We extend Wolff's "local smoothing" inequality to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat direction. An immediate consequence is the L-boundedness of the corresponding Fourier multiplier operators.
In this paper, we formulate necessary conditions for decay rates of L operator norms of weighted oscillatory integral operators R and give sharp L estimates and nearly sharp L estimates.
We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators, generalized Radon transforms and singular oscillatory integrals.
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