Pseudodifferential operators with rough symbols

Details Pseudodifferential operators with rough symbols Pseudodifferential operators with rough symbols

2007-01-29

arxiv.org/abs/math/0701856v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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