Holder function and weak derivatives

Question

Let $B_1$ the $n$-dimensional unit ball and $u\in C^{0,\alpha}(B_1)$ be an $\alpha$-Holder continuous function, for some $\alpha \in (0,1)$, such that $u(0)=0$. Is it true that $$ \int_{B_{1/2}} |x \cdot \nabla u| \,\mathrm{d}x < + \infty? $$ I have no clue about this question I found also some difficulties in the construction of a possible counter-example. I initially thought that the hyphotesi $u\in C^{0,\alpha}(B_1)$ was not enough to ensure the bound of the integral but it was necessary to have some sort of non-degeneracy like $$ 1/c\leq \mbox{sup}\left\{\frac{|u(x)-u(y)|}{|x-y|^\alpha},x,y\in B_{1/2},x\neq y\right\}\leq c, $$ for some $c>1$.