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