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I want to know that the Fourier transform of $\log(|x|)$ behaves like $1/|k|^d$ when tested against Schwartz functions that vanish at $0$? I know one can explicitly compute the Fourier transform in $d=1,2$ (as is done in the book by Vladimirov, https://archive.org/details/vladimirov-equations-of-mathematical-physics, Chapter 2, Section 9) and prove this but I am interested in this statement in arbitrary dimensions.

Can anyone provide me with a simple proof of this statement (is it even true)?

almosteverywhere
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    Since for radial functions, the Fourier transform and its inverse are the same, the answer to this question is given here The Fourier transform of $1/p^3$. Does it answer your question? If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – LL 3.14 Jun 20 '25 at 22:28
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    it answers my question. thank you very much! – almosteverywhere Jun 21 '25 at 12:49

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