The $L^p$ Neumann problem for higher order elliptic equations

We solve the Neumann problem in the half space ${\mathbb{R}}_{+}^{n+1}$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in $L^p$, where $\max (1,\frac{2n}{n+2}-\epsilon )<p<2$. We also establish nontangential and area integral estimates on layer potentials with inputs in $L^p$ or ${\dot{W}}^{\pm 1,p}$ for a similar range of~$p$, based on known bounds for $p\ge 2$; in this case we may relax the requirement of self-adjointess.