Take $\Omega$ to be a bounded domain in $N$ dimensional Euclidean space with smooth boundary and we assume $\Omega$ contains the origin. I am interested is the following equation $$ \Delta \phi(x) + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} \phi_{x_i x_j}(x) = f(x)$$ in $\Omega$ with $ \phi=0$ on $ \partial \Omega$. Here $\gamma$ is such that the operator is elliptic. I would like to get some estimate of the form $$ \sup_{\Omega}| \nabla \phi(x)| \le C \sup_\Omega |f(x)|.$$ I guess there might be some issues at the origin. I have tried cutting the domain off and trying to get estimates independent of the domain cut off; but I was not successful. I guess part of my problem is that I don't truly understand the estimates I use; but normally I can perturb off the Lapclian and not really need to understand the main ideas. **QUESTION**. Is this gradient estimate true and are there any standard methods to try and prove this fact? The method I tried was the $C^{1,\alpha}$ blow up analysis method from Leon Simon; but I couldn't get it to work. thanks