### Session S38 - Geometric Potential Analysis

## Talks

Thursday, July 15, 16:05 ~ 16:35 UTC-3

## A new approach to John—Nirenberg-type spaces

### Oscar Dominguez

#### Universidad Complutense de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study John—Nirenberg-type spaces where oscillations of functions are controlled via covering lemmas. Our methods give new surprising results and clarify classical inequalities.

Joint work with Mario Milman (Instituto Argentino de Matemática, Buenos Aires, Argentina).

Thursday, July 15, 16:40 ~ 17:10 UTC-3

## Log-Sobolev inequalities and the renormalisation group

### Roland Bauerschmidt

#### University of Cambridge, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

We derive a multiscale generalisation of the Bakry--Emery criterion for a measure to satisfy a Log-Sobolev inequality. Our criterion relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry--Emery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, as an application, we prove that the massive continuum Sine-Gordon model with $\beta < 6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. (This is joint work with Thierry Bodineau.)

Joint work with Thierry Bodineau (Ecole Polytechnique, France).

Thursday, July 15, 17:15 ~ 17:45 UTC-3

## Review of some results on limit theorems for densities

### Sergey Bobkov

#### University of Minnesota, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will review several results in the central limit theorem on the convergence to the normal distributions. In particular, classical theorems due to Gnedenko (for the uniform distance), Prokhorov (convergence in total variation), and Barron (the entropic CLT) will be presented as particular cases of a more general statement in Orlicz spaces using a unifying decomposition approach.

Thursday, July 15, 17:50 ~ 18:20 UTC-3

## Fractional Orlicz-Sobolev spaces and applications to nonstandard growth fractional elliptic-type problems

### Julian Fernandez Bonder

#### University of Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I will review the definition and basic properties of Fractional order Orlicz-Sobolev spaces and then show its applications to nonstandard growth fractional elliptic-type problems. In particular I will show regularity for solutions of such problems. The content of this talk is contained in a series of joint works with A. Salort and H. Vivas.

Thursday, July 15, 18:25 ~ 18:55 UTC-3

## The doubling property on compact Lie groups

### Laurent Saloff-Coste

#### Cornell University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

In dimension $n$, all Riemannian metrics with non-negative Ricci curvature are uniformly doubling. Indeed, on any such manifold, the ratio of the volume of a ball of radius $2r$ divided by the volume of the concentric ball of radius $r$ is bounded by $2^n$. This talk is concerned with the conjecture that, for any compact Lie group $G$, there is a constant $C(G)$ such that any left-invariant metric is doubling with constant almost $C(G)$.

Joint work with Laurent Saloff-Coste, Nathaniel Eldredge (University of Northern Colorado) and Maria Gordina (University of Connecticut).

Thursday, July 15, 19:00 ~ 19:30 UTC-3

## Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

### Pablo De Nápoli

#### Universidad de Buenos Aires // IMAS (Conicet), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of the fractional Laplacian on the sphere in terms of fractional powers of the Dirichlet-to-Neumann map in the unit ball. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality.

Joint work with Pablo Raúl Stinga (Iowa State University,Argentina).

Thursday, July 15, 19:35 ~ 20:05 UTC-3

## On Ahlfors' Schwarzian derivatives for curves

### Martin Chuaqui

#### Pontificia Universidad Católica , Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss Ahlfors' Schwarzian derivatives for curves in euclidean space introduced some three decades ago. The definitions consider separate generalizations of the real and imaginary part of the classical operator in the complex plane that have important invariance properties with the respect to the Möbius group in $\mathbb{R}^n$. We will describe some of the applications of the real Schwarzian to the study of simple curves in $\mathbb{R}^n$, to knots in $\mathbb{R}^3$, as well as to the injectivity of the conformal parametrization of minimal surfaces in 3-space. The role of the imaginary Schwarzian will be presented in $\mathbb{R}^3$, highlighting its connection with the osculating sphere, a new transformation law under the Möbius group, and theorems on the existence and uniqueness of parametrized curves with prescribed real and imaginary Schwarzians.

Joint work with Julian Gevirtz, Brad Osgood (Stanford University), and (the late) Peter Duren.

Thursday, July 15, 20:10 ~ 20:40 UTC-3

## On the Compactness of Poincare-Einstein manifolds

### Fang Wang

#### Shanghai Jiao Tong University, China - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will talk about some recent progress on the compactness of Poincar\'{e}-Einstein manifolds and prove that once we get the compactness for one type of conformal compactification, then it also hold for some other type. We consider the conformal compactification with fixed smooth defining function and the adapted (including Fefferman-Graham) compactifications here. The main technique is an improved Schauder estimate for degenerate elliptic equations. This is joint work with Huihuang Zhou.

Joint work with Huihuang Zhou (Shanghai Jiao Tong University).

Friday, July 16, 16:05 ~ 16:35 UTC-3

## Quantitative divergence inequalities and application in kinetic theory

### Cl´ement Mouhot

#### University of Cambridge, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk is about quantitative divergence inequalities and application in kinetic theory.

Friday, July 16, 16:40 ~ 17:10 UTC-3

## Discrete Witten Laplacians and metastability of disordered mean field models

### Giacomo Di Gesù

#### University of Pisa , Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a discrete Schrödinger operator $ H_\varepsilon= -\varepsilon^2\Delta_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb Z^d)$, where $V_\varepsilon$ is defined in terms of a general multiwell energy landscape $f$ on $\mathbb R^d$. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $\mathbb R^d$. Moreover it is unitarily equivalent to a type of discrete diffusion arising in the context of disordered mean field models in Statistical Mechanics, as e.g. the Curie-Weiss model.

In this talk I will present results on the bottom of the spectrum of $H_\varepsilon$ in the semiclassical regime $\varepsilon\ll1$, including the fine asymptotics of the tunnel effect between wells. These results require minimal regularity assumptions on f, are based on microlocalization techniques and permit to recover the Eyring-Kramers formula for the metastable tunneling time of the underlying stochastic process.

Further I will discuss the complex property and the Hodge-type extension of the discrete Witten Laplacian to the full algebra of discrete differential forms. This is inspired by the well-known fact that lifting the continuous space Witten Laplacian to higher forms provides a powerful tool for studying e.g. Morse inequalities and functional inequalities (Brascamp-Lieb, Bakry-Émery, Helffer-Sjöstrand) on manifolds.

Friday, July 16, 17:15 ~ 17:45 UTC-3

## Analysis on manifolds and volume growth

### Alexander Grigor'yan

#### University of Bielefeld, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

I'll give an overview of many results about the relation between the volume growth of a complete non-compact Riemannian manifold and certain properties of solutions of PDEs on such a manifold.

Friday, July 16, 17:50 ~ 18:20 UTC-3

## Positive solutions to the time-independent Schrodinger equation and the existence of the gauge

### Michael Frazier

#### University of Tennessee, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

We survey some joint work with Igor Verbitsky and Fedor Nazarov regarding the time-independent Schrodinger equation $-\triangle u= \omega u$ on a domain $\Omega$, $u=f$ on $\partial \Omega$. Our goal is to find minimal conditions on the Schrodinger potential $\omega$ that guarantee the existence of a solution $u \geq 0$ when $f \geq 0$. We begin with a general result about quasi-metric kernels, which yields matching upper and lower bounds for Green's function of the Schrodinger operator. A condition for the existence of the gauge (the solution $u$ if $f=1$) for Schrodinger operators is given in terms of the exponential integrability of the balayage of $\omega$. Recently, these results were extended to the case of uniform domains, with Martin's kernel replacing the Poisson kernel. We also discuss related results for the fractional Laplacian.

Joint work with Fedor Nazarov (Kent State University, United States of America) and Igor Verbitsky (University of Missouri, United States of America).

Friday, July 16, 18:25 ~ 18:55 UTC-3

## A Sobolev type embedding theorem for Besov spaces defined on doubling metric spaces

### Joaquim Martín

#### Universitat Autònoma de Barcelona, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this tlak we present a Sobolev type embedding result for Besov spaces defined on a doubling measure metric space.

Joint work with Walter A.Ortiz (Universitat Autònoma de Barcelona, Spain).

Friday, July 16, 19:00 ~ 19:30 UTC-3

## Singular Integrals, Geometry of Sets, and Boundary Problems

### Marius Mitrea

#### Baylor University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Presently, it is well understood what geometric features are necessary and sufficient to guarantee the boundedness of convolution-type singular integral operators on Lebesgue spaces. This being said, dealing with other function spaces where membership entails more than a mere size condition (like Sobolev spaces, Hardy spaces, or the John-Nirenberg space BMO) requires new techniques. In this talk I will explore recent progress in this regard, and follow up the implications of such advances into the realm of boundary value problems.

Friday, July 16, 19:35 ~ 20:05 UTC-3

## From affine Poincaré inequalities to affine spectral inequalities

### Julián Haddad

#### UFMG, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We develop the basic theory of $p$-Rayleigh quotients in bounded domains, in the affine case, for $p \geq 1$. We establish $p$-affine versions of the affine Poincaré inequality and introduce the affine invariant $p$-Laplace operator $\Delta_p^{\mathcal A}$ defining the Euler-Lagrange equation of the minimization problem. For $p=1$ we obtain the existence of affine Cheeger sets and study preliminary results towards a possible spectral characterization of John's position.

Joint work with Marcos Montenegro (UFMG, Brazil) and Hugo Jiménez (PUC-Rio, Brazil).

Friday, July 16, 20:10 ~ 20:40 UTC-3

## The fractional Laplacian and fractional gradient operators

### Liguang Liu

#### Renmin University of China, China - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will talk about some recent results related to the fractional Laplacian $\nabla^s_+=(-\Delta)^{\frac s2} $ and the fractional gradient $\nabla^s_-=\nabla (-\Delta)^{\frac {s-1}{2}}$, including also some optimal inequalities of type Hardy-Rellich/Adams-Moser/Morrey-Sobolev, and regularity of the distributional solutions to the dual equations $[\nabla^s_\pm]^\ast u=f$.

Joint work with Jie Xiao (Memorial University, Canada).

Monday, July 19, 16:05 ~ 16:35 UTC-3

## A simple proof of the Alexandrov-Fenchel inequalities for mixed volumes

### Dario Cordero-Erausquin

#### Sorbonne Université, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

We shall give a short proof of the Alexandrov-Fenchel inequalities for convex bodies, which mixes elementary algebraic properties and convexity properties of mixed volumes of polytopes.

Monday, July 19, 16:40 ~ 17:10 UTC-3

## Equivalent definitions of Hardy spaces on product spaces of homogeneous type and applications

### Maria Cristina Pereyra

#### University of New Mexico, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We present an atomic decomposition valid for product Hardy spaces on spaces of homogeneous type where minimal conditions are imposed on the underlying quasi-metric and measure. The Hardy spaces in this context were introduced by co-authors Han, Li and Ward using square functions based on orthogonal wavelet bases of Auscher and Hytönen and underlying reference dyadic grids on the spaces of homogeneous type. This decomposition enables us to show that the product Hardy spaces and their duals (the Carleson Measure Spaces, including BMO and VMO) are independent of the Auscher-Hytönen wavelets chosen and the Hytönen-Kairema dyadic cubes the wavelet are based on.

Joint work with Yongsheng Han (Auburn University, USA), Ji Li (Macquarie University, Australia) and Lesley Ward (University of South Australia, Australia).

Monday, July 19, 17:15 ~ 17:45 UTC-3

## On the fundamental gap of convex sets in hyperbolic space

### Alina Stancu

#### Concordia University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

The difference between the first two eigenvalues of the Dirichlet Laplacian on convex sets of ${\mathbb{R}}^n$ and, respectively ${\mathbb{S}}^n$, satisfies the same strictly positive lower bound depending on the diameter of the domain. In work with collaborators, we have found that the gap of the hyperbolic space on convex sets behaves strikingly different even if a stronger notion of convexity is employed. This is very interesting as many other features of first two eigenvalues behave in the same way on all three spaces of constant sectional curvature.

Monday, July 19, 17:50 ~ 18:20 UTC-3

## On Gaussian Singular Integrals

### Wilfredo Urbina-Romero

#### Roosevelt University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will discuss Gaussian singular integrals, considering their evolution from the seminal work of B. Muckenhoupt, until recent generalizations to new classes of singular integrals. Their boundedness on Lebesgue spaces is considered as well as on Gaussian variable Lebesgue spaces.

Joint work with Ebner Pineda (Escuela Superior Politécnica del Litoral, Ecuador) and Eduard Navas ( Universidad Nacional Experimental Francisco de Miranda, Venezuela).

Monday, July 19, 18:25 ~ 18:55 UTC-3

## Uniqueness Problems in Analysis, Geometry and Topology

### Shihshu Walter Wei

#### University of Oklahoma, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Uniqueness in analysis, geometry and topology has been a very interesting, active and fundamentally important area of research in mathematics.

We will give an overview and discuss uniqueness problems in analysis, geometry, and topology. Some ideas from physics and extrinsic {\it average} variational methods from calculus of variations will be used to unify general theories. Phenomena in geometric mapping theory and generalized Gauge theory will be explored, explained and unified.

Monday, July 19, 19:00 ~ 19:30 UTC-3

## On the dynamical network of interacting particles: from micro to macro

### Ewelina Zatorska

#### Imperial College London, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I will present a derivation of macroscopic model of interacting particles. The population of N particles evolve according to a diffusion process and interacts through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e. with O(N) particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics. The derivation combines the stochastic averaging (over time-scale parameter) and the many particles ($N\to \infty$) limits.

Monday, July 19, 19:35 ~ 20:05 UTC-3

## A Sharp Liouville Principle for $\Delta_m u+u^p|\nabla u|^q\leq 0$ on Geodesically Complete Noncompact Riemannian Manifolds

### Yuhua Sun

#### Nankai University, China - This email address is being protected from spambots. You need JavaScript enabled to view it.

For $(m,p,q)\in (1,\infty)\times\mathbb R\times\mathbb R$, this paper establishes a sharp Liouville principle for the weak solutions to the quasilinear elliptic inequality of second order $\Delta_m u+u^p|\nabla u|^q\leq0$ on the geodesically complete noncompact Riemannian manifolds, which is novel even in the special case of Euclidean space.

Joint work with Jie Xiao (Memorial University of Newfoundland, Canada) and Fanheng Xu (Sun Yat-Sen University, China).

Monday, July 19, 20:10 ~ 20:40 UTC-3

## Hardy spaces associated to the Kohn Laplacian on a family of model domains in ${\bf C}^2$

### Der-Chen Chang

#### Georgetown University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We establish a Hardy space theory on the boundary of a family of model domains of finite type m via a new discrete square function constructed from the heat kernel. We prove that a class of singular integral operators is not only bounded on the Hardy spaces $H^p(M)$, but also bounded from $H^p(M)$ to $L^p(M)$ for $\frac{m+2}{m+2+\vartheta}$.

Joint work with Yongsheng Han (Auburn University, USA) and Xinfeng Wu (China University of Mining && Technology, China).

## Posters

## Coercive Inequalities and $U$-Bounds

### Esther Bou Dagher

#### Imperial College London, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the setting of step-two Carnot groups, we prove Poincaré and $\beta$-Logarithmic Sobolev inequalities for probability measures as a function of various homogeneous norms. To do that, the key idea is to obtain an intermediate inequality called the $U$-Bound inequality (based on joint work with B. Zegarlinski). Using this $U$-Bound inequality, we show that certain infinite dimensional Gibbs measures- with unbounded interaction potentials as a function of homogeneous norms- on an infinite product of Carnot groups satisfy the Poincaré inequality (based on joint work with Y. Qiu, B. Zegarlinski, and M. Zhang). We also enlarge the class of measures as a function of the Carnot-Carathéodory distance that gives us the $q$−Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, we use the Hamilton-Jacobi equation in that setting to prove the $p$−Talagrand inequality and hypercontractivity.

## Sufficient conditions for some two-weighted inequalities for singular integrals

### Álvaro Corvalán

#### Universidad Nacional de General Sarmiento, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

They are quite well known several relationships between $A_p$ Muckenhoupt's weight functions and weighted inequalities for many singular integrals, like Riesz potentials, Riesz transforms, and the Hilbert trans-form. For instance it is a quite straightforward result of E. Stein that the weights for which all the Riesz Transformations are of weak type $(p,p)$ must be $A_p$ weights, or from the Helson-Szeg\"o you can deduce that the weights $w$ for which the Hilbert transform is bounded in $L^2(w)$ are $A2$-weights. In this presentation we give some conditions for pairs of weights involving two-weighted inequalities.

## Higher Order Coercive Inequalities

### Yifu Wang

#### Imperial College London, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the higher order $q$- Poincare and other coercive inequalities for a class probability measures satisfying Adam's regularity condition. Firstly we will introduce an improved version of Adam's inequalities, then obtain the Poincare inequalities in the measures we are discussing. Higher-order Orlicz-Sobolev inequalities and an equivalence of two types of weighted Sobolev norms will then be established. Finally we will also draw a few conclusions about Higher Order Decay to Equilibrium.

Joint work with Boguslaw Zegarlinski, Imperial College.