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PSEUDODIFFERENTIAL OPERATORS AND OPERATOR IDEALS

Final Report Summary - PSEUDODIFFOPERATORS (PSEUDODIFFERENTIAL OPERATORS AND OPERATOR IDEALS)

FINAL PUBLISHABLE SUMMARY REPORT

In our project we are concentrated on the investigation in various independent and intertwined fields in analysis, the theory of pseudo-differential operators, harmonic analysis and the theory of operators ideals. In relation with nuclear operators and Schatten-von Neumann ideals we are interested in finding sufficient and/or necessary conditions for the memberships in such kind of ideals. In particular we consider operators on Lie groups and manifolds by exploiting the notion of full-matrix symbol. The membership of a pseudo-differential operator in a Schatten-von Neumann ideal constitutes a way to measure its regularity. We also apply the theory of pseudo-differential operators to study degenerate elliptic equations, fractional powers of subelliptic operators and Sobolev estimates. In particular we investigate regularity on Sobolev spaces and invertibility. The study of degenerate equations is a field of intensive research with important applications in physics and engineering.


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
- The study of invertibility of degenerate elliptic operators.
- Study of Lp bounds for fractional powers of subelliptic operators.
- Study of formulas for determinants associated to nuclear operators on compact manifolds.
- 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 conditions in terms of kernels have been obtained 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.