Page 489 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 489
HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (MS-28)
quantitative and appealing to harmonic analysts. The talk will be based on several papers coau-
thored with M. Christ, P. Durcik, J. Roos, K. A. Škreb, and C. Thiele.
Optimal Hardy weights on the Euclidean lattice
Marius Lemm, mariusclemm@gmail.com
EPFL, Switzerland
Coauthor: Matthias Keller
We investigate the large-distance asymptotics of optimal Hardy weights on Zd, d ≥ 3, via the
super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is
the familiar (d−2)2 |x|−2 as |x| → ∞. We prove that the inverse-square behavior of the optimal
4
Hardy weight is robust for general elliptic coefficients on Zd in various senses. The proofs
leverage Green’s function estimates rooted in homogenization theory.
Spectral analysis of a confinement model in relativistic quantum
mechanics
Albert Mas, albert.mas.blesa@upc.edu
Universitat Politècnica de Catalunya, Spain
In this talk we will focus on the Dirac operator on domains of R3 with confining boundary
conditions of scalar and electrostatic type. This operator is a generalization of the MIT-bag
operator, which is used as a simplified model for the confinement of quarks in hadrons that has
interested many scientists in the last decades. It is conjectured that, under a volume constraint,
the ball is the domain which has the smallest first positive eigenvalue of the MIT-bag operator.
I will describe our results —in collaboration with N. Arrizabalaga (U. País Vasco), T. Sanz-
Perela (U. Edinburgh and BCAM), and L. Vega (U. País Vasco and BCAM)— on the spectral
analysis of the generalized operator. I will discuss on the parameterization of the eigenvalues,
their symmetry and monotonicity properties, the optimality of the ball for large values of the
parameter, and the connection to boundary Hardy spaces.
Fractional Integrals with Measure in Grand Lebesgue and Morrey spaces
Alexander Meskhi, alexander.meskhi@tsu.ge
A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University,
Kutaisi International University, Georgia
Coauthor: Vakhtang Kokilashvili
In the last two decades, the theory of grand Lebesgue spaces Lp) introduced by T. Iwaniec
and C. Sbordone [2] is one of the intensively developing directions in modern analysis. The
necessity to investigate these spaces emerged from their rather essential role in various fields,
in particular, in the integrability problem of Jacobian under minimal hypotheses. It turns out
that in the theory of PDEs, the generalized grand Lebesgue spaces Lp),θ introduced by Greco,
Iwaniec, and Sbordone [1] are appropriate for treating the existence and uniqueness, as well as
487
quantitative and appealing to harmonic analysts. The talk will be based on several papers coau-
thored with M. Christ, P. Durcik, J. Roos, K. A. Škreb, and C. Thiele.
Optimal Hardy weights on the Euclidean lattice
Marius Lemm, mariusclemm@gmail.com
EPFL, Switzerland
Coauthor: Matthias Keller
We investigate the large-distance asymptotics of optimal Hardy weights on Zd, d ≥ 3, via the
super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is
the familiar (d−2)2 |x|−2 as |x| → ∞. We prove that the inverse-square behavior of the optimal
4
Hardy weight is robust for general elliptic coefficients on Zd in various senses. The proofs
leverage Green’s function estimates rooted in homogenization theory.
Spectral analysis of a confinement model in relativistic quantum
mechanics
Albert Mas, albert.mas.blesa@upc.edu
Universitat Politècnica de Catalunya, Spain
In this talk we will focus on the Dirac operator on domains of R3 with confining boundary
conditions of scalar and electrostatic type. This operator is a generalization of the MIT-bag
operator, which is used as a simplified model for the confinement of quarks in hadrons that has
interested many scientists in the last decades. It is conjectured that, under a volume constraint,
the ball is the domain which has the smallest first positive eigenvalue of the MIT-bag operator.
I will describe our results —in collaboration with N. Arrizabalaga (U. País Vasco), T. Sanz-
Perela (U. Edinburgh and BCAM), and L. Vega (U. País Vasco and BCAM)— on the spectral
analysis of the generalized operator. I will discuss on the parameterization of the eigenvalues,
their symmetry and monotonicity properties, the optimality of the ball for large values of the
parameter, and the connection to boundary Hardy spaces.
Fractional Integrals with Measure in Grand Lebesgue and Morrey spaces
Alexander Meskhi, alexander.meskhi@tsu.ge
A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University,
Kutaisi International University, Georgia
Coauthor: Vakhtang Kokilashvili
In the last two decades, the theory of grand Lebesgue spaces Lp) introduced by T. Iwaniec
and C. Sbordone [2] is one of the intensively developing directions in modern analysis. The
necessity to investigate these spaces emerged from their rather essential role in various fields,
in particular, in the integrability problem of Jacobian under minimal hypotheses. It turns out
that in the theory of PDEs, the generalized grand Lebesgue spaces Lp),θ introduced by Greco,
Iwaniec, and Sbordone [1] are appropriate for treating the existence and uniqueness, as well as
487
