  In this work, we develop $L^p$ boundedness theory for pseudodifferential
operators with rough (not even continuous in general) symbols in the $x$
variable. Moreover, the $B(L^p)$ operator norms are estimated explicitly in
terms of scale invariant quantities involving the symbols. All the estimates
are shown to be sharp with respect to the required smoothness in the $\xi$
variable. As a corollary, we obtain $L^p$ bounds for (smoothed out versions of)
the maximal directional Hilbert transform and the Carleson operator.
