ABSTRACT
Let Ω denote a bounded domain with C∞ smooth boundary. The Cauchy transform associated to Ω is an example of an operator that is not local. A local operator Q on L2(bΩ) would have the property that, given a function u ∈ L2(bΩ) that vanishes on an open arc A in the boundary, then Qu also vanishes on this arc. It is easy to see that the Cauchy transform does not satisfy this property because the transform of a function vanishing on an arc extends holomorphically past that arc. However, the Cauchy transform is an example of a pseudo-local operator. This means that, given an open connected arc contained in the boundary of a bounded domain Ω with C∞ smooth boundary, if a function in L2(bΩ) is C∞ smooth on this arc, then so is its Cauchy transform. In this chapter, we will study this property in more detail and deduce some of its consequences.
