Let $E := \mathbb R^d$. Let $f:E \to \mathbb R_{>0}$ be continuous and integrable. For $r>0$, we define $$ f_r (x) := \frac{f(x)}{ \frac{1}{|B(x, r)|} \int_{B(x, r)} f(y) \, \mathrm{d} y} \quad \forall x \in E. $$
Here $|B(x, r)|$ is the Lebesgue measure of the open ball $B(x, r)$ whose center is $x$ and radius is $r$. So $f_r (x)$ is the ratio of $f(x)$ and the average value of $f$ over $B(x, r)$. I'm interested in how properties of $f$ are translated into those of $f_r$.
Have $f_r$ been a subject of study in some areas of mathematics? Any reference is greatly appreciated!
