What is assumed when one write "$\nabla f$" for $f\in L^p(\mathbb{R}^n)$?
Here is a problem I'm dealing with for weeks.
For a fixed $a\in\mathbb{R}^n$, define $f_{a,r}:= \frac{1}{|B(a,r)|} \int_{B(a,r)} f(x) dx$. Suppose $|\nabla f|\in L^p(\mathbb{R}^n), 1\leq p <n, s>n-p$, and $\limsup_{r\to 0} \frac{1}{r^s}\int_{B(a,r)} |\nabla f(x)|^p dx <\infty$. Prove that $\lim_{r\to 0} f_{a,r} \neq \infty$.
This problem was on a qualifying exam for analysis, but this problem seems somewhat related to PDE. I'm not even sure what conditions on $f$ is assumed. Is it assumed differentiable, so that $\nabla f$ is well-defined? I guessed this, but then realized that if I assume the continuity of $f$, the limit is trivially $f(a)$, so this problem would be nonsense. I'm completely lost.
What is intended and how do I prove this?
For your interest, see the definition of the weak derivative. For your comment, by Lesbegue differentiation theorem, a measurable function has Lesbegue derivative almost everywhere (thus it is totally possible for $f(p)$ not equal its Lesbegue derivative)
The problem is related to Poincare inequality, which is proved in Evans (Chapter 4?). To see the optimal constant $s>n-p$, you can let $f(x)=f(r)=r^{-k}$ be a radial function, then check why $s=n-p$ gives you the desired blow-up cut-off . Mind that $B(a,r)\sim r^{n}$ by homogeneity and $dx\sim r^{n-1}drd\theta$ in all your calculations.