On Pareto functions

Question

The Pareto principle says that the top 20% of wealthy people people hold over 80% of the wealth. Suppose we had a non-negative function on $\mathbb R^n$ that satisfied this principle on every open ball. What can be said about the regularity of the function?

We say a locally integrable, non-negative function $f$ on $\mathbb R^n$ is a Pareto function if there exist $\varepsilon, \delta > 0$ such that for every open ball $B \subset \mathbb R^n$, there exists a subset $E$ of $B$ of measure less than or equal to $(\frac{1}{2} - \varepsilon) |B|$ such that

$$\int_E f \geq \left(\frac{1}{2} + \delta\right) \int_B f.$$

Question: What can be said about the regularity of a Pareto function $f$? Further, what bounds on the relevant norms can be obtained? Weak $L^1$ bounds up to a constant seem possible, but can we obtain weak $L^p$ or $BMO$ regularity?

Answer 1

This is possible only if $f=0$ a.e. Otherwise there exists a point $a$ for which $f(a) >0$ and $a$ is a Lebesgue point of $f$, that is, for $r\to 0$ the average of $|f(x)-f(a)|$ over the ball centered in $a$ of radius $r$ goes to 0. This yields that the average of $f$ over this ball $B$ is $f(a)+o(1)$, but then $\int_E f\geqslant (1/2+\delta+o(1))f(a)\mu (B)$ and \begin{align} & \int_{B\setminus E} f(x)\geqslant \int_{B\setminus E} f(a)-\int_{B\setminus E} |f(x)-f(a)| \\[8pt] \geqslant {} & \int_{B\setminus E} f(a)-\int_{B} |f(x)-f(a)| \\[8pt] \geqslant {} & f(a)(1/2+\varepsilon+o(1))\mu(B), \end{align} summing up these two bounds we get $\int_B f\geqslant (1+\varepsilon+\delta+o(1))\mu(B)f(a)$, contradicting to $\int_B f=(1+o(1))\mu(B)f(a)$.