Year: 2026
Type: preprint
Abstract: We establish an endpoint weighted Fourier estimate under a vanishing restriction condition on the unit sphere. Let $n \geq 2$ and $p_c = 2(n+1)/(n+3)$. We prove that every $f \in L^{p_c}(\mathbb{R}^n)$ with zero spherical Fourier restriction satisfies $$ \left\| \bigl|1-|ξ|^2\bigr|^{-1/2}\widehat{f}(ξ) \right\|_{L^2(\mathbb{R}^n)} \leq C_n \|f\|_{L^{p_c}(\mathbb{R}^n)}, $$ where the restriction condition is understood through the continuous Tomas--Stein restriction operator. This result gives a negative answer to Goldberg's counterexample-existence question and supplies a missing endpoint.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G2371208817
OpenAlex ID: W7223359524