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A Nonlinear Partial Differential Equation and the Unconditional Constant of the Haar System in Lp
  • Burgess DavisEmail author
  • Renming Song
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Our aim here is to identify the best constant in an inequality (see Theorem 1) that has proved useful in the study of singular integrals, stochastic integrals, the structure of Banach spaces, and in several other areas of study. Our work yields the unconditional constant of the Haar system in L P (0,1) and rests partly on solving the nonlinear partial differential equation
$$\left( {p - 1} \right)\left[ {y{F_y} - x{F_x}} \right]{F_{yy}} - {[\left( {p - 1} \right){F_y} - x{F_{xy}}]^2} + {x^2}{F_{xx}}{F_{yy}} = 0$$
(1)
for F nonconstant and satisfying other conditions on a suitable domain of R2.

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Copyright information

© 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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