TY - JOUR
T1 - Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators
AU - Hofmann, Steve
AU - Kenig, Carlos
AU - Mayboroda, Svitlana
AU - Pipher, Jill
N1 - Publisher Copyright:
© 2014 American Mathematical Society.
PY - 2015/4/1
Y1 - 2015/4/1
N2 - 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 Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.
AB - 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 Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.
KW - AMuckenhoupt weights
KW - Dirichlet problem
KW - Divergence form elliptic equations
KW - Harmonic measure
KW - Layer potentials
KW - Non-tangential maximal function
KW - Square function
KW - ∊-approximability
UR - http://www.scopus.com/inward/record.url?scp=84938495508&partnerID=8YFLogxK
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U2 - 10.1090/s0894-0347-2014-00805-5
DO - 10.1090/s0894-0347-2014-00805-5
M3 - Article
AN - SCOPUS:84938495508
SN - 0894-0347
VL - 28
SP - 483
EP - 529
JO - Journal of the American Mathematical Society
JF - Journal of the American Mathematical Society
IS - 2
ER -