In this thesis, we discuss some of the results that were proven by Fabes and Stroock in 1984. Our main purpose is to give a self-contained presentation of the proof of this results. The first result is on the existence of a “reverse H ̈older inequality” for the Green’s function. We utilize the work of Muckenhoupt on the reverse Ho ̈lder inequality and its connection to the A∞ class to establish a comparability property for the Green’s functions. Additionally, we discuss some of the underlying preliminaries. In that, we prove the Alexandrov-Bakelman-Pucci estimate, give a treatment to the Ap and A∞ classes of Muckenhoupt, and establish two intrinsic lemmas on the behavior of Green’s function.

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A1 weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps A∞ weights into the A1 class. We then extend a multiparameter counterexample to the Coifman-Rochberg proposition proposed by Soria (1987) and discuss the difficulties in modifying it into an A∞ counterexample that would disapprove the boundedness of the strong maximal function.

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