Harmonic AnalysisICM Satellite Conference 2018

Porto Alegre

Plenary Speakers

The nonlinear Brascamp—Lieb inequality and applications tooscillatory integrals

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

Abstract: We discuss a certain nonlinear variant of the classical Brascamp—Liebinequality, and describe a recent application in the theory of multilinear oscillatoryintegrals. 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 themore relevant results.

Recent developments in sharp Fourier restriction

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

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

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The Kakeya needle problem for rectifiable sets

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

Abstract: We show that the classical results about rotating a line segment inarbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold ifwe replace the line segment by an arbitrary rectifiable set. This is a joint work withAlan 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 Schrodinger operator L = ∆ + V where ∆ is the LaplaceBeltrami 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 functionalcalculus of Laplace transform type of L which is a singular integral with roughkernel. Our results have 2 parts: (a) We prove that T is of weak type (1, 1), henceby interpolation is bounded on Lp(Rm]Rn) for (1 < p < ∞) and (b) We studyBMOL(Rm]Rn) which is the BMO space associated to L on Rm]Rn and show thatm(L) is bounded from L∞ to BMOL which also implies that m(L) is bounded onLp(Rm]Rn) for (1 < p <∞).

This is joint work with The Anh Bui, Peng Chen, Ji Li, Liang Song and LixinYan.

On L2 × L2 → L1 estimates

Loukas Grafakos (University of Missouri)grafakosl@missouri.edu

Abstract: We discuss how to obtain sharp, and in some cases optimal, L2(Rn)×L2(Rn) → L1(Rn) estimates for a variety of singular and maximal operators. Ourmethods are based on decompositions of functions on R2n via product-type waveletsand elementary combinatorics. As an application, we improve bilinear results con-cerning Bochner-Riesz means and spherical maximal functions.

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Riesz Transforms: Laplace versus Schrodinger

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

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

For the Schrodinger operator L = −∆ + V with V a non-negative function sat-isfying a very mild condition, we consider the corresponding Riesz operators. Thepresence 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 Schrodinger Riesz Transformsthat show their similarities and differences with the standardad Riesz Transforms.

In particular we will review the fundamental work of Shen about continuity onLebesgue 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 eventuallyrequire adding more assumptions on the potential.

Of commutators and Jacobians

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

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

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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 Schrodinger equationwith orthonormal systems of initial data which have extra regularity. The estimatescan be thought of a vector valued generalization of the classical Strichartz estimates.We discuss the optimal range of these estimates and connection to the Strichartzestimates 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 L1(w) →L1,∞(w) norm of the Hilbert transform cannot be improved in general. This is jointwork 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, wherehere 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

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• the Hilbert transform along curve Γ defined by

HΓf(x, y) := p.v.

∫Rf(x− t, y + γ(x, y, t))

dt

t,

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

MΓf(x, y) := supε>0

1

2ε

∫ ε

−ε|f(x− t, y + γ(x, y, t))| dt ,

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

We will insist on the history and motivation for this problem from

• the PDE perspective - the study of constant coefficient parabolic differentialoperators and

• the Harmonic Analysis perspective - connections with singular maximal opera-tors and the conjecture of A. Zygmund on maximal integrals along (Lipschitz)vector fields.

If the time allows we will very briefly outline some of the methods we used in ob-taining our most recent results.

The effect of disorder and irregularities on solutions to bound-ary value problems and spectra of differential operators

Svitlana Mayboroda (University of Minnesota)svitlana@math.umn.edu

Abstract: This talk will address the impact of disorder or irregularities in theambient medium on the behavior of stationary solutions to elliptic partial differentialequations and on spatial distribution of eigenfunctions, as well as the profound andsomewhat surprising connections between these two topics which have been revealedin the past few years.

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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 theaction of a crystal group and at crystal wavelets or frames. The main tool will be therelation between the action of a crystal group and translations of multiply generatedspaces.

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 withCristina Benea, on what we called the helicoidal method. One possible way to thinkof this new, iterative, method, is to view it as a modern and more powerful analogueof Rubio de Francia extrapolation theory. Just as the technique of Rubio de Franciaallows one to obtain weighted norm inequalities for the operator in question and itsmultiple vector valued extensions, so does the helicoidal method allow one to provesparse domination for the corresponding operator and its multiple vector valuedextensions. Using these ideas, in the last few years, we have been able to givecomplete, positive answers, to a number of natural open questions, that have beencirculating for some time. It is also interesting to mention that the fundamentallocalized estimates that lie at the hart of the method, have nontrivial consequenceseven 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 theoremdirectly, without the use of interpolation of trilinear forms, while when applied to thecase of the variational Carleson operator, they provide a significant simplification ofthe proof of Oberlin et al. theorem.

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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 workwith K. Li and J. M. Martell, about a multivariable Rubio de Francia extrapolationtheorem for multilinear Muckenhoupt classes A~p, and also some extensions to moregeneral 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 estimatesobtained in a joint work with K. Li and C. Perez.

Sparse domination and dimensionless estimates for the Rieszvector

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

Abstract: It is known since the 1970s, formulated in the work by Gundy andVaropoulos, that certain classical operators such as the Hilbert or Riesz transformshave stochastic representation using the background noise process and harmonicextensions. On the other hand, their point-wise domination by so-called sparseoperators is known since 2015 by Nazarov-Lerner, Lacey, Conde-Rey, independently.All these principles are based on stopping cubes and carry dimensional constants inseveral parts of the proof. Through a probabilistic argument, a trajectory-wise sparsedomination with continuous parameter can be obtained, thus avoiding all occurrencesof dimensional constants. We give a new proof of a dimensionless Lp estimate for theBakry Riesz vector on Riemannian manifolds with bounded geometry - this includesthe classical Riesz vector on Euclidean space as well as the Riesz vector on theGauss space. Our proof has the advantage that it gives new dimensionless, sharpLp estimates in the weighted setting. (The needed background in probability andgeometry will be given in this talk)

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On the maximal directional Hilbert transform

Malabika Pramanik (University of British Columbia)malabika@math.ubc.ca

Abstract: We prove that the maximal directional Hilbert transform is unboundedon all Lebesgue spaces if and only if the underlying set of drections is infinite. Thisis in stark contrast with directional maximal operators. Quantitative lower boundson the operator norm and their sharpness will also be discussed. This is joint workwith Izabella Laba and Alessandro Marinelli.

On the polynomial Wolff axioms

Keith Rogers (Instituto de Ciencias Matematicas)keith.rogers@icmat.es

Abstract: We will consider unit length δ-tubes, with δ-separated directions, and theKakeya conjecture, which measures the extent to which they can be compressed bypositioning them strategically. This can be reformulated as a bound on the numberof tubes in terms of the measure of any set E ⊂ Rn that contains them. We will seethat such a bound holds when the sets E are obliged to be semialgebraic of boundedcomplexity. This is joint work with Nets Katz.

The Haar system on spaces of Sobolev type

Andreas Seeger (University of Wisconsin - Madison)seeger@math.wisc.edu

Abstract: Let X be a Sobolev space, or more generally, a Triebel-Lizorkin orBesov space. Let Hd be the Haar basis in Rd, in a suitable enumeration, and assumethat Hd ⊂ X. I will report on the current state of the following questions and somequantitative versions of them:

(i) For which spaces X is Hd a basis?

(ii) For which spaces X is Hd an unconditional basis?

In recent work with Gustavo Garrigos and Tino Ullrich we consider all the end-point cases.

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On Compactness of Commutators of Multiplication and Bi-linear Pseudodifferential Operators and a New Subspace ofBMO

Rodolfo Torres (University of Kansas)torres@ku.edu

Abstract: It is known that the compactness of the commutators of pointwise multi-plication with bilinear homogeneous Calderon-Zygmund operators acting on productof Lebesgue spaces is characterized by the multiplying function being in the spaceCMO. This space is the closure in BMO of its subspace of smooth functions withcompact support. It is shown in this work that for bilinear Calderon-Zygmund op-erators arising from smooth (inhomogeneous) bilinear Fourier multipliers or bilinearpseudodifferential operators, one can actually consider multiplying functions in anew subspace of BMO larger than CMO. This is joint work with Qingying Xue.

Two weight norm inequalities for singular and fractional in-tegral operators in Rn

Ignacio Uriarte-Tuero (Michigan State University)ignacio@math.msu.edu

Abstract: I will report on recent advances on the topic, related to proofs of T1 typetheorems in the two weight setting for Calderon-Zygmund singular and fractionalintegral operators, with side conditions, and related counterexamples. Mostly jointwork with Eric Sawyer and Chun-Yen Shen.

Product Hardy space theory on spaces of homogeneous typevia orthonormal wavelet bases

Lesley A. Ward (University of South Australia)Lesley.Ward@unisa.edu.au

Abstract: I will discuss recent joint work with Yongsheng Han, Ji Li and CristinaPereyra. We establish the theory of product Hardy spaces defined on product spaces

of homogeneous type X = X1×X2. Here each factor (Xi, di, µi), i = 1, 2, is a space

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of homogeneous type in the sense of Coifman and Weiss, in other words, a set Xi

equipped with a quasi-metric di and a Borel regular doubling measure µi. We makeextensive use of the orthonormal wavelet expansion of Auscher and Hytonen. Ourresults hold for a range of p determined by the geometric constants of the spaces Xi

and by the Holder continuity parameters of the wavelets. No additional assumptionsare made on the quasi-metrics di, nor on the measures µi.

We show that the wavelet expansion converges for suitable test functions anddistributions. We develop the product Littlewood–Paley theory for appropriate dis-crete and continuous square functions defined in terms of the wavelets. We define the

Hardy spaces Hp(X) and the Carleson measure spaces CMOp(X), including the spe-

cial case BMO(X) = CMO1(X), as well as VMO(X). We prove the duality relations

H1(X)∗ = BMO(X) and VMO(X)∗ = H1(X). We establish the Calderon–Zygmund

decomposition for Hp(X), deduce interpolation theorems, and develop an atomic

decomposition for Hp(X).

It follows as a consequence of the atomic decomposition that the Hardy spacesHp(X)are independent of the particular choice of the basis of Auscher–Hytonen wavelets.The atomic decomposition has also been used by Chen, Li and Ward in establishing

a connection, via averaging, between dyadic and continuous instances of BMO(X),and by Kairema, Li, Pereyra and Ward in proving dyadic structure theorems for

H1(X) and BMO(X).

Variational norm estimates for some oscillatory integrals re-lated to Carleson’s operator

Po-Lam Yung (The Chinese University of Hong Kong)plyung@math.cuhk.edu.hk

Abstract: Carleson’s operator is a useful device in the study of pointwise conver-gence of Fourier series of a general L2 function. Stein and Wainger studied a variantof Carleson’s operator, where a linear phase function is replaced by a polynomialphase. In this talk, we study a variant of the result of Stein and Wainger, where themaximal operator is replaced by its variational norm counterpart. Connections willbe made to square function estimates and local smoothing estimates of the linearSchrodinger equation. This is joint work with Shaoming Guo and Joris Roos.

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Invited Speakers

Boundedness of the Hilbert transform and unconditionalityof wavelet bases

Jorge Antezana (Universidad Nacional de La Plata)jaantezana@yahoo.com.ar

Abstract: Let ψ be a Cr-wavelet with compact support associated to a multireso-lution analysis. Given 1 < p < ∞, it is well known that the corresponding waveletbasis ψjk is unconditional in Lp(u) if and only if u is in the Ap class. On theother hand, in rearrangement invariant spaces (w.r.t. the Lebesgue measure), thewavelet basis is unconditional provided the Boyd indices are “good”. In both cases,the unconditionality of the wavelet basis is related to a condition that is equivalentto the boundedness of the Hilbert transform. In this talk we will discuss some resultsthat attempt to clarify this relationship a bit more.

Energy minimization on the sphere

Dmitriy Bilyk (University of Minnesota)dbilyk@math.umn.edu

Abstract: We explore measures and discrete point distributions which minimizevarious energies on the sphere. Unlike classical examples, such as the Riesz energy,most of the potentials that we consider are minimized when points are orthogonal,thus minimizing energy imposes orthogonal-like structures (bases, frames etc). Inparticular, we consider the so-called p-frame energy: in the well-known case p=2 theminimizers are precisely tight frames (or, more generally, isotropic measures) on thespheres, however, other values of p¿0 are much less understood (except even integers).We provide minimizers in some specific cases, as well as a number of candidates for

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other cases based on extensive numerical computations. We also present partialprogress on the conjecture of Fejes Toth, which states that the pairwise sum of acuteangles between n lines is maximized by the periodically repeated orthonormal basis.The subject is connected to many topics in discrete geometry: lattices, sphericaldesigns and codes, discrepancy etc, while most of the methods are rooted in analysis:spherical harmonics, orthogonal polynomials, positive definite functions etc.

Discrete analogues in harmonic analysis: Directional maximalfunctions

Laura Cladek (University of California at Los Angeles)cladekl@math.ucla.edu

Abstract: The (continuous) directional maximal function,

MV f(x) := supv∈V,N

∣∣∣∣ 1

N

∫ N

0

f(x− v · t)dt∣∣∣∣ ,

is defined for functions f : R2 → C, and collections of directions V ⊂ S1. Under-standing MV is of great importance in Euclidean harmonic analysis; significantly,optimal norm estimates for MV f (in terms of |V |) are directly related to optimalKakeya estimates in the plane.

This work studies discrete analogues of MV in the arithmetic setting of the integerlattice. For instance, model operators we consider are of the form

supv∈V,N

∣∣∣∣∣ 1

N

∑n≤N

f(x− v · pn)

∣∣∣∣∣where p1, p2, . . . are an enumeration of the primes, and V ⊂ x ∈ Z2 : |x| = Afor some fixed radius A. (The fully linear setting follows from the continuous theoryby standard “transference” arguments.)

Beyond establishing analogous norm estimates, this work connects the continuoustheory with an arithmetic incidence problem – a Kakeya-type problem – by way oftime frequency analysis.

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Lack of boundedness of Calderon-Zygmund with totally ir-regular measures

Jose Manuel Conde Alonso (Brown University)jconde@math.brown.edu

Abstract: We consider totally irregular n-dimensional measures µ on Rn+1, that is,

lim supr→0

µ(B(x, r))

(2r)n> 0 and lim inf

r→0

µ(B(x, r))

(2r)n= 0

for µ almost every x. We show that if

Tµf(x) :=

∫K(x, y) f(y) dµ(y)

is an operator whose kernel K(·, ·) is the gradient of the fundamental solution for auniformly elliptic operator in divergence form associated with a matrix with Holdercontinuous coefficients, then Tµ is not bounded on L2(µ). This extends a celebratedresult proved previously by Eiderman, Nazarov and Volberg for the n-dimensionalRiesz transform.

Based on joint work with Mihalis Mourgoglou and Xavier Tolsa.

Sharp maximal directional estimates along polynomial k-manifolds

Francesco Di Plinio (University of Virginia)fd4v@virginia.edu

Abstract: We compute the sharp - up to epsilon losses - growth order of the L2

norm of the maximal directional averaging operator along a finite subset of a poly-nomial manifold of arbitrary dimension k. Our result is fully sharp in the case ofthe two-dimensional sphere. This extends results of Cordoba and Katz (k = 1)and of Demeter (for the particular case of the 2-sphere) and implies L2 estimatesfor Kakeya type maximal functions with tubes restricted to point along polynomialdirections. Our proof technique is novel and relies on an iterated polynomial parti-tioning scheme in the vein of Guth, Katz et. al. Joint work with Ioannis Parissis (UBasque Country).

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Uncertainty Principles and Sphere Packings

Felipe Goncalves (University of Alberta)felipe.goncalves@ualberta.ca

Abstract: In this talk we describe new uncertainty principle for functions connected

with sphere packings: If both f and f are radial and eventually nonnegative, both

having nonpositive total mass, then it is impossible for both f and f be nonnegativeoutside an arbitrarily small neighborhood of the origin. This uncertainty principle(now called +1 uncertainty) was discovered first in 2010 by Bourgain, Clozel andKahane with applications to number theory. However, recently, in joint work withHenry Cohn, we discovered that it possess a twin uncertainty principle (now called−1 uncertainty) which is intrinsically connected with the linear programming boundsfor sphere packings developed by Cohn and Elkies in 2003. Using the techniquesand constructions via modular functions from Viazovska’s solution of the 8 and24 dimensional sphere packing problem we were able to solve completely the +1uncertainty principle in dimension 12. Producing for the first time an extremizer ofa uncertainty inequality via quasi-modular functions and Eisenstein series. We willalso discuss some unknown numerical phenomenas and conjectures connected withthe topic. This talk is aimed for the general audience and the relevant backgroundwill (tentatively) be given.

Lower bound of the Riesz transform kernel on stratified Liegroups, commutators and applications

Ji Li (Macquarie University)ji.li@mq.edu.au

Abstract: It is well-known that the commutator of Riesz transform (Hilbert trans-form in dimension 1) and a symbol b is bounded on L2(Rn) if and only if b is in theBMO space BMO(Rn) (Coifman–Rochberg–Weiss).

Inspired by this result, it is natural to ask whether it holds for commutator ofRiesz transform on Heisenberg groups Hn. Note that in the setting of several complexvariables, the Heisenberg group Hn is the boundary of the Siegel upper half space,whose roles are holomorphically equivalent to the unit sphere and the unit ball inCn respectively, and hence the role of Riesz transform on Hn is similar to that ofHilbert transform on the real line R.

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We answer this question in the setting of stratified Lie groups G, which is moregeneral than the Heisenberg group Hn. We first obtain a suitable version of lowerbound for the kernel of the Riesz transform on G, and then establish a characterisationfor the boundedness of the Riesz commutator, i.e., the commutator of Riesz transformand a symbol b is bounded on L2(G) if and only if b is in the BMO space BMO(G)studied by Folland and Stein.

In the mean time we also establish characterisations for the endpoint boundednessof Riesz commutators, including the weak type (1, 1), H1(G) → L1(G), and L∞(G)to BMO(G), where H1(G) is the Hardy space studied by Folland and Stein.

As applications of the Riesz commutators, we introduce the curl operator on Gand then establish the div-curl lemma with respect to H1(G).

The results we provide here are based on the joint works with Xuan Thinh Duong,Hong-Quan Li and Brett D. Wick.

On The Fractional Maximal Operators

Jose Ramon (International Center for Theoretical Physics)jmadrid@ictp.it

Abstract: In this talk we will discuss the regularity properties of the fractionalmaximal operators when these act on Sobolev spaces and on spaces of functions ofbounded variation

Bilinear bi-parameter singular integrals: boundedness prop-erties and commutator estimates

Henri Martikainen (University of Helsinki)henri.martikainen@helsinki.fi

Abstract: We study general bilinear bi-parameter singular integrals T via moderndyadic methods and representation theorems. We discuss how these methods cane.g. be applied to show weighted estimates T : Lp(w1) × Lq(w2) → Lr(v3), where1 < p, q <∞, 1/2 < r <∞, 1/p+ 1/q = 1/r, w1 ∈ Ap(Rn×Rm), w2 ∈ Aq(Rn×Rm)

and v3 := wr/p1 w

r/q2 . Estimates for commutators and iterated commutators, like

[b1, T ]1 and [b2, [b1, T ]1]2, where b1 and b2 are little BMO functions, can also beobtained.

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Bumping matrix weights

Kabe Moen (University of Alabama)kabe.moen@ua.edu

Abstract: We will present some new two weight bump conditions in the matrixweighted setting. These result hold for maximal functions, Calderon-Zygmund op-erators, and fractional integral operators and perfectly match the scalar setting.We will also present some endpoint results for the maximal function and Calderon-Zygmund operators.

Self-similar measures: asymptotic bounds for the dimensionand Fourier decay of smooth images

Carolina A. Mosquera (Universidad de Buenos Aires)caroalemosquera@gmail.com

Abstract: R. Kaufman and M. Tsujii proved that the Fourier transform of self-similar measures has a power decay outside of a sparse set of frequencies. We presenta version of this result for homogeneous self-similar measures, with quantitative esti-mates, and derive several applications: (1) non-linear smooth images of homogeneousself-similar measures have a power Fourier decay, (2) convolving with a homogeneousself-similar measure increases correlation dimension by a quantitative amount, (3)the dimension and Frostman exponent of (biased) Bernoulli convolutions tend to 1as the contraction ratio tends to 1, at an explicit quantitative rate.

The results are based on a joint work with Pablo Shmerkin.

Leibniz-type rules associated to bilinear pseudodifferentialoperators

Virginia Naibo (Kansas State University)vnaibo@ksu.edu

Abstract: We will present various Leibniz-type rules for Coifman-Meyer multi-plier operators as well as for bilinear pseudodifferential operators with homogeneoussymbols and with symbols in the bilinear Hormander classes. The estimates will bediscussed in the settings of Triebel-Lizorkin and Besov spaces based on quasi-Banach

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spaces such as weighted Lebesgue, Lorentz and Morrey spaces. These results extendand complement the well-known fractional Leibniz rules corresponding to the caseof a multiplier identically equal to one and, even in such situation, they lead to newestimates.

Continuous solutions for divergence-type equations associ-ated to elliptic systems of complex vector fields

Tiago Picon (Universidade de Sao Paulo - Ribeirao Preto)picon@ffclrp.usp.br

Abstract: In this talk, we characterize all the distributions F ∈ D′(U) such thatthere exists a continuous weak solution v ∈ C(U,Cn) (with U ⊂ Ω) to the divergence-type equation

L∗1v1 + ...+ L∗nvn = F,

where L1, . . . , Ln is an elliptic system of linearly independent vector fields withsmooth complex coefficients defined on Ω ⊂ RN . In case where (L1, . . . , Ln) is theusual gradient field on RN , we recover the classical result for the divergence equationproved by T. De Pauw and W. Pfeffer. Its proof is based on the closed range theoremand inspired by [1] and [3] in the classical case. Our method slightly differs fromtheirs by relying on the Banach-Grothendieck theorem and introducing tools frompseudodifferential operators, useful in our local setting of a system of complex vectorfields with variable coefficients( this is a joint work with Prof. Laurent Moonens(University of Paris-Sud, Orsay).

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Bibliography

[1] J. Bourgain and H. Brezis, On the equation div Y = f and application to controlof phases, J. Amer. Math. Soc. 16 (2003), 393–426.

[2] L. Moonens and T. Picon, Continuous solutions for divergence-type equationsassociated to elliptic systems of complex vector fields, J. of Function Analysis,275, 5 (2018), 1073–1099.

[3] T. De Pauw and W. Pfeffer, Distribution for which div u = F has a continuoussolution, Comm. Pure Appl. Math LXI (2008), 230–260.

Degenerate Poincare-Sobolev Inequalities

Ezequiel Rela (Universidad de Buenos Aires)erela@dm.uba.ar

Abstract: In this talk I will present some recents results on weighted weightedPoincare and Poincare-Sobolev type inequalities with an explicit analysis on thedependence on the Ap constants of the involved weights. We obtain inequalities ofthe form (

1

w(Q)

∫Q

|f − fQ|qw) 1

q

≤ Cw`(Q)

(1

w(Q)

∫Q

|∇f |pw) 1

p

,

with different quantitative estimates for both the exponent q and the constant Cw.We will derive those estimates together with a large variety of related results as aconsequence of a general selfimproving property shared by functions satisfying theinequality

1

|Q|

∫Q

|f − fQ|dµ ≤ a(Q),

for all cubes Q ⊂ Rn and where a is some functional that obeys a specific discretegeometrical summability condition. We introduce a Sobolev-type exponent p∗w > p

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associated to the weight w and obtain further improvements involving Lp∗w norms on

the left hand side of the inequality above. For the endpoint case of A1 weights wereach the classical critical Sobolev exponent p∗ = pn

n−p which is the largest possible

and provide different type of quantitative estimates for Cw. We also show that thisbest possible estimate cannot hold with an exponent on the A1 constant smaller than1/p.

As a consequence of our results (and the method of proof) we obtain further exten-sions to two weights Poincare inequalities and to the case of higher order derivatives.We also apply our method to obtain similar estimates in the scale of Lorentz spaces.

This is a joint work with Carlos Perez Moreno from BCAM.

Minimal energy on fractals

Alexander Reznikov (Florida State University)reznikov@math.fsu.edu

Abstract: In recent years the problem of minimal discrete Riesz energy on smoothmanifolds got a lot of attention; in particular, there are a lot of positive results on thedistributional properties of the configurations that minimize the energy. As it turnsout, the situation on fractals (for example, on the 1/3-Cantor set) is significantlydifferent. We will survey some recent positive and negative results and discuss theopen problems in this area.

A1 theory for singular operators

Israel P. Rivera Rıos (Universidad Nacional del Sur)petnapet@gmail.com

Abstract: In this talk we present results around the quantitative dependence onthe A1 constant for certain singular operators such as rough singular integrals orcommutators.

Certain fractional type operators with Hormander conditions

Marıa Silvina Riveros (Universidad Nacional de Cordoba)sriveros@famaf.unc.edu.ar

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Abstract: We study fractional type operators with more than one kernel, definedby

Tα,mf(x) =

∫Rk1(x− A1y)k2(x− A2y) . . . km(x− Amy)f(y)dy,

where, for 1 ≤ i ≤ m, each ki satisfies a fractional size condition and generalizedfractional Hormander condition, and Ai are invertibles matrices. We obtain weightedCoifman type estimates, strong and weak type inequalities and BMO estimates forthis operator. We also present some examples different from those in the literature.

This work “Certain fractional type operators with Hormander conditions” is incollaboration with Lic. Gonzalo H. Ibanez-Firnkorn and has been accepted in An-nales Academiae Scientiarum Fennicae Mathematica (03-2018).

An extension problem in Heisenberg type groups and appli-cations to Hardy inequalities

Luz Roncal (Basque Center for Applied Mathematics)lroncal@bcamath.org

Abstract: In their influential paper [1], Caffarelli and Silvestre interpret fractionalpowers (−∆)s of the Euclidean Laplacian ∆ as Dirichlet-to-Neumann maps for cer-tain boundary problems. This observation has had an enormous impact on the studyof the non-local operators (−∆)s, as it suddenly allowed to use local methods.

As observed by Frank et al [2], the relevant boundary value problem has ananalogue in the Heisenberg group. This analogue has been also studied by Mollers etal [3], and they obtained the explicit solution operator and investigated its properties.

In this talk we study the mentioned boundary value problem in the context ofso-called H-type groups, which are a generalisation of the Heisenberg group. For anyH-type group we state a certain boundary value problem, find the explicit integralkernel of its solution operator and obtain the conformally invariant powers of thesublaplacian as the Dirichlet-to-Neumann map. We further find an isometry propertyof the solution operator which generalises the isometry properties previously found byMollers et al [3] for Euclidean space and for the Heisenberg group. As an application,we find Hardy and trace Hardy inequalities on H-type groups with different weightfunctions.

Joint work with S. Thangavelu (Indian Institute of Science of Bangalore).

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Bibliography

[1] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractionalLaplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.

[2] R. L. Frank, M.d.M. Gonzalez, D. D. Monticelli and J. Tan, An extension problemfor the CR fractional Laplacian, Adv. Math. 270 (2015), 97–137.

[3] J. Mollers, B. Ørsted and G. Zhang, On boundary value problems for some con-formally invariant differential operators, Comm. Partial Differential Equations41 (2016), no. 4, 609–643.

Regularity of maximal functions on Hardy-Sobolev spaces

Mateus Sousa (Universidad de Buenos Aires)msousa@dm.uba.ar

Abstract: In this talk we will prove boundedness of maximal functions of convolu-tion type associated to smooth kernels on the Hardy-Sobolev spaces ·H1,p(Rd) when1/p < 1 + 1/d. We will also prove this range is sharp up to the endpoint.

Characterizations of two-weight inequalities between Lp spaces

Emil Vuorinen (University of Helsinki)emil.vuorinen@helsinki.fi

Abstract: We discuss the problem of characterizing two-weight inequalities of theHilbert transform between Lp spaces, where p is strictly between one and infinity.We describe one approach using a so-called quadratic testing condition.

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Poster Session

Poincare inequalities for John domains

Marıa Eugenia Cejas (Universidad Nacional de La Plata)mec.eugenia@gmail.com

Abstract: The goal is to study Poincare type inequalities: weighted fractionalPoincare for domain satisfying the Boman chain condition and the weighted improvedPoincare inequality for unbounded John domains.

Fractional Poincare inequality: The unweighted fractional (q, p) Poincare in-equality was obtained in [3] for bounded John domains using an appropriate decom-position of the domain in cubes and their properties. Our goal is to give sufficientconditions on the weight to extend this inequality to the case of weighted norms.

Weighted improved Poincare inequality: In [1] we study the weighted im-proved Poincare inequality. Namely, in the case of bounded John domains we givesufficient conditions on the weight for the weighted improved Poincare inequality.This result allows us to obtain a bigger class of weights than the well-known class ofMuckenhoupt of Ap. As a consequence we get solvability of divergence in weightedSobolev spaces for weights which are not necessarily in Ap. Then, it is natural to con-sider the case of unbounded John domain. In [2], the author study the unweightedcase. The idea is to decompose the unbounded domain by union of bounded s-John domains and apply the theory for bounded John domain. We will adapt thistechnique to obtain sufficient conditions on the weight for the weighted improvedPoincare inequality for the unbounded case.

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Bibliography

[1] Acosta, G. ; Cejas, M. E.; Duran, R, Improved Poincare inequalities andsolutions of the divergence in weighted forms., Ann. Acad. Sci. Fenn. Math.42 (2017), no. 1, 211-226.

[2] Hurri-Syrjanen, Ritva, An improved Poincare inequality, Proc. Amer.Math. Soc. 120 (1994), no. 1, 213-222.

[3] Hurri-Syrjanen, Ritva; Vahakangas, Antti V., On fractional Poincareinequalities, J. Anal. Math. 120 (2013), 85-104.

Using Wavelets to Buld a Core Inflation Measure for Brazil’sIPCA

Erik Filomena (Federal University of Rio Grande do Sul)eriksef@hotmail.com

Abstract: We present an estimation of the core inflation for the official Brazil-ian CPI using Discrete Wavelet Transform and Maximum Overlap Discrete WaveletTransform. This is done by obtaining a Multi Resolution Analysis based on theDaubechies 2 wavelet and estimating the underlying inflation rate by either remov-ing detail levels completely or applying a threshold algorithm. Lastly, a few tests ofquality of measurement proposed by the literature are performed. This is done withthe full set of data and a restricted set, obtained with a wavelet method for detectingstructural breaks in time series. .

Harmonic Analysis of the Spherical Means on HeisenbergGroup

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Sajith Govindankuttymenon (Sullamussalam Science College)sajseem@hotmail.com

Abstract: Reconstruction of a function from its averages over lower dimensionalmanifolds, a basic problem of Integral Geometry is revisited in the context of Heisen-berg Groups. The spherical mean value operator on the Heisenberg group Hn isstudied by Thangavelu, Agranowsky et al. The case of the Gelfand pairs (H,K)where K is a compact subgroup of U(n) is considered . Using the spectral theory ofLaplacians, we were able to get some Injectivity results for the general K−sphericalmean value operator . Analysis on the H−Type groups(Kaplan) is initiated byCowling,Dooley Korany and Ricci. Theory of Gelfand pairs in this case is furtherdeveloped by Linda Saal ,Goday et al. Rank 1 simple groups naturally takes toH−type groups as Iwasawa N groups. We would also like to discuss some recentdevelopments in some specific cases.

Sharp bounds for fractional operator with Lα,r′-Hormander

conditions

Gonzalo H. Ibanez-Firnkorn (Universidad Nacional de Cordoba)gibanez@famaf.unc.edu.ar

Abstract: We prove the sharp boundedness for a fractional type operator given bya kernel that satisfy a Lα,r

′-Hormander conditions and a fractional size condition,

where 0 < α < n and 1 < r′ ≤ ∞. To prove this result we use a new appropriatesparse domination which we provide in this work. For the case r′ = ∞ we recoverthe sharp boundedness for the fractional integral, Iα, proved in [?].

This work “Sharp bounds for fractional operator with Lα,r′-Hormander condi-

tions” is in collaboration with Dra. M. Silvina Riveros and Dr. Raul Vidal.

Differentiation along rectangles

Laurent Moonens (Universite Paris-Sud)Laurent.Moonens@math.u-psud.fr

Abstract: Lebesgue’s differentiation theorem states that, when f is a locally inte-grable function in Euclidean space, its average on the ball B(x, r) centered at x withradius r, converges to f(x) for almost every x, when r approaches zero.

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Many questions arise when the family of balls B(x, r) is replaced by a differen-tiation basis B =

⋃x Bx (where, for each x, Bx is, roughly speaking, a collection of

sets shrinking to the point x). In this case, one looks for conditions on B such thatthe average of f on sets belonging to Bx are known to converge to f(x) for a.e. x,when those sets shrink to the point x.

Many interesting phenomena happen when sets in B have a rectangular shape(Lebesgue’s theorem may or may not hold in this case, depending on the geometricalproperties of sets in B). In this poster, we shall present some of the history aroundthis problem, as well as recent results obtained with E. D’Aniello and J. Rosenblattin the planar case, when the rectangles in B are only allowed to lie along a fixedsequence of directions.

The dyadic fractional diffusion kernel as a central limit

Federico Morana (Instituto de Matematica Aplicada del Litoral)fmorana@santafe-conicet.gov.ar

Abstract: The fundamental solution kernel of fractional diffusions associated withdyadic differentiation in R+ is given by Kt(x, y) =

∑h∈H e

−t|I(h)|−sh(x)h(y), wheres is the order of differentiation, t is the time, H is the Haar system and I(h) is thesupport of the wavelet h ∈ H (see [1]). It can be proved that this kernel is a dyadicdistance dependant Markov kernel which also satisfies a stability condition, meaningthat the behavior at infinity is like a power law of the dyadic distance.

We show that this special kernel can be obtained as the central limit of adequatesimultaneous iteration and mollification of any dyadic stable Markov kernels with thesame stability parameters. Furthermore, only two alternatives to “centrality” arisesfrom these mollification and iteration processes applied to dyadic Markov kernels:concentration (which leads to approximate identities) and dissipation. And againthe stability property of the initial kernels is the discriminant between these threecases.

The main tool is provided by Fourier analysis through the Haar wavelets, whichgives an unconditional basis of the Lebesgue spaces Lp for 1 < p <∞, and thereforethe convergence holds in that sense. We present our results in the basic settingof the space R+ with the usual (regular) dyadic tiling and the associated metric.Nevertheless the results can be extended to spaces of homogeneous type (see [3]).

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Bibliography

[1] Marcelo Actis and Hugo Aimar, Dyadic nonlocal diffusions in metric measurespaces, Fract. Calc. Appl. Anal. 18 (2015), no. 3, 762-788. MR 3351499.

[2] Hugo Aimar, Ivana Gomez and Federico Morana, The dyadic fractional dif-fusion kernel as a central limit. Czechoslovak Math. J. (2017). In press.https://arxiv.org/abs/1702.02866.

[3] Federico Morana, Difusion fraccionaria diadica. Lımite central y aproximacionde la identidad. Thesis (Ph.D.), UNL, 2018.

[4] Hugo Aimar, Ivana Gomez and Federico Morana. Heavy Tailed Approx-imate Identities and σ-stable Markov Kernels. Potential Anal. (2017).https://doi.org/10.1007/s11118-017-9644-8

Lp → Lq estimates for cube skeletons maximal operators Andrea

Olivo (Universidad de Buenos Aires)aolivo@dm.uba.ar

Abstract: A problem recently studied is the relation between sizes of sets B, S ⊆ R2

when B contains the boundary of a square with center in every point of S and sidesparallel to the axis. By size, we mean Lebesgue measure and some fractal dimensions.

More generally, in higher dimensions, the sets consider are B, S ⊆ Rn when Bcontains the k-skeleton of an n-dimensional cube around every point of S.

This type of problems have associated a maximal operator. In this work we studya possible maximal operator and present results about its behaviour from Lp → Lq,for 1 ≤ p ≤ ∞. Whit this bounds we recover several results for the sizes of sets B.

Weighted inequalities of Fefferman-Stein type for Riesz-Schrodingertransforms

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Pablo Quijano (Univesidad Nacional del Litoral)pabloquijanoar@gmail.com

Abstract: In this work we are concerned with Fefferman-Stein type inequalities.More precisely, given an operator T , a weight w and some p, 1 < p <∞, we look foroperators M such that the inequality∫

|Tf |pw ≤ C

∫|f |pMw

holds true. Specifically, we are interested in the case of T being any first or secondorder Riesz transform associated to the Schrodinger operator L = −∆ + V , with Va non-negative function satisfying an appropriate reverse-Holder condition. For theRiesz-Schrodinger transforms ∇L−1/2 and ∇2L−1 we make use of a result due to C.Perez where this problem is solved for classical Calderon-Zygmund operators.

Trilinear Restriction Inequality for the Saddle with Sharp de-pendence on Transversality

Felipe Ponce Vanegas (Universidad Nacional de Colombia)feponcev@gmail.com

Abstract: The multilinear approach to the Stein’s restriction conjecture has provento be very successful, and one of the major results is the multilinear inequality ofBennett, Carbery and Tao, which allows to control efficiently transversal contribu-tions of the adjoint restriction operator. Unfortunately, they did not get the optimaldependence on transversality. Recently, Ramos got the sharp dependence for subsetsof the paraboloid in R3. We extend this result to subsets of the saddle.

Regularity of Fractional Maximal Functions Through FourierMultipliers

Joao Pedro Ramos (University of Bonn)joaopgramos95@gmail.com

Abstract: In a paper from 2003, Kinnunen and Saksman studied the regularisingproperties of the standard fractional maximal function associated to balls. Morespecifically, they prove that, for any Lp function f and 1 < p < n, the fractional

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maximal function Mβf(x) = supr1

rn−β

∫B(x,r)

|f(y)|dy has a weak derivative, and this

lies in a certain Lq space (with q depending on p, n, β).Although some contribution to the subject has been made in the past few years,

the question whether replacing ball by spherical averages had not been addressed,for the geometry of the problem then changes drastically from one case to the other.This hinted that another kind of method should be used.

The purpose of this poster is to give partial answers to this question in the casewhere we replace ball averages by spherical ones. As suggested by the title, weshall reformulate the problem in terms of the boundedness of a maximal multiplieroperator, which then makes us turn our attention to some central themes in Fourieranalysis, such as Littlewood-Paley theory, Bessel functions, multiplier operators andthe local smoothing conjecture.

Oscillatory integrals related to Carleson’s theorem: variation-norm estimates and discrete analogues

Joris Roos (University of Wisconsin - Madison)jroos@math.wisc.edu

Abstract: In a joint work with Shaoming Guo and Po-Lam Yung we prove variation-norm estimates for certain oscillatory integrals related to Carleson’s theorem. Corre-sponding maximal operators were first studied by Stein and Wainger. Our estimatesare sharp in the range of exponents, up to endpoints. The proof relies on squarefunction estimates for a family of Schrodinger-like equations due to Lee, Rogers andSeeger and local smoothing estimates for these equations. These variational esti-mates can be combined with certain number-theoretic estimates and an argument inthe spirit of Bourgain’s multi-frequency maximal lemma to prove bounds for discreteanalogues of the Stein-Wainger maximal operator (cf. recent work of Ben Krause).

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