Multi Layer Potentials and Boundary Problems : for Higher Order Elliptic Systems in Lipschitz Domains 1 Introduction 2 Smoothness scales and Caldeón Zygmund theory in the scalar valued case 3 Function spaces of Whitney arrays 4 The double multi layer potential operator 5 The single multi layer potential operator 6 Functional analytic properties of multi layer potentials and boundary value problems Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems including boundary integral methods variational methods harmonic measure techniques and methods based on classical harmonic analysis. When the differential operator is of higher order (as is the case e.g. with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón one of the founders of the modern theory of Singular Integral Operators advocated the use of layer potentials for the treatment of higher order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach. This research monograph lays for the first time the mathematical foundation aimed at solving boundary value problems for higher order elliptic operators in non smooth domains using the layer potential method and addresses a comprehensive range of topics dealing with elliptic boundary value problems in non smooth domains including layer potentials jump relations non tangential maximal function estimates multi traces and extensions boundary value problems with data in Whitney Lebesque spaces Whitney Besov spaces Whitney Sobolev based Lebesgue spaces Whitney Triebel Lizorkin spaces Whitney Sobolev based Hardy spaces Whitney BMO and Whitney VMO spaces