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.

  • 2
    $\begingroup$ 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? $\endgroup$ – Robert Israel May 14 at 0:40
  • $\begingroup$ ( 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 ) $\endgroup$ – Calvin Khor May 14 at 0:42
  • $\begingroup$ @RobertIsrael Sorry, my fault, you are right. The question is now edited. $\endgroup$ – Hopf eccentric May 14 at 0:59
  • $\begingroup$ @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. $\endgroup$ – 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.


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