Harmonic Analysis and Its Applications
Auburn University, USA
It was well known that the methods of harmonic analysis can be used to solve problems in PDE. In particular, since Bourgain, Tao and Kenig solved many longstanding open problems,people pay more attention on harmonic analysis. As Nirenberg pointed out that to solve the Navier-Stokes equation, one needs more harmonic analysis. The main purpose of this lecture is to describe how the methods of harmonic analysis can be used in PDE. Namely, we will talk about the following.
Hardy-Littlewood maximal function
To study the boundary valued problem in Laplace, Heat and Wave equations, the Hardy-Littlewood maximal function is a necessary tool. We will study the Lp and the weak L1 boundeness of the Hardy-Littlewood maximal function and other kind of maximal functions.
Calderón-Zygmund operator theory
To study partial differential equations with constant coefficients, Calderón and Zygmund developed the first generation of Calderón-Zygmund operators, the convolution singular integral operators. We will study the Lp and the weak L1 boundeness of such operators. Particularly, we will introduce the Calderón-Zygmund Lemma.
Littlewood-Paley theory and function spaces
Function spaces, such as the Hölder and Lp spaces, are crucial for the study of PDE Littlewood-Paley theory provides a unified method to study these spaces. We will study the Lp; Hardy and Besov spaces in terms of using the Littlewood-Paley theory.