A function is in a space $W^{1,2}_{-\tau}$ if $\int_{\mathbb{R}^n} f^2|x|^{2\tau-n}<\infty$ and $\int_{\mathbb{R}^n} |\partial_kf|^2|x|^{2\tau+2-n}<\infty$ for all $k=1,2,\dots n$.

Given a vector field $Z$ in a space of $W^{1,2}_{-\tau}$. Assume ($\mathbb{R}^n,\ g=g_{Euc}$), I would like to know if the following inequality holds. $$\int_{\mathbb{R}^n} |Z|^2||x|^{2\tau-n} \leq C \int_{\mathbb{R}^n} |L_Zg|^2|x|^{2\tau+2-n}.$$ Where $L_{Z}g$ refers to Lie derivative.

This inequality is actually found in a paper by J. Corvino and R. Schoen's paper on VEE (P.196-198). https://projecteuclid.org/download/pdf_1/euclid.jdg/1146169910 But for the vector field part, they omit the proof.

Thank you very much!


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