Abstract: The talk is twofold. We try to introduce singular integral theory and its application in boundary value problem. More precisely, in our first part, we will talk some recent progress about Calderón type commutator with non-smooth kernel. Specially the weighted weak type (1, 1) boundedness and a weak type (1, 1) criterion will be discussed here. In the second part, we will give an application of Calderón type commutator. Thus we will show that the Neumann problem of Laplacian equation on a bounded $C^1$ domain is uniquely solvable if boundary data belongs to weighted $L^p$ space.
