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 $n\ge 2$ 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 $${\Vert |1-|\xi {|}^2{|}^{-1/2}\hat{f}(\xi )\Vert }_{L^2({\mathbb{R}}^n)}\le C_n\Vert f{\Vert }_{L^{p_c}({\mathbb{R}}^n)},$$ 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.