Some Extermal Problems in Martingale Theory and Harmonic Analysis

Part of the Selected Works in Probability and Statistics book series (SWPS)


The main new results are Theorem 6.3 and its corollary for stochastic integrals, Theorem 6.6. However, we begin with a little background with roots in some of the work of Kolmogorov and M. Riesz of more than seventy years ago. Let n be a positive integer, D a domain of ℝ n , and ℍ a real or complex Hilbert space. Suppose that u and v are harmonic on D with values in ℍ. Let \(\left| {\nabla u} \right| = {\left( {{{\sum\nolimits_{k = 1}^n {\left| {\partial u/\partial {x_k}} \right|} }^2}} \right)^{1/2}}\). Then v is differentially subordinate to u if, for all xD,
$$\left| {\nabla \upsilon \left( x \right)} \right| \leqslant \left| {\nabla u\left( x \right)} \right|.$$


Extremal Problem Harmonic Measure Stochastic Integral Complex Hilbert Space Good Constant 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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