# Analysis seminar

## Upcoming talks

## Upcoming Talks

- Wed February 27th
**Matthew Romney****Title:**TBA - Wed March 3rd
**Augusto Gerolin****Title:**TBA - Wed March 13th
**Jarmo Jääskeläinen****Title:**TBA - Wed March 20th
**Ilmari Kangasniemi (Helsinki)****Title:**TBA - Wed March 27th
**Russell Brown (Kentucky)****Title:**TBA## Spring 2019

- Mon February 4th
**Juha Lehrbäck****Title:**Muckenhoupt A_p properties of distance functions and applications**Abstract:**Let E be a subset of a metric measure space X. I will discuss sharp conditions, based on the Assouad (co)dimension of E, which imply that certain powers of the distance function dist(x,E) are Muckenhoupt A_p weights in X. When general results on weighted embeddings are applied with such weights, we obtain various Hardy-Sobolev type inequalities for the space X. However, many of the results are new even in the Euclidean setting.

This talk is based on a joint work with Bartlomiej Dyda, Lizaveta Ihnazyeva, Heli Tuominen and Antti Vähäkangas. - Wed January 30th
**Danka Lucic****Title:**On the universal infinitesimal Hilbertianity of Riemannian manifolds**Abstract:**A metric measure space is said to be infinitesimally Hilbertian provided its associated Sobolev space is Hilbert. The advantage of working with this class of spaces is that their associated abstract differential structure has nice functional analytic properties.

The aim of this talk is to show that Riemannian manifolds are universally infinitesimally Hilbertian, i.e. infinitesimally Hilbertian when endowed with any locally finite Borel measure. The result will be achieved as a consequence of the following statement: the abstract tangent module (that will be defined during the talk) associated to a weighted Finsler manifold can be isometrically embedded into the space of all 2-integrable sections of its tangent bundle. A key tool we proved in order to build such embedding result is the density in energy of compactly supported smooth functions in the Sobolev space (regardless of the chosen measure on the Finsler manifold).

This is a joint work with Enrico Pasqualetto. - Wed January 23rd
**Enrico Pasqualetto****Title:**Infinitesimal Hilbertianity of Hadamard spaces.**Abstract:**Hadamard spaces are complete metric spaces having non-positive sectional curvature (in the sense of Alexandrov). The aim of the talk is to show that any Hadamard space is 'universally infinitesimally Hilbertian', meaning that its associated Sobolev space is Hilbert (regardless of the chosen reference measure).

The result is achieved by constructing an isometric embedding of the space of 'abstract' vector fields, defined via a suitable notion of derivation, into the space of 'concrete' vector fields, which are the sections of the bundle having the tangent cones as fibers. The main tools that we need are the superposition principle for normal metric 1-currents and the rigidity properties of barycenters on Hadamard spaces.

The interest towards this problem is motivated by the study of Lipschitz regularity of harmonic maps from finite-dimensional RCD spaces (i.e. metric measure spaces with synthetic lower bounds on the Ricci curvature and upper bounds on the dimension) to Hadamard spaces.

This is a joint work with Simone Di Marino, Nicola Gigli and Elefterios Soultanis.

## Fall 2018

**5.12.2018 Darya Apushkinskaya (Peoples' Friendship University of Russia, Moscow)**When Hopf's lemma remains valid?

Title:**Abstract:**The Hopf lemma, known also as the boundary point principle, is one of the important tools in qualitative analysis of partial differential equations. This lemma states that a supersolution of a partial differential equation with a minimum value at a boundary point, must increase linearly away from its boundary minimum provided the boundary is smooth enough. For general operators of non-divergence type with bounded measurable coefficients this result was established in elliptic case independently by E. Hopf and O. Oleinik (1952) and in parabolic case by L. Nirenberg (1953). The first result for elliptic equations with divergence structure was proved by R. Finn and D. Gilbarg (1957). Later the efforts of many mathematicians were aimed at the extension of the classes of admissible opeartors and at the reduction of the boundary smoothness. We present several versions of the Hopf lemma for general elliptic and parabolic equations in divergence and non-divergence forms under the sharp requirements on the coefficients of equations and on the boundary of a domain. Also we provide a new sharp counterexample.**28.11.2018 Yi-Hsuan Lin**On fractional inverse problems

Title:We demonstrate recent progress in the fractional inverse problems, especially for the famous Calderón type problem, where one tries to determine an unknown coefficient in an anisotropic Schrödinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness and approximation properties, which turn out to yield strong results in related inverse problems.

Abstract:**21.11.2018 Zheng Zhu**The product of Sobolev extension domains

Title:If u and V are W^{1,p}-extension domain, then U\times V is also a W^{1,p}-extension domain. Conversely, if U\times V is a W^{1,p}-extension domain, then both U and V must be W^{1,p}-extension domains. This is a joint work with Prof. Pekka Koskela.

Abstract:**14.11.2018 Aleksis Koski**Sobolev homeomorphic extensions

Title:**Abstract:**In the mathematical theory of nonlinear elasticity one typically represents elastic bodies as domains in Euclidean space, and the main object of study are deformations (mappings) between two such bodies. The class of acceptable deformations one considers usually consists of Sobolev homeomorphisms between the respective domains, for example, with some given boundary values. It is hence a fundamental question in this theory to ask whether a given boundary map admits a homeomorphic extension in the Sobolev class or not. We share some recent developements on this subject, including sharp existence results and counterexamples.**7.11.2018 Olli Tapiola**Carleson measures, rectifiability and epsilon-approximability of harmonic functions in L^p

Title:**Abstract:**The study of uniform rectifiability started as a hunt for optimal geometric conditions for various aspects of Calderón-Zygmund type harmonic analysis but more recently, the emphasis has been on exploring its connections to partial differential equations and other types of analysis. In this talk, we discuss some recent trends in uniform rectifiability, related topics and an L^p version of \epsilon-approximability of Hytönen and Rosén that generalizes the L^\infty type approximability property of Varopoulos. The talk is partially based on my previous and on-going work with S. Hofmann and S. Bortz.**31.10.2018 Rami Luisto**Regularity of the inverse of BV homeomorphisms in dimension three

Title:**Abstract:**By the classical inverse function theorem the inverse of a C^1 homeomorphism is again a C^1 homeomorphism when the Jacobian is strictly positive. It is natural to ask if similar results hold under milder regularity assumptions; natural candidates being Sobolev- and BV-homeomorphisms which are commonly used in the study of nonlinear elasticity. The problem of the weak regularity of the inverse has attracted much attention in the past decade, starting from the planar results of Hencl, Koskela and Onninen (2006, 2007) and extending to higher dimensions with the work of Csörnyei, Hencl and Maly (2010). These results show that for a Sobolev (1,n-1) homeomorphism f the inverse is a BV-mapping, and belongs to W^(1,n-1) when the mapping f is of finite distortion. In this talk we discuss our recent result in [arxiv:1804.03449], joint with Stanislav Hencl and Aapo Kauranen, which states that in dimension three the inverse of a BV-homeomorphism is BV if and only if the distributional adjoint of the mapping is a finite Radon measure. The distributional adjoint defined here is a new concept motivated both by our characterization and the success of the distributional Jacobian in the study of nonlinear elasticity.**24.10.2018 Marco Caroccia (University of Lisbon)**Perimeter with densities: An energy for epitaxial growth

Title:**Abstract:**In this talk I will introduce a free energy, first studied by Ratz and Voigt, defined as the integration over the reduced boundary of a crystal E of a suitable psi(u(x)) where psi is a convex function and u denotes the adatom density (the density of atoms not attached to the crystal and free to move on the surface). This energy leads the surface diffusion evolution in absence of the elastic stress and for small adatom densities. We treat the variational point of view, by analyzing the semicontinuity issues and the relaxad functional.**17.10.2018 Ivan Yaroslavtsev (Technical University of Delft, Netherlands.)**The Hilbert transform and orthogonal martingales in Banach spaces

Title:**Abstract:**Due to a classical argument of Doob it is well-known that the periodic Hilbert transform has a representation in terms of stochastic integrals with respect to a 2-dimensional Brownian motion. These stochastic integrals happen to be orthogonal martingales, so any estimates for orthogonal martingales lead to the same estimates for the periodic Hilbert transform. The goal of this talk is to show the converse dependence. Namely, we show that for any Banach space $X$ and any convex continuous functions $\Phi, \Psi:X \to \mathbb R_+$ one has that for any orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$

\[

\mathbb E \Psi(N_t) \leq C_{\Phi, \Psi} \mathbb E \Phi(M_t),\;\;\; t\geq 0,

\]

where the sharp constant C_{\Phi, \Psi} coincides with the $\Phi, \Psi$-norm of the periodic Hilbert transform.

This estimate has a lot of applications. In particular, it will allow us to show that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform coincide for all $1<p<\infty$ and for any Banach space $X$. This extends the result of Bañuelos and Kwaśnicki, who showed in 2017 that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform are equal in the real-valued setting, which had been an open problem for 90 years. This talk is based on joint work with Adam Osękowski (University of Warsaw).**10.10. 2018 Atte Lohvansuu**Duality of moduli in regular metric spaces

Title:**3.10.2018 Christian Webb (Aalto University)**Title: On the statistical behavior of the Riemann zeta function

Abstract: A notoriously difficult problem of analytic number theory is to describe the behavior of the Riemann zeta function on its so-called critical line. After reviewing some basic facts about thezeta function, I will discuss what can be said if the problem is relaxed slightly, namely if one is only interested in the behavior of the zeta function in the vicinity of a typical point on the critical line. Time permitting, I will also discuss how this problem is related to various models of probability theory and mathematical physics. The talk is based on joint work with Eero Saksman.**26.9.2018 Rakesh (University of Delaware)**Inverse problems for the wave equation

Title:**Abstract:**We describe some problems and results for formally determined (not over-determined) inverse (coefficient determination) problems for the perturbed wave equation. The problems have little geometry since the wave speed is constant. The difficulties are in the analysis associated with proving the injectivity of non-linear or even linear maps. Even some basic problems remain unsolved.**19.9.2018 Anna Kausamo**On the Monge problem in Multi-Marginal Optimal Mass Transportation

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- Mon February 4th