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We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in\mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complexelliptic, bounded and measurable. In addition, we suppose that $A$ satisfiessome additional regularity in the direction transverse to the boundary, namelythat the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition ofFefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions,we establish a full range of boundedness results for double and single layerpotentials in $L^p$, Hardy, Sobolev, BMO and H\"older spaces. Furthermore, weprove solvability of the Dirichlet problem for $L$, with data in$L^p(\mathbb{R}^n)$, $BMO(\mathbb{R}^n)$, and $C^\alpha(\mathbb{R}^n)$, andsolvability of the Neumann and Regularity problems, with data in the spaces$L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n)$ and$L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n)$ respectively, with the appropriaterestrictions on indices, assuming invertibility of layer potentials in for the$t$-independent operator $L_0:= -\nabla\cdot(A(\cdot,0)\nabla)$.

Layer potentials and boundary value problems for elliptic equations with  complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm  condition