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
Dasgupta A
(2021)
Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality
in Journal of Fourier Analysis and Applications
Akylzhanov R
(2020)
L-L multipliers on locally compact groups
in Journal of Functional Analysis
Kirilov A
(2021)
Global hypoellipticity and global solvability for vector fields on compact Lie groups
in Journal of Functional Analysis
Botchway L
(2020)
Difference equations and pseudo-differential operators on Z n
in Journal of Functional Analysis
Ruzhansky M
(2019)
Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations
in Journal of Inverse and Ill-posed Problems
Ruzhansky M
(2019)
A local-to-global boundedness argument and Fourier integral operators
in Journal of Mathematical Analysis and Applications
Akylzhanov R
(2019)
Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and L-L Fourier multipliers on compact homogeneous manifolds
in Journal of Mathematical Analysis and Applications
Ruzhansky M
(2022)
Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature
in Journal of Mathematical Analysis and Applications
Ruzhansky M
(2020)
Factorizations and Hardy-Rellich inequalities on stratified groups
in Journal of Spectral Theory
Delgado J
(2017)
-BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS
in Journal of the Institute of Mathematics of Jussieu
Akylzhanov R
(2019)
Contractions of group representations via geometric quantization
in Letters in Mathematical Physics
Hasanov A
(2020)
Hypergeometric Expansions of Solutions of the Degenerating Model Parabolic Equations of the Third Order
in Lobachevskii Journal of Mathematics
Ruzhansky M
(2020)
Self-Similar Solutions of Some Model Degenerate Partial Differential Equations of the Second, Third, and Fourth Order
in Lobachevskii Journal of Mathematics
Delgado J
(2018)
The bounded approximation property of variable Lebesgue spaces and nuclearity
in MATHEMATICA SCANDINAVICA
Laptev A
(2019)
Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I
in MATHEMATICA SCANDINAVICA
Kalmenov T
(2019)
On spectral and boundary properties of the volume potential for the Helmholtz equation
in Mathematical Modelling of Natural Phenomena
Ruzhansky M
(2018)
On a Very Weak Solution of the Wave Equation for a Hamiltonian in a Singular Electromagnetic Field
in Mathematical Notes
Delgado Julio
(2017)
Schatten classes and traces on compact groups
in MATHEMATICAL RESEARCH LETTERS
Delgado J
(2017)
Schatten classes and traces on compact groups
in Mathematical Research Letters
Garetto C
(2018)
Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness.
in Mathematische annalen
Avetisyan Z
(2021)
Approximations in $$L^1$$ with convergent Fourier series
in Mathematische Zeitschrift
Daher R
(2019)
Titchmarsh theorems for Fourier transforms of Hölder-Lipschitz functions on compact homogeneous manifolds
in Monatshefte für Mathematik
Ruzhansky M
(2018)
Convolution, Fourier analysis, and distributions generated by Riesz bases
in Monatshefte für Mathematik
Cardona D
(2022)
Determinants and Plemelj-Smithies formulas
in Monatshefte für Mathematik
Ruzhansky M
(2022)
A comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups
in Nonlinear Analysis
Ruzhansky M
(2019)
Weighted anisotropic Hardy and Rellich type inequalities for general vector fields
in Nonlinear Differential Equations and Applications NoDEA
Kassymov A
(2019)
Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups
in Nonlinear Differential Equations and Applications NoDEA
Ruzhansky M
(2017)
Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups
in Nonlinear Differential Equations and Applications NoDEA
Ruzhansky M
(2019)
Green's Identities, Comparison Principle and Uniqueness of Positive Solutions for Nonlinear p-sub-Laplacian Equations on Stratified Lie Groups
in Potential Analysis
Ruzhansky M
(2018)
Rellich inequalities for sub-Laplacians with drift
in Proceedings of the American Mathematical Society
Ruzhansky M
(2019)
Limiting cases of Sobolev inequalities on stratified groups
in Proceedings of the Japan Academy, Series A, Mathematical Sciences
Esposito M
(2019)
Pseudo-differential operators with nonlinear quantizing functions
in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Ruzhansky M
(2019)
Hardy inequalities on metric measure spaces.
in Proceedings. Mathematical, physical, and engineering sciences
Ruzhansky M
(2021)
Direct and inverse problems for time-fractional pseudo-parabolic equations
in Quaestiones Mathematicae
Ruzhansky M
(2018)
Weighted $$L^{p}$$ L p -Hardy and $$L^{p}$$ L p -Rellich inequalities with boundary terms on stratified Lie groups
in Revista Matemática Complutense
Ruzhansky M
(2020)
Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups
in Semigroup Forum
Mantoiu M
(2018)
Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits
in The Journal of Geometric Analysis
Ozawa T
(2019)
L p -Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups
in The Quarterly Journal of Mathematics
Dasgupta A
(2018)
Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations
in Transactions of the American Mathematical Society, Series B
Kirilov A
(2021)
Global Properties of Vector Fields on Compact Lie Groups in Komatsu Classes
in Zeitschrift für Analysis und ihre Anwendungen
Ruzhansky M
(2021)
Principal Frequency of $p$-Sub-Laplacians for General Vector Fields
in Zeitschrift für Analysis und ihre Anwendungen
Ruzhansky M
(2020)
Van der Corput lemmas for Mittag-Leffler functions
Altybay A
(2020)
The heat equation with strongly singular potentials
Ben-Artzi M
(2018)
Spectral identities and smoothing estimates for evolution operators
Ruzhansky M
(2018)
Hypoelliptic functional inequalities
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/ |