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

Abstract

We investigate singular integral operators with variable Calderón--Zygmund kernels and their commutators with $VMO$ functions on Orlicz spaces. After revisiting the classical $L^p$ theory, we establish boundedness results in $L^{\Phi }$ under the standard ${\Delta }_2$ and ${\nabla }_2$ conditions on the Young function. The analysis combines decomposition techniques with weak-type estimates to derive the main operator bounds. As an application, these results provide a functional-analytic framework for establishing a priori estimates and proving the interior regularity of solutions to higher-order elliptic equations and systems with discontinuous coefficients.

17 pages