Square function and maximal function estimates for operators beyond divergence form equations

TL;DR

This paper extends square function and maximal function estimates to a broader class of differential operators beyond divergence form equations, using novel techniques that do not rely on interpolation or $L_p$ estimates.

Contribution

It introduces new $L_2$ estimates for operators of the form $B_1D_1+D_2B_2$ with independent bounded accretive operators, broadening the scope of previous results.

Findings

Established $L_2$ square function estimates for new operator class.

Proved non-tangential maximal function estimates with minimal resolvent decay assumptions.

Extended the framework from the Kato square root problem to more general operators.

Abstract

We prove square function estimates in $L_2$ for general operators of the form $B_1D_1+D_2B_2$, where $D_i$ are partially elliptic constant coefficient homogeneous first order self-adjoint differential operators with orthogonal ranges, and $B_i$ are bounded accretive multiplication operators, extending earlier estimates from the Kato square root problem to a wider class of operators. The main novelty is that $B_1$ and $B_2$ are not assumed to be related in any way. We show how these operators appear naturally from exterior differential systems with boundary data in $L_2$. We also prove non-tangential maximal function estimates, where our proof needs only off-diagonal decay of resolvents in $L_2$, unlike earlier proofs which relied on interpolation and $L_p$ estimates.