We obtain some sharp Lᵖ weighted Fourier restriction estimates of the form \|Ef\|_Lᵖ(Bⁿ⁺¹(0,R),Hdx) Rβ\|f\|₂, where E is the Fourier extension operator over the truncated paraboloid, and H is a weight function on Rⁿ⁺¹ which is n-dimensional up to scale √ R.
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Du et al. (2024) studied this question.