Endpoint weighted Fourier estimate with vanishing spherical trace

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 $$ \left\| \bigl|1-|\xi|^2\bigr|^{-1/2}\widehat{f}(\xi) \right\|_{L^2(\mathbb{R}^n)} \leq C_n \|f\|_{L^{p_c}(\mathbb{R}^n)}, $$ 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.