  Recently, Auscher and Axelsson gave a new approach to non-smooth boundary
value problems with $L^{2}$ data, that relies on some appropriate weighted
maximal regularity estimates. As part of the development of the corresponding
$L^{p}$ theory, we prove here the relevant weighted maximal estimates in tent
spaces $T^{p,2}$ for $p$ in a certain open range. We also study the case
$p=\infty$.
