Layer potentials and boundary value problems for elliptic equations with complex L^∞ coefficients satisfying the small Carleson measure norm condition

We consider divergence form elliptic equations Lu:=∇{dot operator}(A∇u)=0 in the half space R₊ⁿ⁺¹:={(x,t)∈Rⁿ×(0,∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x, t)-A(x, 0) satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in L^p, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L, with data in L^p(Rⁿ), BMO(Rⁿ), and C^α(Rⁿ), and solvability of the Neumann and Regularity problems, with data in the spaces L^p(Rⁿ)/H^p(Rⁿ) and L₁^p(Rⁿ)/H^1,p(Rⁿ) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t-independent operator L₀:=-∇{dot operator}(A({dot operator}, 0)∇).