A critical weighted Fourier estimate on the spherical zero-trace subspace

Let n ≥ 2 and p = 2(n + 1)/(n + 3). We establish a critical weighted Fourier estimate for functions in Lᵖ(ℝⁿ) whose Fourier transforms have zero spherical trace in the Stein–Tomas sense. The main ingredient is a Hilbert-valued sequence estimate combining unconditional cancellation with critical decay of the Gram matrix. Fixed-spacing sampling identifies the zero-trace condition with unconditional cancellation, and windowed Fourier reconstruction connects the singular normal energy to the discrete block energy. We also derive a radius scaling law, a necessary upper bound on the exponent, an exact finite-support distance formula, and failure of the discrete estimate at its limiting exponents. Status: This manuscript is a preprint and has not undergone peer review. Declaration of generative AI use: OpenAI Codex was used to assist with drafting, translation, manuscript organization, proof exposition, checking mathematical arguments, reference searches and checks, and LaTeX preparation. The assistance extended to the discrete argument and its continuous transfer and was not limited to language editing. AI-assisted checks do not constitute independent verification. The author retains full responsibility for reviewing and validating the mathematical arguments, references, and final content. Further details are provided in the manuscript.