Global Marcinkiewicz estimates for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg" display="inline" id="d1e26"> <mml:mi>p</mml:mi> </mml:math> -Laplace equations in composite media: A new free-scaling approach via distributi

We study the regularity of solutions to nonlinear elliptic equations of p -Laplace type modeled from composite materials. The main difficulty comes from the geometric structures of the composite, specifically the disjoint Reifenberg flat subdomains Ω j , their boundaries ∂ Ω j , and the BMO smallness properties of each tensor coefficient A j that pose significant challenges. In this paper, we develop a novel free-scaling approach to establish the local decay estimates for level sets of the gradient of the weak solutions. This approach is of independent technical interest, and it is flexible enough to be applied for deriving improved gradient regularity in a larger class of rearrangement-invariant function spaces.