Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it.
As you can see in Leoni (or in Lieb & Loss), the decreasing and the increasing rearrangement of a measurable function $u:\Omega \to [0,\infty[$ ($\Omega \subseteq \mathbb{R}^N$ measurable and bounded) are respectively defined by: $$\begin{split} u^*(s) &:= \inf \{ t>0:\ \mathcal{L}^N (\{ u>t\}) \leq s\} \\ u_*(s) &:= u^*(\mathcal{L}^N(\Omega) -s)\; .\end{split}$$
A remarkable property of rearrangements is the following chain of inequalities, known as Hardy-Littlewood inequalities:
Let $u,v:\Omega \to [0,\infty[$ be measurable.
Whenever the three integrals make sense, we have: $$\tag{HL} \int_0^{\mathcal{L}^N(\Omega)} u_*(s)v^*(s)\ \text{d} s \leq \int_\Omega u(x)v(x)\ \text{d} x\leq \int_0^{\mathcal{L}^N(\Omega)} u^*(s)v^*(s)\ \text{d} s\; .$$
Such inequalities are the continuous versions of the classical rearrangement inequalities for discrete sets of numbers.
I am interested in the equality case in (HL).
As far as the rightmost inequality is concerned, there is a vast literature on the subject. For example, it is known that if equality occurs, then the level sets of $u$ and $v$, i.e. the sets $\{u>t\}$ and $\{v>\tau\}$, are mutually nested; in other words:
If equality holds in $\int_\Omega u(x)v(x)\ \text{d} x\leq \int_0^{\mathcal{L}^N(\Omega)} u^*(s)v^*(s)\ \text{d} s$, then forall $t>0$ there exists $\tau >0$ s.t. $\{u>t\}=\{v>\tau\}$ (up to a null set).
Moreover, it is also known that if $v^*$ (say) is strictly decreasing, then the only function attaining equality in $\int_\Omega u(x)v(x)\ \text{d} x\leq \int_0^{\mathcal{L}^N(\Omega)} u^*(s)v^*(s)\ \text{d} s$ is: $$u_v(x):= u^*\Big( \mathcal{L}^N(\{ v>v(x)\})\Big)\; .$$
(The latter part is a generalization of a statement in Lieb & Loss.)
On the other hand, as far as the leftmost inequality is concerned, I found almost nothing interesting in the literature.
I'm induced to conjecture that equality holds in $\int_0^{\mathcal{L}^N(\Omega)} u_*(s)v^*(s)\ \text{d} s \leq \int_\Omega u(x)v(x)\ \text{d} x$ if the level set of $u$ are nested into the complements of level sets of $v$, viz. that $\{u>t\}=\{v\leq \tau\}$ or something like that.
But until now I was not able to prove it... I was trying to mimic the proof of Alvino, Lions & Trombetti, but it seems not to work.
Any hints? Thanks a lot!
