The method of layer potentials in L^p and endpoint spaces for elliptic operators with L^∞ coefficients

We consider layer potentials associated to elliptic operators Lu=−div(A∇u) acting in the upper half-space Rⁿ⁺¹₊ for n≥2, or more generally, in a Lipschitz graph domain, where the coefficient matrix A is L^∞- and t-independent, and solutions of Lu=0 satisfy interior estimates of De Giorgi/Nash/Moser type. A ‘Calderón–Zygmund’ theory is developed for the boundedness of layer potentials, whereby sharp L^p and endpoint space bounds are deduced from L²-bounds. Appropriate versions of the classical ‘jump relation’ formulae are also derived. The method of layer potentials is then used to establish well-posedness of boundary value problems for L with data in L^p and endpoint spaces.