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A Concentration Inequality for Harmonic Measures on the Sphere

  • Gideon Schechtman
  • Michael Schmuckenschläger
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

Let μ be the normalized Lebesgue measure on S n −1. For x = (x 1,…,x n ) with ||x||2 < 1 we denote by μ x the probability measure on S n −1 given by \( \frac{{1 - {{\left\| x \right\|}^2}}} {{{{\left\| {y - x} \right\|}^n}}}d\mu \left( y \right) \). We recall that if f is an integrable function on S n -1 then u(x) = \( u(x) = {\int_{{S^{n - 1}}} {f(y)d\mu }^x}(y) \) is a harmonic function whose radial limits are equal μ-almost everywhere to f.

Keywords

Brownian Motion Lipschitz Function Lipschitz Constant Harmonic Measure Absolute Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Gideon Schechtman
    • 1
  • Michael Schmuckenschläger
    • 1
    • 2
  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.J. Kepler UniversitätLinzAustria

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