Revista Matemática Iberoamericana
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Published online: 2016-06-08
Estimates for Fourier transforms of surface measures in $\mathbb R^3$ and PDE applicationsMichael Greenblatt (1) University of Illinois at Chicago, USA
An explicit local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in $\mathbb R^3$, as well as to provide new proofs of some known estimates. These are then used to give $L^q$ bounds on solutions to certain PDE problems in terms of the $L^p$ norms of their initial data for various values of $p$ and $q$.
Keywords: Oscillatory integral, Fourier transform
Greenblatt Michael: Estimates for Fourier transforms of surface measures in $\mathbb R^3$ and PDE applications. Rev. Mat. Iberoam. 32 (2016), 419-446. doi: 10.4171/RMI/890