Eli’s Impact: A Case Study

This chapter illustrates the continuing powerful influence of Eli Stein's ideas. It starts by recalling his ideas on Littlewood–Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. Before Eli, Littlewood–Paley theory was one of the deepest parts of the classical study of Fourier series in one variable. Stein, however, found the right viewpoint to develop Littlewood–Paley theory and went on to develop Littlewood–Paley theory on any compact Lie group, and then in any setting in which there is a reasonable heat kernel. Afterward, the chapter discusses the remarkable recent work of Gressman and Strain on the Boltzmann equation, and explains in particular its connection to Stein's work.