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

Abstract

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 $X=L^{\ddot{I}}ˆ$ is treated first. Under suitable assumptions on the Young function $\ddot{I}ˆ$ and on the coefficients, the optimal associated space for the lower-order datum is identified. An \textit{a priori} estimate in $W^{1,\ddot{I}ˆ}$ 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.