## Abstract

We consider divergence form elliptic operators (formula presented), defined in the half space (formula presented), where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu= 0, and we then combine these estimates with the method of “ϵ-approximability” to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A_{∞}with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in L^{p}, for some p < ∞). Previously, these results had been known only in the case n = 1.

Original language | English (US) |
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Pages (from-to) | 483-529 |

Number of pages | 47 |

Journal | Journal of the American Mathematical Society |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2015 |

### Bibliographical note

Publisher Copyright:© 2014 American Mathematical Society.

## Keywords

- AMuckenhoupt weights
- Dirichlet problem
- Divergence form elliptic equations
- Harmonic measure
- Layer potentials
- Non-tangential maximal function
- Square function
- ∊-approximability