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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

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

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?

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?

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

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

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?

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

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

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?

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 measurable 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

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

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?

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

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

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?