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

Journal article, 2013

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.