  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
$\max\bigl(1,\frac{2n}{n+2}-\varepsilon\bigr) < p < 2$.
  We also establish nontangential and area integral 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 $p\geq2$; in this case we may relax the
requirement of self-adjointess.
