Stochastic homogenization: a short proof of the annealed Calderón–Zygmund estimate

Abstract

A building block for many field theories in continuum physics are second-order elliptic operators in divergence form, as given through a coefficient field which may be assimilated to a metric tensor field on \(\mathbb {R}^d\). The mapping properties of these linear operators are a crucial ingredient for analysis. In this paper, we focus on Calderón–Zygmund estimates, that is, on the boundedness of the corresponding Helmholtz projection in \(\mathrm{{L}}^p(\mathbb {R}^d)\)-spaces. Even when the coefficient field is uniformly smooth, this estimate may fail for p not close to 2. We seek an intrinsic criterion on the validity of the Calderón–Zygmund estimate in the whole range of \(p\in (1,\infty )\); intrinsic in the sense that it is formulated in terms of the scalar and vector potentials of the harmonic coordinates. We seek genericity in form of a statistical statement, and thus consider general ensembles of coefficient fields. Our criterion comes in form of finite stochastic moments for the potentials, or rather their corrections from being affine. In line with this, the Calderón–Zygmund estimates we obtain are annealed as opposed to quenched, meaning that there is an inner norm in form of a stochastic moment next to the (outer) \(\mathrm{{L}}^p\)-norm in space. This result grows out of recent progress in quantitative stochastic homogenization; it is ultimately inspired by the classical large-scale regularity theory of Avellaneda and Lin (Commun Pure Appl Math 40(6):803–847, 1987). More specifically, we provide an easier version of the proof given by us in Josien and Otto (J Funct Anal, 2022), albeit under stronger assumptions. Annealed Calderon–Zygmund estimates were first established in Duerinckx and Otto (Stoch Partial Differ Equ Anal Comput 8(3):625–692, 2020), and are a very convenient tool for error estimates in stochastic homogenization.