# A limit of an integral of a quotient related to fractional Sobolev space

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.

• Are you sure that shouldn't be $|f(x+h) - f(x)|^p$ instead of $|f(x+h)-f(h)|^p$ in your second integral? – Robert Israel May 14 at 0:40
• ( The first condition is saying that a Gagliardo seminorm is finite, i.e. $f\in W^{\alpha,p}(\mathbb R^n)$, c.f. e.g. arxiv.org/pdf/1104.4345.pdf ) – Calvin Khor May 14 at 0:42
• @RobertIsrael Sorry, my fault, you are right. The question is now edited. – Hopf eccentric May 14 at 0:59
• @CalvinKhor Yeah I'm aware of that. In fact I encounter this problem when studying the $W^{\alpha,p}$ norm. But i'm still not sure how is that related to my question. – Hopf eccentric May 14 at 1:08

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.