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In the Lieb-Loss's book Analysis, they present the Riesz rearrangement in Section 3, Theorem 3.9 (page 93). Note that the functions $f, g, h,$ are all nonnegative. I want to ask whether the nonnegativity condition can be removed in order to deal with e.g. $g(x)=-\ln(x)$, because in some cases, such as for the fundamental solution of $-\Delta$ in $\mathbb{R}^2$, functions like $-\ln x$ have to be considered. In these cases, does the Riesz's Rearrangement inequality still holds?

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$\newcommand\R{\mathbb R}$The Riesz rearrangement inequality $$\iint_{\mathbb{R}^n\times \mathbb{R}^n} f(x) g(x-y) h(y) \, dx\,dy \\ \le \iint_{\mathbb{R}^n\times \mathbb{R}^n} f^*(x) g^*(x-y) h^*(y) \, dx\,dy\tag{1}$$ will hold for $g$ of any signs if $f,g,h$ are integrable.

Indeed, then, with any constant $N>0$ in place of $g$, both sides of (1) will equal the same real number. Also, (1) will hold with the nonnegative function $\max(0,N+g)$ in place of $g$. So, (1) will hold with the function $g_N:=\max(-N,g)=\max(0,N+g)-N$ in place of $g$. We also have $|g_N|\le|g|$, $g_N\to g$ pointwise as $N\to\infty$, and $(g_N)^*=(g^*)_N$. So, (1) follows by dominated convergence.

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  • $\begingroup$ Thanks a lot. So without the nonnegative condition of g, if g is strictly symmetric decreasing, then whether the "=" holds if and only if f, h is a transtation of f^* and h^*?(compared to <Analysis>, Theorem 3.9, strict rearrangement inequality) $\endgroup$
    – W.J.
    Apr 4, 2021 at 15:58
  • $\begingroup$ @wanjie : I don't have the book. Anyhow, if you have any additional questions, it is better to ask them in separate posts. $\endgroup$ Apr 4, 2021 at 16:17
  • $\begingroup$ I believe OP has done that, mathoverflow.net/questions/389317/… $\endgroup$ Apr 4, 2021 at 23:17
  • $\begingroup$ @GerryMyerson : Thank you for letting me know. $\endgroup$ Apr 5, 2021 at 0:27
  • $\begingroup$ @wanjie : So, are you satisfied with this answer? $\endgroup$ Apr 8, 2021 at 19:20

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