This book offers an overview of the research results presented by group members and guests of the Ghent Analysis & PDE Center during the weekly seminar that took place from 2021 to 2022. It is an informal event of the Analysis & PDE Center and associated researchers, where everyone can present their work or relevant literature for about 20-30 minutes. The seminar aims to exchange ideas and foster effective learning and collaboration.
In this book, group members and guests summarise their results presented during the seminar and provide outlooks for future work. In this way, the book also provides an overview of the recent developments in the Ghent Analysis & PDE Center. The main topics are functional analysis, Fourier analysis, noncommutative analysis, geometric analysis, partial differential equations of different types, harmonic analysis, functional inequalities, pseudo-differential operators, fractional derivatives, special functions, microlocal analysis, inverseproblems and imaging. The target audience of this book is any researcher working in the above fields.

Covers topics in analysis and partial differential equations Fosters effective learning and collaboration Provides an overview of the recent developments of the Ghent Analysis & PDE Center

Autorentext
Michael Ruzhansky is a Senior Full Professor of Mathematics at Ghent University in Belgium, and a Professor of Mathematics at Queen Mary University of London in the United Kingdom. His research interests mainly lie in Partial Differential Equations, Microlocal and Harmonic Analysis, and Pseudo-Differential Operators on Lie Groups and Manifolds. Previously, he had appointments at Utrecht University, Johns Hopkins University, University of Edinburgh, and Imperial College London. He is the recipient of various awards and fellowships, notably, the Ferran Sunyer i Balaguer Prize in 2014 and 2018, Daiwa Adrian Prize in 2010, and the ISAAC award in 2007. He is serving as the head of the Ghent Analysis & PDE Center of Ghent University. Dr Karel Van Bockstal obtained his PhD (in mathematical engineering) in 2015 at Ghent University, Belgium, and is currently a postdoctoral researcher (Ghent Analysis & PDE Center) at the Department of Mathematics: Analysis, Logicand Discrete Mathematics, Ghent University. His area of specialisation is related to mathematical analysis, evolutionary partial differential equations and the development of numerical algorithms and their implementation. This research focus concerns direct and inverse problems with applications in heat transfer, elasticity, electromagnetism and thermo-elasticity. He authors 37 publications included in ISI Web of Science. In addition, he was awarded the EAIP Young Scientist Award of the 8th International Conference "Inverse Problems: Modelling and Simulation", May 2016.

Klappentext

Part I Analysis.- A note on a Capelli operator and its resonance.- Schatten-von Neumann classes Sp on the torus for 0 < p 2.- Log-Sobolev and Nash inequalities on graded groups.- One-sided Hardy-Littlewood maximal function on generalised Lorentz spaces.- Remarks on gradient Yamabe solitons.- Boundedness of Fourier multipliers on fundamental domains of lattices.- Pointwise domination and weak L1 boundedness of Littlewood-Paley operators via sparse operators.- Hp Lp boundedness of Fourier multipliers on graded Lie groups.- On a reverse integral Hardy inequality on polarisable metric measure space.- Logarithmic Sobolev inequalities of fractional order on noncommutative tori.- The Prabhakar fractional q-integral and q-differential operators.- A note on weak type boundedness of pseudo-differential operators on rank one symmetric spaces of noncompact type.- Lp-Lq norms ofspectral multipliers.- Estimates for oscillatory integrals with discontinuous amplitude.- The unitary dual of the Heisenberg group over Rp.- Critical Sobolev-type identities and inequalities on stratified Lie groups.- Part II Partial differential equations.- Anisotropic Picone type identities for general vector fields and some applications.- An equivalence between the Neumann problem and its boundary domain integral equation systems for Stokes equations.- Short note on generalised bivariate Mittag-Leffler-type functions.- Inverse problems for time-fractional mixed equation involving the Caputo fractional derivative.- Time dependent inverse source problems for integrodifferential Kelvin-Voigt system.- A nonlocal initial conditional boundary value problem on metric graph.- Second-order semiregular non-commutative harmonic oscillators: the spectral zeta function.- Global well-posedness with loss of regularity for a class of singular hyperbolic Cauchy problems.- On a mixed equationinvolving Prabhakar fractional order integral-differential operators.- Inverse problem of determining a time-dependent source in a fractional Langevin-type partial differential equation.- Very weak solution of the discrete wave equation for harmonic oscillator.- An estimate for the multivariate Mittag-Leffler function.- Part III Mathematical modelling.- Mathematical modelling of the Lomb-Scargle method in astrophysics.- The application of physics informed networks to solve hyperbolic partial differential equations with nonconvex flux function and diffusion term.- Fractional differential equations: a primer for structural dynamics applications.- Text matching as time series matching.- Performing Particle Image Segmentation on an Extremely Small Dataset.- Two-dimensional dispersed composites on a square torus.

Inhalt
Part I Analysis.- A note on a Capelli operator and its resonance.- Schatten-von Neumann classes Sp on the torus for 0 < p 2.- Log-Sobolev and Nash inequalities on graded groups.- One-sided Hardy-Littlewood maximal function on generalised Lorentz spaces.- Remarks on gradient Yamabe solitons.- Boundedness of Fourier multipliers on fundamental domains of lattices.- Pointwise domination and weak L 1 boundedness of Littlewood-Paley operators via sparse operators.- Hp Lp boundedness of Fourier multipliers on graded Lie groups.- On a reverse integral Hardy inequality on polarisable metric measure space.- Logarithmic Sobolev inequalities of fractional order on noncommutative tori.- The Prabhakar fractional q -integral and q -differential operators.- A note on weak type boundedness of pseudo-differential operators on rank one symmetric spaces of noncompact type.- Lp - Lq norms ofspectral multipliers.- Estimates for oscillatory integrals with discontinuous amplitude.- The unitary dual of the Heisenberg group over Rp.- Critical Sobolev-type identities and inequalities on stratified Lie groups.- Part II Partial differential equations.- Anisotropic Picone type identities for general vector fields and some applications.- An equivalence between the Neumann problem and its boundary domain integral equation systems for Stokes equations.- Short note on generalised bivariate Mittag-Leffler-type functions.- Inverse problems for time-fractional mixed equation involving the Caputo fractional derivative.- Time dependent inverse source problems for integrodifferential Kelvin-Voigt system.- A nonlocal initial conditional boundary value problem on metric graph.- Second-order semiregular non-commutative harmonic oscillators: the spectral zeta function.- Global well-posedness with loss of regularity for a class of singular hyperbolic Cauchy problems.- On a mixed equationinvolving Prabhakar fractional order integral-differential operators.- Inverse problem of determining a time-dependent source in a fractional Langevin-type partial differential equation.- Very weak solution of the discrete wave equation for harmonic oscillator.- An estimate for the multivariate Mittag-Leffler function.- Part III Mathematical modelling.- Mathematical modelling of the Lomb-Scargle method in astrophysics.- The application of physics informed…