韩永生教授，美国奥本大学

The geometric considerations enter in a decisive way in many questions of harmonic analysis. For example, the estimation of the Fourier transform of surface-carried measure, which played a key role in averages over lower-dimensional varieties, restriction theorems, in connection with the study of Fourier integral operators, and in applications to linear and non-linear dispersive equations.
In this talk, our focus will be on singular integral and the geometry that describe the singularities of the kernels of these operators, and which controls the releveant estimates that are made. The kind of geometry that arises is local in nature and is based on a distance = d(x, y). This metric controls what happens when y is near x. However, the exact size of d(x, y) is not crucial but what matters is the order of magnitude of d as y → x.
We will describe how distants deduce the singular integrals in the Euclidean space; spaces of homogeneous type in the sense of Coifman and Weiss, and the Dunkl setting which is associated with finite reflection groups on the Euclidean space.