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Marie Curie Project Pseudo-differential operators and operator ideals

FP7-PEOPLE-2011-IIF Project No 301599 - Acronym: PseudodiffOperatorS

2012-2014

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 Name of Researcher: Dr Julio DELGADO

Scientist in Charge: Prof Michael Ruzhansky

Imperial College London

 

PROJECT ABSTRACT

 

This project is 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. We apply the theory of pseudo-differential operators to study degenerate elliptic equations, degenerate hyperbolic operators, fractional powers of subelliptic operators, Sobolev estimates. In particular we investigate regularity on Sobolev spaces, invertibility and the Cauchy problem for degenerate hyperbolic equations. The study of degenerate equations is a field of intensive research with important applications in physics and engineering. A second field of interest is the study of pseudo-differential operators on compact Lie groups applying techniques of the Weyl-Hormander calculus. Concerning 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 study different ideals of operators on certain Lie groups and investigate the case of pseudo-differential operators. The study of traces is important in its own, traces of pseudo-differential operators play an essential role in the study of geometric and topological invariants. The membership of a pseudo-differential operator in a Schatten-von Neumann ideal constitutes a way to measure its regularity, and the case of localization operators is relevant in time-frequency analysis. 

 

PAPERS (2012-present)

1.      Delgado J., Ruzhansky M., Schatten classes and traces on compact Lie groups, submitted for publication, arxiv

2.      Delgado J., Ruzhansky M., Kernel and symbol conditions for Schatten classes on compact manifolds, submitted.

3.      Delgado J., Ruzhansky M., Fourier multipliers, symbols and nuclearity on compact manifolds, arxiv

4.     Delgado J., Ruzhansky, M., Quantization on compact manifolds and nuclearity, preprint.

5.     Delgado J., On the r-nuclearity of some integral operators on Lebesgue spaces, to appear in Tohoku Mathematical Journal.

6.     Delgado J., A class of invertible subelliptic operators in S(m, g)-calculus, submitted.

7.     Delgado J., Lp bounds in S(m,g) calculus, submitted.

8.     Delgado J., Ruzhansky, M., Schatten classes on compact manifolds: Kernel conditions, Journal of Functional Analysis 267 (2014), 772-798.

9.     Delgado J., Ruzhansky M., Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups, Journal Math Pures Appl.102 (2014), 153-172. arxiv

10. Delgado J., Trace formulas for nuclear operators in spaces of Bochner integrable functions, Monatshefte fur Mathematik 172 (2013), 259-275.

11. Delgado, J, Wong, M. W., Lp-nuclear pseudo-differential operators on Z and S1, Proc. Amer. Math. Soc. 141 (2013), 3935-3942.