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

Abstract

We prove square function estimates in L 2 for general operators of the form B 1 D 1 + D 2 B 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.