From Littlewood–Paley Theory to Quantitative Geometric Measure Theory

Beginning with the work of Littlewood and Paley, harmonic analysts showed various ways that one could understand a function by decomposing it into pieces at different scales/frequencies and then recombining those pieces through "square functions". This perspective, now known as Littlewood–Paley theory, has become a basic tool throughout modern analysis. In this talk I will sketch some of that history, including later developments of Fefferman and Stein, where square functions and related maximal-function methods helped illuminate the endpoint spaces H^1 and BMO. I will then explain how these ideas helped shape what is now called quantitative geometric measure theory, where analytic information is used to detect and measure geometric structure. Finally, I will discuss how this circle of ideas appears in my recent work on free boundary problems for caloric measure.