In this paper we establish square-function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m, with variable t-independent coefficients in the upper half-space. This generalizes known results for variable-coefficient second-order operators, and also for constant-coefficient higher-order operators.
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Acknowledgements We would like to thank the American Institute of Mathematics for hosting the SQuaRE workshop on “Singular integral operators and solvability of boundary problems for elliptic equations with rough coefficients,” at which many of the results and techniques of this paper were discussed. Steve Hofmann is partially supported by the NSF grant DMS-1361701. Svitlana Mayboroda is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, and the NSF Materials Research Science and Engineering Center Seed Grant.
- Elliptic equation
- Primary: 35J30; Secondary: 31B10
- higher-order differential equation
- layer potentials
- square function estimates