Harmonic analysis, operator theory and control

The aim of this course is to provide analysis tools for the study of PDEs. – First we introduce the main harmonic analysis tools to study the class of singular integral operators. – In the second part, we aim at introducing tools from operator theory usefull in control theory. *

• Harmonic analysis (Ph. Jaming) – $L^p$ and weak-$L^p$ spaces, interpolation. – Some Fourier analysis, Sobolev spaces, Paley-Wiener spaces. – Hardy-Litllewood maximal function: covering lemma, boundedness of the maximal function, application to Lebesgue’s differentiation theorem. Harmonic functions on the half-space, Poisson kernel and boundary behavior. – Hilbert and Riesz transforms. – Singular integrals, Calder\’on-Zugmund decomposition. – BMO. – Littlewood-Paley and H\”ormander’s multiplier theorem.
• Complements of theory of operators and semi-groups (M. Tucsnak) – Extrapolation and very weak solutions of linear evolution equations. Applications to wave and heat equations with inhomogeneous edge conditions. – Holomorphic semigroups: equivalent definitions, generator characterization, resolvent estimates, fractional powers, maximal regularity, perturbations. – Control and observation operators: admissibility in an abstract framework, applications to systems described by linear PDEs. – Some concepts of controllability and observability in infinite dimension.