Square function estimates on layer potentials for higher-order el- liptic equations

In this paper we continue towards the goal of resolving the Dirichlet and Neumann problems for general divergence form higher order elliptic operators with L data. The investigation of the second-order case has spanned the past three decades in the subject, drawing from the field of harmonic analysis and giving back to it many tools, and by now the real coefficient case is relatively well understood. However, there are still open problems in the theory of even the simplest higher order operators, such as the bilaplacian; for instance, the sharp range of p for which the Dirichlet problem for the bilaplacian is well-posed in L is still not known in high dimensions. Even less is known in the case of more complicated operators; indeed, we are not aware of any L well-posedness results that are currently available in the general variable coefficient case. In this work we aim to develop the method of layer potentials for general divergence form higher order elliptic operators. The main results of the present paper are square function estimates for single and double layer potentials in L and the corresponding Sobolev spaces. We remark that one of the key difficulties in this context lies in the definition of suitable layer potentials and, more generally, of Dirichlet and Neumann boundary data, as in the higher order case there is considerable ambiguity, some choices leading to ill-posed problems. Our approach is new even in the constant coefficient context, but is carefully crafted to handle the general case. Let us discuss the background and the results in more detail. In this project we study elliptic differential operators of the form