Maximal Functions in Analysis
by Robert Fefferman (scribe: Philip Ascher) · University of Chicago VIGRE REU
Eleven-page self-contained REU lecture notes building from the fundamental theorem of calculus to the Hardy-Littlewood maximal theorem, proving the Vitali covering lemma, the weak-type L1 bound, the Lp case, and an elementary introduction to Calderón-Zygmund theory.
More resources on Maximal Functions
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
245A, Notes 5: Differentiation Theorems
Graduate real-analysis lecture notes published as a blog post, proving the Hardy-Littlewood maximal inequality with covering and rising-sun arguments, then deducing the Lebesgue differentiation theorem, absolute continuity and bounded variation results, with many embedded exercises.
Real Analysis: Measure Theory, Integration, and Hilbert Spaces
Volume III of the Princeton Lectures in Analysis; chapter 3 develops differentiation of the integral through the Vitali covering lemma, the Hardy-Littlewood maximal function, its weak-type bound, and the Lebesgue differentiation theorem, with exercises and problems.
247A Lecture Notes 3: The Hardy-Littlewood Maximal Inequality
Fifteen pages of Terence Tao's UCLA graduate harmonic analysis notes covering the Hardy-Littlewood maximal inequality: Vitali-type covering arguments, the weak-(1,1) bound, Marcinkiewicz interpolation to Lp, dyadic maximal functions, and the Lebesgue differentiation theorem.