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For $d\\ge3$ we prove that the Sobolev type estimate $\\|u\\|_{L^q(\\mathbb{R}^d)}\\le C \\|P(D)u\\|_{L^p(\\mathbb{R}^d)}$ holds with $C$ independent of the first order and the constant terms of $P(D)$ if and only if $1/p-1/q=2/d$ and $\\frac{2d(d-1)}{d^2+2d-4}<p<\\frac{2(d-1)}d$. We also obtain restricted weak type endpoint estimates for the critical $(p,q)=(\\frac{2(d-1)}{d},\\frac{2d(d-1)}{(d-2)^2})$, $(\\frac{2d(d-1)}{d^2+2d-4}, \\frac{2(d-1)}{d-2})$. 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For $d\\ge3$ we prove that the Sobolev type estimate $\\|u\\|_{L^q(\\mathbb{R}^d)}\\le C \\|P(D)u\\|_{L^p(\\mathbb{R}^d)}$ holds with $C$ independent of the first order and the constant terms of $P(D)$ if and only if $1/p-1/q=2/d$ and $\\frac{2d(d-1)}{d^2+2d-4}<p<\\frac{2(d-1)}d$. We also obtain restricted weak type endpoint estimates for the critical $(p,q)=(\\frac{2(d-1)}{d},\\frac{2d(d-1)}{(d-2)^2})$, $(\\frac{2d(d-1)}{d^2+2d-4}, \\frac{2(d-1)}{d-2})$. 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