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.
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