  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 the negative smoothness
space $\dot W^{-1,p}$, where $\max(0,\frac{1}{2}-\frac{1}{n}-\varepsilon)
<\frac{1}{p} <\frac{1}{2}$. Our arguments are inspired by an argument of Shen
and build on known well posedness results in the case $p=2$.
  We use the same techniques to establish nontangential and square function
estimates on layer potentials with inputs in $L^p$ or $\dot W^{\pm1,p}$ for a
similar range of $p$, based on known bounds for $p$ near $2$; in this case we
may relax the requirement of self-adjointess.
