On the harmonic and geometric maximal operators

Abstract

In this dissertation, we attempt to characterize the boundedness of two operators over certain spaces of functions. The operators in question, the harmonic maximal operator and the geometric maximal operator, arise naturally from consideration of the paradigmatic maximal operator, the Hardy-Littlewood maximal function, and from consideration of well-known analogues of the simple arithmetic mean. In particular, we seek to improve upon earlier work by removing certain unwieldy assumptions, thus moving closer to a complete characterization of the $L^p$-boundedness of both our operators for a pair of weights. This work primarily concerns weighted norm inequalities for two operators, the harmonic maximal operator and geometric maximal operator, given either a dyadic basis or a general basis. For any basis of cubes $Q$, we define those operators as follows. The harmonic maximal operator, M_{-1}f(x)=\sup_{Q\ni x} \left(\avgQ |f|^{-1} \right)^{-1} and the geometric maximal operator M_{0}f(x)=\sup_{Q\ni x} \exp \left(\avgQ \log |f| \right) \, . We begin by considering our two operators restricted to the dyadic basis. The geometry of this basis allows us to decompose and recompose integrals of our operators taken over $\R^n$, a kind of discretization that simplifies the methods necessary for characterization. Within this framework we are able to remove the technical obstacles that lead to the unwieldy doubling conditions found in previous work. Having achieved a complete characterization in our well-behaved dyadic basis, we turn to a more difficult situation: that of a general basis. The principles necessary to treat our maximal operators in the standard basis of cubes or the dyadic basis can be abstracted further, leading to more general results that apply to a large extent to any measure regardless of its geometry. We conclude by investigating the necessary geometric theory to recover the standard basis results from our dyadic results. (Published By University of Alabama Libraries