In the Lieb-Loss's book , they present the Riesz rearrangement in Section3, Theorem 3.9(Page 93). Note that the functions f,g,h are all nonnegative. I want to ask whether the nonnegative condition can be removed, such as g(x)=-ln(x). Because in come cases, such as the fundamental solution of -\Delta in \mathbb{R}^2 is -ln x. In this cases, does the Riesz's Rearrangement inequality still holds?

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?