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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 coefficienthomogeneous first order self-adjoint differential operators with orthogonalranges, and $B_i$ are bounded accretive multiplication operators, extendingearlier estimates from the Kato square root problem to a wider class ofoperators. The main novelty is that $B_1$ and $B_2$ are not assumed to berelated in any way. We show how these operators appear naturally from exteriordifferential systems with boundary data in $L_2$. We also prove non-tangentialmaximal function estimates, where our proof needs only off-diagonal decay ofresolvents in $L_2$, unlike earlier proofs which relied on interpolation and$L_p$ estimates.

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