INDIVIDUAL FELLOWSHIPS
Project n°: 301599
Project Acronym: PseudodiffOperatorS
Project Full Name: Pseudo-Differential Operators
and Operator Ideals
Marie Curie Actions
IIF Final Report
Period covered: from 18/06/2012 to 17/06/2014
Date of preparation: 07/07/2014
Start date of project: 18/06/2012
Date of submission (SESAM): _____
Project coordinator name: Dr. Julio Delgado
Project coordinator organisation name:
IMPERIAL COLLEGE OF SCIENCE,
TECHNOLOGY AND MEDECINE
Version: 1
Final Report
PROJECT FINAL REPORT
Grant Agreement number:
301599
Project acronym:
PseudodiffOperatorS
Project title:
Pseudo-Differential Operators and
Operator Ideals
Funding Scheme:
FP7-PEOPLE-2011-IIF
Period covered-start date:
18/06/2012
Period covered-end date:
17/06/2014
Name, title and organisation of the person
in charge of the project for the beneficiary:
Prof. Michael Ruzhansky
Tel:
+44 2075948500
Fax:
+44 2075948500
E-mail:
m.ruzhansky@imperial.ac.uk
Project website address:
http://www2.imperial.ac.uk/~ruzh/Project-Delgado.htm
1. Progress and achievements during the project
Please provide a concise overview of the progress of the work in line with the structure of Annex I
of the Grant agreement.
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A
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A summary of the progress of the researcher training activities/transfer of knowledge
activities/integration activities (as it applies for the MC action);
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Highlight clearly significant results;
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If applicable, explain the reasons for deviations from Annex I and their impact on other
tasks as well as on available resources and planning;
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If applicable, explain the reasons for failing to achieve critical objectives and/or not being
on schedule and explain the impact on other tasks as well as on available resources and
planning (the explanations should be coherent with the declaration by the project
coordinator) ;
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A statement on the use of resources, in particular highlighting and explaining deviations in
Annex 1 (Description of Work)
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Management costs and overheads (2007 and 2008 calls) section 3;
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If applicable, propose corrective actions.
summary
of
progress
towards
objectives
and
details
for
each
task;
FINAL PUBLISHABLE SUMMARY
In our project, we apply the theory of pseudo-differential operators to the study of several and intertwined
problems in harmonic analysis. We have exploited the notion of full matrix-symbol on compact Lie groups
in order to obtain a characterisation of invariant operators in the ideals of Schatten-von Neumann and rnuclear operators. We have exhibited some applications to the analysis of differential operators. We also
found sharp sufficient conditions on kernels for the membership to Schatten-von Neumann ideals. We
developed a notion of full matrix-symbol for operators on compact manifolds and applied it to the study of
Schaten-von Neumann ideals and nuclearity. Conditions on symbols characterising operators in those ideals
have been obtained. The concept of full matrix-symbol and the corresponding notion of pseudo-differential
operator have played an instrumental role for the achievement of those objectives. A crucial motivation was
the application to the study of different differential operators on compact Lie groups and compact manifolds.
The activities carried out during the project can be summarised as follows:
- Analysis of operators on compact Lie groups in terms of matrix-symbols
- Introduction of a matrix-symbol notion for operators on compact smooth manifolds
- Analysis of operators on closed manifolds in terms of matrix-symbols
- Analysis of kernels of operators on closed manifolds
- Study of the r-nuclear operators on Lp Lebesgue spaces and the Grothendieck-Lidskii formula
- A study for the introduction of Weyl-Hormander calculus on the torus
- Lp bounds and invertibility of subelliptic operators in the setting of weyl-Hormander calculus
- Presentation of the results in several conferences and writing of publications
The more significant results of the project are listed below:
1. Characterisation of Schatten classes of invariant operators on compact topological groups in
terms of the full matrix-symbol and applications to the analysis of differential operators.
2. For operators on Lebesgue spaces sufficient conditions have been obtained for the membership
to the ideal of r-nuclear operators and related the Lidskii's formula to the symbol for operators
on Lp.
3. Introduction of a new notion of full matrix-symbol for operators on compact manifolds.
4. Characterisation of Schatten classes of invariant operators on closed manifolds.
5. Sharp sufficient kernel conditions for the membership to Schatten classes.
6. A characterisation of nuclearity and a trace formula have been established for operators on spaces
of Bochner integrable functions.
7. Plemelj-Smithies formulas for the determinants of operators by applying the introduced
notion of matrix-symbol.
8. Invertibility for a class of subelliptic operators in the setting of Weyl-Hormander calculus.
9. Lp bounds for fractional powers of Grushin operators in the setting of Weyl-Hormander calculus.
10. Characterisation of the ideal of Hilbert-Schmidt operators for symbols adapted to boundary
problems.
Publications:
-J. Delgado and M. Ruzhansky. Schatten classes on compact manifolds: Kernel conditions, Journal of
Functional Analysis. Vol. 267, no. 3, 772–798, 2014.
-J. Delgado and M. Ruzhansky . Schatten classes and trace formula on compact Lie groups, iv:1303.3914v1
and submitted to a Journal.
-J. Delgado and M. Ruzhansky. Lp-Nuclearity, traces, and Grothendieck-Lidskii
formula on compact Lie groups, Journal des Mathematiques Pures et Appl. Vol. 102 , no. 1, 153-172, 2014.
-J. Delgado. Trace formulas for nuclear operators in spaces of Bochner integrable functions, Monatshefte fur
Mathematik. Vol. 172, no. 3, 259–275, 2013.
-J. Delgado. On the r-nuclearity of some integral operators on Lebesgue spaces, to appear in Tohoku
Mathematical Journal.
-J. Delgado and M. Ruzhansky. Fourier multipliers, Symbols and Nuclearity on compact manifolds,
submitted.
-J. Delgado and M. Ruzhansky . Kernel and symbol criteria for Schatten classes and r-nuclearity on compact
manifolds, submitted.
-J Delgado. A class of invertible subelliptic operators in S(m, g)-calculus, submitted.
-J. Delgado. Lp bounds in S(m, g)-calculus, submitted.
-J. Delgado and M. Ruzhansky. Plemelj-Smithies formulas for determinants of operators on compact
manifolds. Preprint.
-J. Delgado and M. Ruzhansky. Quantization on compact manifolds and Nuclearity. Preprint.
Conferences:
- Spectral properties on compact groups, Strobl14 : Modern Time–Frequency Analysis, Strobl, Austria, 2-6
June 2014.
- Spectral properties on compact groups, London Analysis and Probability Seminar, King’s College London,
UK, 22 May 2014.
- Spectral properties on compact Lie groups, LMS Joint Research Groups Workshop, Swansea University,
UK, 10-11 April 2014.
- Ideals of operators and spectral properties on compact Lie groups, 16th Workshop on Applications and
Generalizations of Complex Analysis University of Aveiro, Portugal, 21-22 March 2014.
- On some Ideals of operators on compact groups, Analysis seminar, Loughborough University,
Loughborough, UK, 4 December, 2013.
- Ideals of operators on compact topological groups, Centre de Recerca Matematica, UAB, Barcelona, Spain,
4-8 November, 2013.
- Schatten classes and r-nuclearity on compact Lie groups, 9th ISAAC Congress, Krakow, Poland, August 510, 2013.
- Schatten classes and Grothendieck-Lidskii formula for operators on compact Lie groups, Universidade
Federal do Parana, Curitiba, Brazil, July 25-26, 2013.
- Schatten classes and Grothendieck-Lidskii formula for operators on compact Lie groups, Universidade de
Sao Paulo, Sao Carlos, Brazil, July 22-24, 2013.
- Some Ideals of Operators on Compact Lie Groups, Colombian National Congress, UniNorte, Barranquilla,
Colombia, July 15-19, 2013.
- Schatten classes and Lp-nuclearity of operators on compact Lie groups. Generalized Functions and
Nonlinear Problems, University of Campinas, Brazil, July 10-12, 2013.
- Singular numbers of operators on compact Lie groups. Linear and Nonlinear Hyperbolic Equations,
Centro di Ricerca Matematica, Pisa, Italy, July 1-4, 2013.
- On some operator ideals on compact Lie groups, University of Sheffield, UK, May 28, 2013.
- Schatten classes and r-nuclearity on compact Lie groups, Complex Analysis and Dynamical Systems VI,
Nahariya, Israel, May 19-24, 2013.
- Nuclearity test for operators on compact manifolds. Conference in Harmonic Analysis and PDEs on
manifolds, Chuo University, Tokyo, Japan, April 19-20, 2013.
- On some ideals of operators on compact manifolds. Workshop on Geometric and Singular Analysis,
University of Potsdam, Germany, March 25-29, 2013.
- Schatten classes of operatos on compact Lie groups. Intensive European School, University of Aveiro,
Aveiro, Portugal, March 5-14, 2013.
- An Introduction to the Weyl-Hormander calculus(Three lectures). Imperial Analysis Seminar, Imperial
College London, London, January 11,18,25, 2013.
- Nuclear operators on compact Lie groups. Modern problems of applied mathematics and information
technologies AL-KHOREZMY 2012, National University of Uzbekistan, Tashkent, Uzbekistan, December
15-19, 2012.
- On the nuclearity of pseudo-differential operators on compact Lie groups. Workshop on Phase space
methods for pseudo-differential operators, ESI 12, Erwin Schrodinger Institute, Vienna, Austria, October 15-
- Nuclear operators on compact Lie groups. Modern problems of applied mathematics and information
technologies AL-KHOREZMY 2012, National University of Uzbekistan, Tashkent, Uzbekistan, December
15-19, 2012.
- On the nuclearity of pseudo-differential operators on compact Lie groups. Workshop on Phase space
methods for pseudo-differential operators, ESI 12, Erwin Schrodinger Institute, Vienna, Austria, October
15-19, 2012.
- On the traceability and the asymptotic behaviour of the eigenvalues of a class of integral operators on
Lebesgue spaces. Microlocal and Time-frequency Analysis. University of Novi sad, Republic of Serbia,
September 3-8, 2012.
Transfer of knowledge:
The fellow has contributed by transferring his knowledge on ideals of operators in order to intertwine
with the theory of pseudo-differential operators on Lie groups and the expertise of Prof. Michael
Ruzhansky. In that way the fellow has gained insight in the theory of Lie groups and its applications to the
analysis of differential operators. The fellow has also lectured a mini-course on Weyl-Hormander calculus
in Imperial College.
2. ADDITIONAL INFORMATION
3. PROJECT MANAGEMENT
Please use this section to summarise management activities during the period:
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Project planning and status – from management point of view;
Problems which have occurred and how they were solved or envisaged solutions;
Impact of possible deviations from the planned milestones and deliverables, if any;
Development of the project website (if applicable);
Gender issues; Ethical issues;
Justification of subcontracting (if applicable);
Justification of real costs (management costs);
Other
The PseudodiffOperatorS project has been run very successfully. The management of the project
has gone well and according to the plan. The objectives were achieved as well as several
important additionals.
Attachments:
Date:_____
Date:____________
Signature Scientist in Charge:
Signature Researcher:
____________
Michael Ruzhansky
____________
Julio Delgado