Selected Works of Donald L. Burkholder

  • Burgess Davis
  • Renming Song

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

Table of contents

  1. Front Matter
    Pages i-xxv
  2. Burgess Davis, Renming Song
    Pages 1-22
  3. Burgess Davis, Renming Song
    Pages 23-30
  4. Burgess Davis, Renming Song
    Pages 31-46
  5. Burgess Davis, Renming Song
    Pages 47-56
  6. Burgess Davis, Renming Song
    Pages 57-62
  7. Burgess Davis, Renming Song
    Pages 63-66
  8. Burgess Davis, Renming Song
    Pages 67-75
  9. Burgess Davis, Renming Song
    Pages 76-82
  10. Burgess Davis, Renming Song
    Pages 97-107
  11. Burgess Davis, Renming Song
    Pages 164-180
  12. Burgess Davis, Renming Song
    Pages 199-216
  13. Burgess Davis, Renming Song
    Pages 217-240
  14. Burgess Davis, Renming Song
    Pages 259-283
  15. Burgess Davis, Renming Song
    Pages 309-323

About this book


This book chronicles Donald Burkholder's thirty-five year study of martingales and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well- known open problems, one involving the two dimensional Hilbert transform and its connection to quasiconformal mappings and the other a conjecture in the calculus of variations concerning rank-one convex and quasiconvex functions. This book includes reprints of many of Burkholder's papers, together with two commentaries on his work and its continuing impact.


Banach spaces Hardy spaces martingales singular integrals stochastic analysis

Editors and affiliations

  • Burgess Davis
    • 1
  • Renming Song
    • 2
  1. 1., Department of Mathematics andPurdue UniversityWest LafayetteUSA
  2. 2., Department of MathematicsUniversity of Illinois, UrbanaUrbanaUSA

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-7244-6
  • Online ISBN 978-1-4419-7245-3
  • About this book