# fa.functional analysis – Strict Riesz’s rearrangement inequality

(Continue to the last question Riesz rearrangement inequality) In the Lieb-Loss’s book , they present the strict Riesz rearrangement inequality in Section3, Theorem 3.9(Page 93). They say that when the functions f,g,h are all nonnegative, and if g is strictly symmetric decreasing, then Riesz rearrangement inequality holds and the “=” holds iff f and h are translation of $$f^*, h^*$$. Namely if f,g,h are all nonnegative, then
$$iint_{mathbb{R}^ntimes mathbb{R}^n} f(x) g(x-y) h(y) , dx,dy \ le iint_{mathbb{R}^ntimes mathbb{R}^n} f^*(x) g^*(x-y) h^*(y) , dx,dytag{1}$$
and if g is strictly symmetric decreasing, then there is a equality only of $$f=T(f^*), h=T(h^*)$$ for some translation $$T$$. I want to ask when remove the nonnegative condition, such as g(x)=-ln(x), whether the “=” holds iff f and h are translation of $$f^*, h^*$$. For example, let $$g(x)=-ln x$$, which is strictly symmetric decreasing. In this cases, we know that (1) still holds. Does the equality holds in (1) only if f and h are a translation of $$f^*, h^*$$?