Interior a priori estimate for higher order elliptic systems in Orlicz spaces

Abstract

We study singular integral operators with variable Calderón--Zygmund kernels and their commutators with $VMO$ functions in the framework of Orlicz spaces. After revisiting the classical $L^p$ theory, we establish boundedness results in $L^Φ$ under standard $Δ_2$ and $\nabla_2$ conditions on the Young function. The proofs rely on decomposition techniques and weak-type estimates. As an application, these results provide a functional-analytic foundation for a priori estimates and interior regularity of solutions to higher-order elliptic operators with discontinuous coefficients.