My intention is to understand how to apply the lemma below
Lemma:Let $\phi(s)$ be a non-negative and non-decreasing function. Suppose that \begin{equation} \phi(r) \le C_1 \left[\Bigl(\dfrac{r}{R}\Bigr)^\alpha + \mu \right]\phi(R) + C_2 R^\beta] \end{equation} for all $r\le R \le R_0$, with $C_1,\alpha,\beta$ positive constants. Then, for any $\sigma<\min\{\alpha\beta\}$ there exists a constant $\mu_0 = \mu_0(C_1,\alpha,\beta,\sigma)$ such that if $\mu<\mu_0$, then for for all $r\le R\le R_0$ we have \begin{equation} \phi(r)\le C_3\Bigl(\dfrac{r}{R}\Bigr)^\sigma \left[\phi(R) + C_2R^\sigma \right] \end{equation} where $C_3=C_3(C_1,\sigma-\min\{\alpha,\beta\})$ is a positive constant. In turn, \begin{equation} \phi(r)\le C_4r^\sigma, \end{equation} where $C_4=C_4(C_2,C_3,R_0,\phi,\sigma)$ is a positive constant.
To obtain the theorem 1.1 in here on the page 4. The part is the following: on the page 19, we get $$ \phi(r) \le C \Bigl( \dfrac{r}{R}\Bigr)^{n+p\alpha} \phi (R) + CR^{n + p \gamma /(p-\gamma)} + CR^{n+p(q-n)/(p-1)q} $$ where $\phi(r) = \int_{B_r}|\nabla u - (\nabla u)_{r}|^p dX$. I'd like that someone help me to explain the facts
- In view of Lemma above and $W^{1,p}$ bounds of $u$ we conclude \begin{equation} \int_{B_r(X_0)} | \nabla u- (\nabla u)_{r}|p \le Cr^\alpha, \end{equation}
2.Finally Campanato's embedding Theorem (see for example [17]) gives the desired Hölder continuity of the gradient of $u$.
3.(On page 20) In view of Lemma 2.7 we obtain \begin{equation} \int_{B_r(X_0)}| \nabla u - (\nabla u)_r|^pdX \le Cr^n \end{equation}
