  We consider layer potentials associated to elliptic operators $Lu=-{\rm
div}(A \nabla u)$ acting in the upper half-space $\mathbb{R}^{n+1}_+$ for
$n\geq 2$, or more generally, in a Lipschitz graph domain, where the
coefficient matrix $A$ is $L^\infty$ and $t$-independent, and solutions of
$Lu=0$ satisfy interior estimates of De Giorgi/Nash/Moser type. A
"Calder\'on-Zygmund" theory is developed for the boundedness of layer
potentials, whereby sharp $L^p$ and endpoint space bounds are deduced from
$L^2$ 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.
