Orateur
Description
Since work of Hilbert in 1908, singular integrals have played an important role in the study of PDEs. In the 1960s, Kohn and Nirenberg (building on work of Seeley, Unterberger, Bokobza) introduced pseudodifferential operators, which simplified many arguments that relied on singular integral operators. Since then, pseudodifferential operators have been used to great effect in the study of PDEs; especially elliptic PDEs. Because of their powerful algebraic structure, pseudodifferential operators have largely replaced singular integral operators in many applications.
However, there are modern settings where pseudodifferential operators do not apply and singular integrals once again become a central tool.
In this mini-course we:
• Define singular integral operators with motivating examples from Euclidean space.
• Describe function spaces from harmonic analysis and their connections with singular integral operators.
• Present maximally subelliptic PDEs as a case study where singular integrals give sharp results and standard pseudodifferential operators do not.
• Show how singular integrals allow us to not only study linear subelliptic PDEs, but also fully nonlinear subelliptic PDEs, subelliptic PDEs with rough coefficients, and subelliptic boundary value problems.
This course is designed for graduate students with some functional analysis (L^p spaces, Fourier transform). Some familiarity with distributions and pseudodifferential operators will be useful, but not necessary.