Suppose $N \ge 3$ and let $\Phi(x):= C\_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$. 


Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \partial \Omega$. Let $ \delta(x):=dist(x,\partial \Omega)$. 


Question I am interested in. I am interested in obtaining bounds on $ \| \nabla u \|\_{L^p(\Omega)}$ for certain ranges of $p$ assuming, for instance, $ \| f \delta^\alpha \|\_{L^1(\Omega)}$ is finite and where $ \alpha \in [0,1]$.


If $ \alpha=0$ it seems (at least formally) that one can just play with Newtonian potential of $f$ to get the desired estimates. If $ \alpha=1$ i understand the duality proof on how to obtain $L^p$ bounds on the solution $u$; but to be its not clear whether one can use the Newtonian potential to obtain bounds on gradient of $u$. 


So for a precise question set $v(x):= (f \ast \Phi)(x)$ and assume $ \| f \delta^\alpha \|\_{L^1(\Omega)}$ is finite. For what values of $ \alpha$ can we obtain a gradient $L^p$ bound on the full domain $\Omega$. Moreover do the estimates hold on solutions of the above pde with $u=0$ on $\partial \Omega$. 
I am not very familiar with Newtonian potentials and so i am sorry if the above is complete nonsense. 
regards
greg



Gradient bounds on Newtonian potentials
Take a look at


[1] J. I. Díaz and J.-M. Rakotoson, “On the differentiability of very weak solutions with right hand side data integrable with respect to the distance to the boundary,” J. Funct. Anal., vol. 257, no. 3, pp. 807–831, 2009.


