BMO space and log-lipschitz regularity Let $\Omega$ an open bounded domain in $R^n$ with smooth boundary and consider a function $u \in W^{1,p}(\Omega) (2<p < +\infty)$ . Suppose that for every ball $B \subset \subset  \Omega$ exists a constant $A$ that does not depend on $B$ such that
$$\int_{B} |\nabla u - (\nabla u )_{B}|^p \leq A (r_{B})^{n},$$
where $r_B$  is the radius of the ball $B$ and
$$ (\nabla u )_{B}:= \left( \frac{1}{|B|} \int_{B} \frac{\partial u}{ \partial x_1},..., \frac{1}{|B|} \int_{B} \frac{\partial u}{\partial x_n}\right)$$
I am reading a paper and the author says that if the function satisfy the things above, then $u$ is locally log - lipschitz, that is, for every ball  $B \subset \subset  \Omega$ the is a constant K = K(B) such that
$$ |u(x) - u(y)| \leq K |x-y||log|x-y||.$$
The author says that this is true by theorem 3 of this paper, but I am not seeing how to apply such theorem ...
Someone could help me to understand the justification of the author or justify the affirmation  by other way?

The John-Nirenberg inequality implies that every BMO function $f$ is  in exponentially integrable (on domains of suitable shape: cubes or balls), meaning there is $\lambda>0$ such that $\exp(f/\lambda)\in L^1$. 
The exponential function is an Orlicz function with its conjugate being $\psi(t)=t\log t$.    Theorem 3 in Cianchi's paper yields the modulus of continuity $C\psi^{-1}(|x-y|^{-n})$. Since $\psi^{-1}(t)\sim t/\log t$ when $t \to\infty$, this is exactly the log-Lipschitz continuity.
