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

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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.

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Authors and Affiliations

  1. Matematiska vetenskaper, Chalmers tekniska högskola och Göteborgs universitet, 412 96, Göteborg, Sweden

    Andreas Rosén

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  1. Andreas Rosén
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Correspondence to Andreas Rosén.

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Formerly Andreas Axelsson. Supported by Grant 621-2011-3744 from the Swedish research council, VR.

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Rosén, A. Square function and maximal function estimates for operators beyond divergence form equations. J. Evol. Equ. 13, 651–674 (2013). https://doi.org/10.1007/s00028-013-0195-1

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