Let $0<\alpha<1$ and $1\leq p<\infty$. Suppose $f\in L^p(\mathbb{R}^n)$ satisfies \begin{align} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n+\alpha p}}dxdy<\infty \end{align} Is it true that \begin{align} \int_{B(0,r)}\frac{|f(x+h)-f(x)|^p}{|h|^{n+\alpha p}}dh\to 0\qquad\text{as}\qquad r\to 0 \end{align} for a.e. $x\in\mathbb{R}^n$?
Since this is an integration over the ball $B(0,r)$ (of radius $r$), at first I tried to look at Lebesgue-Besicovitch differentiation theorem, but the theorem actually involves the average (i.e. we need to take care of $|B(0,r)|^{-1}$ too). Now I have no idea where to start.
Any hint, comment and answer are greatly appreciated.
Writing $y = x+h$, this is $$ \int_{\mathbb R^n} \int_{\mathbb R^n} \frac{|f(x+h) - f(x)|^p}{|h|^{n+\alpha p}}\; dx \; dh < \infty$$ Therefore for a.e. $x$ we have $$ \int_{\mathbb R^n} \frac{|f(x+h) - f(x)|^p}{|h|^{n+\alpha p}}\; dh < \infty $$ Now use Dominated Convergence.