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Here, for the weighted version of the Hardy Inequality, I refer to Muckenhoupt's formulation in Theorem 1 of 1

Sobolev Inequality: $$C_d \int_{\mathbb{R}^d} \vert \nabla \phi \vert^2 \geq \left( \int_{\mathbb{R}^d} \vert \phi(x)\vert^{2d/(d-2)} \right)^{\frac{d-2}{d}} \qquad $$

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  • $\begingroup$ Theorem 1 in Muckenhoupt's paper is not the hardy inequality but a conditions equivalent to it. In any case Muckenhoupt deals with inequalities in dimension 1 while your inequality is in dimension $d$. You should find references to similar higher dimensional Hardy type inequalities and the literature is vast. $\endgroup$
    – Piotr Hajlasz
    Commented Dec 16, 2018 at 21:37

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