difference quotient methods in L1

Question

Usually for some pde, if we want to prove that a given solution has higher derivatives, we proceed by difference quotient method.
For elliptic pdes, the result is usually that some derivative belongs to $L^2$.
I was wondering if there are examples of pdes, where we can apply the difference quotient method and get a derivative in a weaker space than $L^2$ (for instance, on a bounded domain, some $L^q$ for $q<2$) ?