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Let $\Omega \subset R^n$ an open bounded domain and consider $B_r(x_0) \subset \Omega$ an open ball. Let $u \in W^{1,p}(\Omega)$ ($p \geq 2$). Let $s > n$ and suppose that $\int_{B_r(x_0)} |\nabla u| ^s < \infty$.Does the following inequality is true ?

$$ |u(x) - u(x_0)| \leq C(n,p,\Omega) |x - x_0| ^{1 - \frac{n}{s}} (\int_{B_{r}(x_0)} |\nabla u| ^s )^{1/s} $$

I am asking this because of the corolary 27.6 of this lectures (it is a general $C¹(R^n) $ case of the above inequality ): http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Sobolev%20Inequalities.pdf

I dont need a proof , just a reference.

Any help will be apreciated!

Thanks in advance

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Look in Chapter 5 of Partial Differential Equations by Evans. Of course the inequality requires $|x - x_0| \le r$ but the proof in Evans may have a stricter requirement.

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  • $\begingroup$ I think the problem is whether $p^*$ greater then $s$ or not. $\endgroup$ – spatially Dec 11 '14 at 0:33
  • $\begingroup$ You're thinking of the Sobolev inequality, not Morrey's inequality. $\endgroup$ – Umberto P. Dec 11 '14 at 14:47
  • $\begingroup$ What I was worried is that $u$ may not good enough to get in $L^s$ so that you can not apply Morrey's inequality. $\endgroup$ – spatially Dec 12 '14 at 1:50
  • $\begingroup$ But $|\nabla u| \in L^s$ is given. $\endgroup$ – Umberto P. Dec 12 '14 at 2:33

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