Pseudodifferential Operators with Rough Symbols

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 ξ variable. As a corollary, we obtain L ^p bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator.