Duke Mathematical Journal
- Duke Math. J.
- Volume 165, Number 12 (2016), 2331-2389.
Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions
Let , , be a uniformly rectifiable set of dimension . Then bounded harmonic functions in satisfy Carleson measure estimates and are -approximable. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure and surface measure.
Duke Math. J. Volume 165, Number 12 (2016), 2331-2389.
Received: 6 August 2014
Revised: 2 July 2015
First available in Project Euclid: 6 September 2016
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
Secondary: 28A78: Hausdorff and packing measures 31B05: Harmonic, subharmonic, superharmonic functions 35J25: Boundary value problems for second-order elliptic equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX]
Hofmann, Steve; Martell, José María; Mayboroda, Svitlana. Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions. Duke Math. J. 165 (2016), no. 12, 2331--2389. doi:10.1215/00127094-3477128. https://projecteuclid.org/euclid.dmj/1473186402