Selected Works of Donald L. Burkholder (Selected Works in Probability and Statistics)

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. Front Matter....Pages i-xxv Donald Burkholder’s Work in Martingales and Analysis....Pages 1-22 Don Burkholder’s work on Banach spaces....Pages 23-30 On a Class of Stochastic Approximation Processes....Pages 31-46 Sufficiency in the Undominated Case....Pages 47-56 Iterates of Conditional Expectation Operators....Pages 57-62 On the Order Structure of the Set of Sufficient Subfields....Pages 63-66 Semi-Gaussian Subspaces....Pages 67-75 Successive Conditional Expectations of an Integrable Function....Pages 76-82 Maximal Inequalities as Necessary Conditions for Almost Everywhere Convergence....Pages 83-96 Martingale Transforms....Pages 97-107 Extrapolation and Interpolation of Quasi-Linear Operators on Martingales....Pages 108-163 A Maximal Function Characterization of the Class H p ....Pages 164-180 Integral Inequalities for Convex Functions of Operators on Martingales....Pages 181-198 Distribution function inequalities for the area integral....Pages 199-216 The 1971 Wald Memorial Lectures....Pages 217-240 Boundary Behaviour of Harmonic Functions in a Half-Space and Brownian Motion....Pages 241-258 One-Sided Maximal Functions and H p ....Pages 259-283 Exit Times of Brownian Motion, Harmonic Majorization, and Hardy Spaces....Pages 285-308 Boundary Value Estimation of the Range of an Analytic function....Pages 309-323 A Sharp Inequality for Martingale Transforms....Pages 324-329 Weak Inequalities for Exit Times and Analytic Functions....Pages 330-337 Brownian Motion and the Hardy Spaces H p ....Pages 97-359 A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional....Pages 360-374 Martingale Transforms and the Geometry Of Banach Spaces....Pages 375-390 Research Announcements....Pages 391-395 A Geometric Condition that Implies the Existence of Certain Singular Integrals of Banach-Space-Valued Functions....Pages 396-412 Special Invited Paper....Pages 413-468 An Elementary Proof of an Inequality of R. E. A. C. Paley....Pages 469-473 An Extension of a Classical Martingale Inequality....Pages 474-483 Martingales and Fourier Analysis in Banach Spaces....Pages 484-531 A Sharp and Strict L p -Inequality for Stochastic Integrals....Pages 532-537 A proof of Pelczyński’s conjecture for the Haar system....Pages 538-542 Differential Subordination of Harmonic Functions and Martingales....Pages 543-565 Explorations in Martingale Theory and Its Applications....Pages 566-630 Strong Differential Subordination and Stochastic Integration....Pages 631-661 Sharp Norm Comparison of Martingale Maximal Functions and Stochastic Integrals....Pages 662-677 Some Extermal Problems in Martingale Theory and Harmonic Analysis....Pages 678-694 The Best Constant in the Davis Inequality for the Expectation of the Martingale Square Function....Pages 695-709 Joseph L Doop....Pages 710-710 Joseph Leo Doob, 1910-2004....Pages 711-722 Obituary: Frank B Knight....Pages 723-723 Comments on (5), (14), (22), (30), and (31)....Pages 724-729