Beijing Institute of Mathematical Sciences and Applications

In this course, we begin by exploring foundational concepts such as the Hardy-Littlewood maximal function, the Lebesgue differentiation theorem, and the Marcinkiewicz interpolation theorem. We then delve into the Calderón-Zygmund decomposition, which effectively separates functions and sets into "good" and "bad" parts, allowing for distinct treatment through techniques in real variable theory and harmonic analysis.

Next, we examine the space of functions of bounded mean oscillation (BMO), which consists of functions whose mean oscillation over cubes is uniformly bounded. BMO is significant in the regularity theory of nonlinear partial differential equations. We rigorously analyze the proofs of key theorems, including the John-Nirenberg inequality, the Fefferman-Stein theorem, and the Stampacchia interpolation theorem.

Additionally, we study Muckenhoupt’s $A_p$ weights and their associated weighted norm inequalities, which provide characterizations of BMO functions. A primary objective of the course is to demonstrate that the Hardy-Littlewood maximal operator is of weighted strong type $(p,p)$ for $1<p<\mathrm{\infty }$ if and only if the weight satisfies Muckenhoupt’s $A_p$ condition.

Finally, we examine the statement and proof of the celebrated Gehring Lemma, demonstrating how this lemma establishes that every $A_p$ weight satisfies a reverse Hölder inequality.