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$L^p$ integrability of functions with Fourier support on a smooth space curve
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abstract
We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of Agranovsky and Narayanan, and the exponents are sharp in all dimensions. In the process, we develop a mechanism that should lead to further progress on related problems.
Forward citations
Cited by 2 Pith papers
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Weighted $L^2$ restriction and comparison of nondegeneracy conditions for quadratic manifolds of arbitrary codimensions
The authors give sharp uniform Fourier decay and weighted L2 restriction for all quadratic manifolds, and an almost complete implication diagram among nondegeneracy conditions.
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$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve
If a function lies in L^p and its Fourier transform is supported on a small fractal set on the moment curve, then for p up to a sharp threshold (d^2+d+2α)/(2α) for d≥3, and 4/α for d=2, the function is identically zero.
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