by Burgess Davis
Don was born shortly before the Great Depression into a farming family in eastern Nebraska. Though the farm was not engulfed in the Dust Bowl, he would have seen huge dust clouds far away, as my father did from a Missouri farm. Perhaps growing up in all this trouble had something to do with Don’s ferocious work ethic, which was passed on to his two sons, now both outstandingly successful academics.
Joe Doob was at the University of Illinois when Don arrived there, and he certainly influenced Don’s interest in martingales. Don worked with only a few co-authors, but he and Dick Gundy made a great team. They began their joint work by finding comparisons of \( L_p \) norms of martingales’ square and maximal functions. This had been an open problem and their solution, called good-lambda, is remarkably easy to understand and now used to compare norms all over mathematics. Not much later Don with Dick Gundy and Marty Silverstein used probabilistic methods to complete results of Hardy and Littlewood by giving a maximal function characterization of the class of \( H^p \) of harmonic functions in the upper half plane. Before this paper, but not after it, many mathematicians did not think probability was real math. This was the last research paper Don published with a collaborator, but he wrote many more alone. Some of the best involved solving a tough martingale problem he had thought up and then voilà his solution is important not just for martingales. I encourage the reader to google “Bellman functions in harmonic analysis A. Volberg” for a great short discussion of an example.
Don was very kind and encouraging to young mathematicians. He was an excellent dissertation advisor and a good friend. He received many honors, but the only one he ever talked about (and loved to) is that he was president of his senior class in high school because the other three students in his class had already been president.
Don’s wife Jean was elected to the Urbana School Board and served for 22 years, and also was first chair of the Urbana Human Relations Committee. In addition to that and more civic engagement, Jean found time to be a strong support to Don and his career. They were an amazing couple.
Burgess Davis was an undergraduate at Ohio State and Don Burkholder was his doctoral advisor at the University of Illinois, Urbana-Champaign. He worked at Rutgers and then Purdue and is now Emeritus Professor of Mathematics and Statistics at Purdue.
