The Lebesgue Fundamental Theorem of Calculus

Abstract

This thesis provides a modern and self–contained study of integral differentiation with respect to axis–aligned rectangles. It focuses on the classical Jessen–Marcinkiewicz–Zygmund (JMZ) Theorem and the later extension by Zygmund. Throughout, our aim is to make the underlying theory and its results accessible to undergraduate–level readers. Every intermediate result is proved in full, ensuring all the essential details are appreciated. After a brief review of the preliminaries, including measure theory and Lp , Lp,∞ and L(log L)k spaces, we then discuss dyadic intervals and the Hardy–Littlewood maximal operator. An important intermediate result is the Lebesgue Differentiation Theorem (LDT). We provide a proof of the LDT that avoids any covering lemmas and instead uses the properties of dyadic intervals. Both the weak– L1 and strong Lp bounds for maximal operators are also presented. Together with the Lebesgue Differentiation Theorem, they form the basis for the main part of the thesis. A detailed and step–by–step reconstruction of the 1935 JMZ Theorem forms the core of this thesis. The theorem extends the Lebesgue Differentiation Theorem from balls and cubes to the more general axis–aligned rectangles. We give a complete proof of the theorem and also show that the condition f ∈L(log+ L)d−1(Rd) is sharp. Further, we revisit Zygmund’s 1967 extension, which considers rectangles with 1 ≤k≤ddistinct side lengths. This naturally leads to a discussion of Zygmund’s Conjecture, which tries to push the limits of Zygmund’s extension of the JMZ Theorem. We analyse the counterexamples by F. Soria and G. Rey, and the examples by A. C´ordoba and F. Soria. Each step in the construction of their examples and counterexamples is explained in detail. Finally, we close the thesis with a few concluding remarks and an outlook for possible future research on the conditions under which Zygmund’s Conjecture holds – or fails.