Regularity in affiliated von Neumann algebras and applications to partial differential equations
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
The proposed research will concentrate on the development of the regularity theory in affiliated von Neumann algebras and its subsequent applications to several areas of analysis and the theory of partial differential equations.
The subject of the regularity of spectral and Fourier multipliers has been now a topic of intensive continuous research over many decades due to its importance to many areas. Indeed, solutions to main equations of mathematical physics such as Schrödinger, wave, Klein-Gordon, relativistic Klein-Gordon, and many other equations can be written as spectral multipliers, i.e. functions of the operator governing the equation (e.g. the Laplacian). Multiplier theorems and their further dependence (decay) for large times has been a building block of the so-called dispersive estimates, implying further Strichartz estimates, nowadays being the main tool for investigating the global in time well-posedness of nonlinear equations. This scheme has many variants motivated by a variety of settings of the mathematical physics, with different operators replacing the Laplacian, different types of potentials, and different types of nonlinearities.
The present project aims at bringing the modern techniques of von Neumann algebras into these investigations. Indeed, several results known in the simplest Euclidean setting allow for their interpretation in terms of the functional subspaces of affiliated von Neumann algebras, or rather of spaces of (densely defined) operators affiliated to the von Neumann algebra of the space. This can be the group von Neumann algebra if the underlying space has a group structure, or von Neumann algebras generated by given operators on the space, such as the Dirac operator of noncommutative geometry or the one in the setting of quantum groups.
In this approach we can think of multipliers as those operators that are affiliated to the given von Neumann algebra (the affiliation is an extension of the inclusion, setting up a rigorous framework, after John von Neumann, for doing spectral analysis or functional calculus of unbounded operators with complicated spectral structure). We are interested in developing a new approach to proving multiplier theorems for operators on different function spaces by looking at their regularity in the relevant scales of regularity in the affiliated von Neumann algebras. The aim of the project is two-fold: to make advances in a general theory, but keeping in mind all the particular important motivating examples of settings (groups, manifolds, fractals, and many others that are included in this framework) and of evolution PDEs, with applications to the global in time well-posedness for their initial and initial-boundary problems. As such, it will provide a new approach to establishing dispersive estimates for their solutions, the problem that is long-standing and notoriously difficult in the area of partial differential equations with variable coefficients or in complicated geometry.
This is important, challenging and timely research with deep implications in theories of noncommutative operator analysis and partial differential equations, as well as their relation to other areas and applications.
The subject of the regularity of spectral and Fourier multipliers has been now a topic of intensive continuous research over many decades due to its importance to many areas. Indeed, solutions to main equations of mathematical physics such as Schrödinger, wave, Klein-Gordon, relativistic Klein-Gordon, and many other equations can be written as spectral multipliers, i.e. functions of the operator governing the equation (e.g. the Laplacian). Multiplier theorems and their further dependence (decay) for large times has been a building block of the so-called dispersive estimates, implying further Strichartz estimates, nowadays being the main tool for investigating the global in time well-posedness of nonlinear equations. This scheme has many variants motivated by a variety of settings of the mathematical physics, with different operators replacing the Laplacian, different types of potentials, and different types of nonlinearities.
The present project aims at bringing the modern techniques of von Neumann algebras into these investigations. Indeed, several results known in the simplest Euclidean setting allow for their interpretation in terms of the functional subspaces of affiliated von Neumann algebras, or rather of spaces of (densely defined) operators affiliated to the von Neumann algebra of the space. This can be the group von Neumann algebra if the underlying space has a group structure, or von Neumann algebras generated by given operators on the space, such as the Dirac operator of noncommutative geometry or the one in the setting of quantum groups.
In this approach we can think of multipliers as those operators that are affiliated to the given von Neumann algebra (the affiliation is an extension of the inclusion, setting up a rigorous framework, after John von Neumann, for doing spectral analysis or functional calculus of unbounded operators with complicated spectral structure). We are interested in developing a new approach to proving multiplier theorems for operators on different function spaces by looking at their regularity in the relevant scales of regularity in the affiliated von Neumann algebras. The aim of the project is two-fold: to make advances in a general theory, but keeping in mind all the particular important motivating examples of settings (groups, manifolds, fractals, and many others that are included in this framework) and of evolution PDEs, with applications to the global in time well-posedness for their initial and initial-boundary problems. As such, it will provide a new approach to establishing dispersive estimates for their solutions, the problem that is long-standing and notoriously difficult in the area of partial differential equations with variable coefficients or in complicated geometry.
This is important, challenging and timely research with deep implications in theories of noncommutative operator analysis and partial differential equations, as well as their relation to other areas and applications.
Planned Impact
As it is often the case with pure mathematics, the main impact will be academic. However, the range of the academic beneficiaries will be potentially very wide as the area of the operator analysis and its applications to partial differential equations influences advances in a variety of subjects. As it is written more specifically in the "Academic Beneficiaries" section, the expected impact on mathematics (and possibly on theoretical physics) is expected to be substantial. Besides these, there is a link to a range of applications of nonlinear PDEs through the planned work in the direction of dispersive and Strichartz estimates, and thus part of our research will be applicable there. The impact to this end is specified in more detail in the Pathways to Impact supplement to this application.
In addition to the academic aspects, in order to maximise the impact and exploitation of the EPSRC investment in this research, and to increase the knowledge transfer, we plan to organise an intensive workshop/conference devoted to the topic of the grant. A high-profile meeting would be extremely useful, to communicate the obtained results to the leading experts in the field of noncommutative analysis, the main topic of the EPSRC grant, to colleagues working in its applications, to discuss the achievements and future developments, thus also increasing the long-term influence of the conducted research. Inviting the leading mathematicians from a variety of countries working in the field will certainly significantly contribute to the worldwide academic advancement of the area highlighting in a unique way the results obtained during our project. Communicating the research findings in an especially designed meeting to an internationally wide-spread selection of world leaders in the field would be an ideal way to facilitate and to maximise the knowledge transfer related to this research. A minicourse given by the PI/RA planned in the framework of the meeting will contribute to the training of highly skilled researchers and the participation of PhD students and young postdocs will be very useful for improving teaching and learning. The meeting will also serve as an excellent way of identifying further research areas that would be influenced by the conducted research in a longer run. Consequently, we will consider editing and producing a volume of research papers originating from the meeting to increase its visibility and impact.
In addition to the academic aspects, in order to maximise the impact and exploitation of the EPSRC investment in this research, and to increase the knowledge transfer, we plan to organise an intensive workshop/conference devoted to the topic of the grant. A high-profile meeting would be extremely useful, to communicate the obtained results to the leading experts in the field of noncommutative analysis, the main topic of the EPSRC grant, to colleagues working in its applications, to discuss the achievements and future developments, thus also increasing the long-term influence of the conducted research. Inviting the leading mathematicians from a variety of countries working in the field will certainly significantly contribute to the worldwide academic advancement of the area highlighting in a unique way the results obtained during our project. Communicating the research findings in an especially designed meeting to an internationally wide-spread selection of world leaders in the field would be an ideal way to facilitate and to maximise the knowledge transfer related to this research. A minicourse given by the PI/RA planned in the framework of the meeting will contribute to the training of highly skilled researchers and the participation of PhD students and young postdocs will be very useful for improving teaching and learning. The meeting will also serve as an excellent way of identifying further research areas that would be influenced by the conducted research in a longer run. Consequently, we will consider editing and producing a volume of research papers originating from the meeting to increase its visibility and impact.
Organisations
People |
ORCID iD |
Michael Ruzhansky (Principal Investigator) |
Publications
Ruzhansky M
(2021)
Time-Dependent Wave Equations on Graded Groups
in Acta Applicandae Mathematicae
Ben-Artzi M
(2020)
Spectral identities and smoothing estimates for evolution operators
in Advances in Differential Equations
Ben-Artzi M.
(2020)
Spectral Identities And Smoothing Estimates For Evolution Operators
in Advances in Differential Equations
Ben-Artzi Matania
(2020)
SPECTRAL IDENTITIES AND SMOOTHING ESTIMATES FOR EVOLUTION OPERATORS
in ADVANCES IN DIFFERENTIAL EQUATIONS
Ruzhansky M
(2017)
Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups
in Advances in Mathematics
Akylzhanov R
(2020)
Re-expansions on compact Lie groups
in Analysis and Mathematical Physics
Kassymov A
(2021)
Reverse integral Hardy inequality on metric measure spaces
in Annales Fennici Mathematici
Ruzhansky M
(2020)
Subelliptic geometric Hardy type inequalities on half-spaces and convex domains
in Annals of Functional Analysis
Altybay A
(2021)
The heat equation with strongly singular potentials
in Applied Mathematics and Computation
Altybay A.
(2019)
Wave equation with distributional propagation speed and mass term: Numerical simulations
in Applied Mathematics E - Notes
Altybay Arshyn
(2019)
Wave Equation With Distributional Propagation Speed And Mass Term: Numerical Simulations
in APPLIED MATHEMATICS E-NOTES
Ruzhansky M
(2020)
Hardy and Rellich inequalities for anisotropic p-sub-Laplacians
in Banach Journal of Mathematical Analysis
Kassymov A
(2020)
Sobolev, Hardy, Gagliardo-Nirenberg, and Caffarelli-Kohn-Nirenberg-type inequalities for some fractional derivatives
in Banach Journal of Mathematical Analysis
Ruzhansky M
(2021)
Van der Corput lemmas for Mittag-Leffler functions. II. a-directions
in Bulletin des Sciences Mathématiques
Kirilov A
(2020)
Partial Fourier series on compact Lie groups
in Bulletin des Sciences Mathématiques
Ruzhansky M
(2020)
Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups
in Bulletin of Mathematical Sciences
Ruzhansky M
(2018)
A comparison principle for nonlinear heat Rockland operators on graded groups A COMPARISON PRINCIPLE FOR NONLINEAR HEAT ROCKLAND OPERATORS
in Bulletin of the London Mathematical Society
Ruzhansky M
(2020)
Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations
in Calculus of Variations and Partial Differential Equations
Altybay A
(2021)
Fractional Klein-Gordon equation with singular mass
in Chaos, Solitons & Fractals
Ruzhansky M
(2021)
Fourier multipliers on graded Lie groups
in Colloquium Mathematicum
Ruzhansky M
(2021)
Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states
in Communications in Contemporary Mathematics
Akylzhanov R
(2018)
Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups
in Communications in Mathematical Physics
Cardona D
(2023)
Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups
in Complex Variables and Elliptic Equations
Rottensteiner D
(2020)
The Harmonic Oscillator on the Heisenberg Group
in Comptes Rendus. Mathématique
Ruzhansky M.
(2019)
On nonlinear damped wave equations for positive operators. I. Discrete spectrum
in Differential and Integral Equations
Ruzhansky Michael
(2019)
ON NONLINEAR DAMPED WAVE EQUATIONS FOR POSITIVE OPERATORS. I. DISCRETE SPECTRUM
in DIFFERENTIAL AND INTEGRAL EQUATIONS
Kassymov A
(2022)
Reverse Stein-Weiss, Hardy-Littlewood-Sobolev, Hardy, Sobolev and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups
in Forum Mathematicum
Ruzhansky M
(2020)
Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions
in Fractional Calculus and Applied Analysis
Ruzhansky M
(2018)
Hardy-Littlewood, Bessel-Riesz, and Fractional Integral Operators in Anisotropic Morrey and Campanato Spaces
in Fractional Calculus and Applied Analysis
Ruzhansky M
(2020)
On a Non-Local Problem for a Multi-Term Fractional Diffusion-Wave Equation
in Fractional Calculus and Applied Analysis
Kanguzhin B
(2017)
On convolutions in Hilbert spaces
in Functional Analysis and Its Applications
Ruzhansky M
(2018)
Elements of Potential Theory on Carnot Groups
in Functional Analysis and Its Applications
Karimov Erkinjon
(2020)
Non-local initial problem for second order time-fractional and space-singular equation
in HOKKAIDO MATHEMATICAL JOURNAL
Huang J
(2023)
Intrinsic Image Transfer for Illumination Manipulation.
in IEEE transactions on pattern analysis and machine intelligence
Kumar V
(2020)
Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds
in Indagationes Mathematicae
Ruzhansky M
(2018)
Sobolev Type Inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund Spaces on Homogeneous Groups
in Integral Equations and Operator Theory
Ruzhansky M
(2019)
Bitsadze-Samarskii type problem for the integro-differential diffusion-wave equation on the Heisenberg group
in Integral Transforms and Special Functions
Kassymov A
(2019)
Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups
in Integral Transforms and Special Functions
Altybay A
(2020)
A parallel hybrid implementation of the 2D acoustic wave equation
in International Journal of Nonlinear Sciences and Numerical Simulation
Kumar V
(2023)
L p - L q Boundedness of (k, a) -Fourier Multipliers with Applications to Nonlinear Equations
in International Mathematics Research Notices
Delgado J
(2018)
Fourier multipliers, symbols, and nuclearity on compact manifolds
in Journal d'Analyse Mathématique
Dasgupta A
(2017)
Erratum to "The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups"
in Journal d'Analyse Mathématique
Muñoz J
(2019)
Wave propagation with irregular dissipation and applications to acoustic problems and shallow waters
in Journal de Mathématiques Pures et Appliquées
Kumar V
(2021)
A note on K -functional, Modulus of smoothness, Jackson theorem and Bernstein-Nikolskii-Stechkin inequality on Damek-Ricci spaces
in Journal of Approximation Theory
Ruzhansky M.
(2021)
Geometric Hardy Inequalities on Starshaped Sets
in Journal of Convex Analysis
Ruzhansky Michael
(2021)
Geometric Hardy Inequalities on Starshaped Sets
in JOURNAL OF CONVEX ANALYSIS
Garetto C
(2020)
Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II: Microlocal analysis
in Journal of Differential Equations
Ruzhansky M
(2020)
Very weak solutions to hypoelliptic wave equations
in Journal of Differential Equations
Ruzhansky M
(2022)
Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups
in Journal of Differential Equations
Ruzhansky M
(2018)
Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups
in Journal of Differential Equations
Description | The general procedure has been developed for deriving Lp-Lq estimates for Fourier and spectral multipliers. In the setting of locally compact groups and compact quantum groups the technique relies heavily on the von Neumann algebras theory that becomes instrumental in handling the spectral properties of the appearing operators. The research continues into the direction of simplification of the proofs as well as in its extensions to other settings. The main research is accompanied by the related sub Riemannian research allowing one to obtain additional properties of the sub elliptic operators. |
Exploitation Route | The developed techniques are now actively applied in a variety of related settings (non-harmonic analysis, symmetric spaces, Jacobi operators, anharmonic oscillators). |
Sectors | Other |
URL | https://analysis-pde.org/von-neumann/ |