We shall present several classical boundary value problems for the Laplacian on Graph Lipstchitz Domains in the complex plane which can be solved using Harmonic Analysis techniques via the use of Muckenhoupt weights. We will start by introducing these weights and their main properties, present the original technique due to C. Kenig in the 80?s dealing with the Dirichlet problem and move up to a recent result related with the solution of a transmission problem. This is a joint work with Teresa Luque (Univ. Complutense de Madrid, Spain), Virginia Naibo (Kansas State University), Carmen Ortiz-Caraballo (Univ. of Extremadura, Spain) and your former PhD student Mar?a Soria (Rutgers University).