Schauder--Orlicz-Type Estimates for Divergence-Form Elliptic Equations with Lower-Order Terms

Schauder Orlicz-type estimates are derived for weak solutions to second-order linear elliptic equations in divergence form with lower-order terms. The Orlicz setting $X=L^{\psi }$ is treated first. Under suitable assumptions on the Young function $\psi$ and on the coefficients, the optimal associated space for the lower-order datum is identified. An $\mathit{\text{a}}\mathit{\text{priori}}$ estimate in $W^{1,\psi }$ is then obtained. The discussion is next extended to rearrangement-invariant Banach function spaces. A class $(\mathcal{C})$ is introduced to characterize the spaces $X$ for which a corresponding associated space $Y$ yields Schauder-type estimates. Lorentz spaces are finally examined as concrete examples.