Let $f:\mathbb{R}^N \to \mathbb{R}_+$. What does the following condition imply on the behavior of $f$?
$$\limsup_{r \to 0} r^{-p}⨍_{B(x_0,r)} f^{q}(x) \ \mathrm{d}x < \infty,$$ $$p,q\ge 1, x_0 \in \mathbb{R}^N.$$
I thought we could use Lebesgue's differentiation lemma in some way, but I'm not clear how.
To make sense of the statement assume that $f$ is locally in $L^q(\mathbb{R}^n)$, so $f^q$ is locally in $L^1(\mathbb{R}^n)$. By Lebesgue Differentiation theorem we have that, a.e. $$ f(x)^q = \lim_{r \to 0^+} ⨍_{B(x,r)} f(y)^q \,d\,y = \limsup_{r \to 0^+} ⨍_{B(x,r)} f(y)^q \,d\,y $$ But, by hypothesis, you have that $$ ⨍_{B(x,r)} f(y)^q \,d\,y \leq C \, r^p $$ for $r$ small enough and $C$ large. Thus, $\, f(x)^q = 0$ almost everywhere.
Remark: The Hölder condition I was referring to appears when you substract the original function. I.e. the following condition is equivalent to be Hölder of exponent $0 < \alpha < 1$ $$ \Big| \, f(x) - ⨍_{B(x,r)} f(y) \, d \, y \, \Big| \leq C r^\alpha. $$