Riesz rearrangement inequality

Question

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?

Answer 1

$\newcommand\R{\mathbb R}$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.