Sur) sheldyombrosi@gmail.com

Abstract: In this talk we present some recent results obtained in a joint work with K. Li and J. M. Martell, about a multivariable Rubio de Francia extrapolation theorem for multilinear Muckenhoupt classes A~p, and also some extensions to more general classes of weights.

To illustrate the power of extrapolation methods we will present some applica- tions of the beforementioned results and some mixed weak-type weighted estimates obtained in a joint work with K. Li and C. Pérez.

Sparse domination and dimensionless estimates for the Riesz vector

Stefanie Petermichl (Université Paul Sabatier) stefanie.petermichl@gmail.com

Abstract: It is known since the 1970s, formulated in the work by Gundy and Varopoulos, that certain classical operators such as the Hilbert or Riesz transforms have stochastic representation using the background noise process and harmonic extensions. On the other hand, their point-wise domination by so-called sparse operators is known since 2015 by Nazarov-Lerner, Lacey, Conde-Rey, independently. All these principles are based on stopping cubes and carry dimensional constants in several parts of the proof. Through a probabilistic argument, a trajectory-wise sparse domination with continuous parameter can be obtained, thus avoiding all occurrences of dimensional constants. We give a new proof of a dimensionless Lp estimate for the Bakry Riesz vector on Riemannian manifolds with bounded geometry - this includes the classical Riesz vector on Euclidean space as well as the Riesz vector on the Gauss space. Our proof has the advant