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Specifically, let $\\{\\Delta_j\\}$ be regions of the unit sphere $\\mathbb{S}^{n-1}$ and let $S_j f$ be the smooth Fourier restriction of $f$ to the conical region $\\{\\xi\\in\\mathbb{R}^n:\\xi/|\\xi|\\in\\Delta_j\\}$. We are interested in the following estimate\n  $$\\Big\\|(\\sum_j|S_jf|^2)^{1/2}\\Big\\|_p\\lesssim_\\epsilon \\delta^{-\\epsilon}\\|f\\|_p.$$\n  The first result is: when $\\{\\Delta_j\\}$ is a set of disjoint $\\delta$-balls, then the estimate holds for $p=4$. 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