Proof of the Fourier extension conjecture on the paraboloid in d>2 by decomposing smooth Alpert projections, applying a bilinear reduction, and bounding the resulting oscillatory integral with periodic amplitude via lattice averaging and stationary phase.
A probabilistic analogue of the Fourier extension conjecture
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abstract
We prove a probabilistic Fourier extension theorem that says Fourier extension holds when averaged over certain smooth Alpert multipliers. The proofs use smooth Alpert wavelets with the classical techniques of stationary phase and interpolation of L^2 and L^4 estimates. The correct L^4 bounds for resonant forms require an expectation over Alpert multipliers.
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Discusses two alternative proofs of Fefferman's Fourier extension theorem using decoupling and wavelet decompositions, with one method extended to higher dimensions.
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The Fourier extension conjecture for the paraboloid
Proof of the Fourier extension conjecture on the paraboloid in d>2 by decomposing smooth Alpert projections, applying a bilinear reduction, and bounding the resulting oscillatory integral with periodic amplitude via lattice averaging and stationary phase.
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A discussion of two new proofs of Fefferman's Fourier extension theorem in the plane
Discusses two alternative proofs of Fefferman's Fourier extension theorem using decoupling and wavelet decompositions, with one method extended to higher dimensions.