Bounded weak solutions with Orlicz space data: an overview

It is well known that non-negative solutions to the Dirichlet problem $\mathrm{\Delta }u=f$ in a bounded domain $\mathrm{\Omega }$, where $f\in L^q(\mathrm{\Omega })$, $q>\frac{n}{2}$, satisfy $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^q(\mathrm{\Omega })}$. We generalize this result by replacing the Laplacian with a degenerate elliptic operator, and we show that we can take the data $f$ in an Orlicz space $L^A(\mathrm{\Omega })$ that, in the classical case, lies strictly between $L^{\frac{n}{2}}(\mathrm{\Omega })$ and $L^q(\mathrm{\Omega })$, $q>\frac{n}{2}$.