Exponent Sets and Muckenhoupt Ap-weights

Abstract: In the study of the weighted p-Laplace equation, it is often important to acquire good estimates of capacities. One useful tool for finding such estimates in metric spaces is exponent sets, which are sets describing the local dimensionality of the measure associated with the space. In this thesis, we limit ourselves to the weighted Rn space, where we investigate the relationship between exponent sets and Muckenhoupt Ap-weights - a certain class of well behaved functions. Additionally, we restrict our scope to radial weights, that is, weights w(x) that only depend on |x|. First, we determine conditions on α such that |x|α ∈ Ap(μ) for doubling measures μ on Rn. From those results, we develop weight exponent sets - a tool for making Ap-classifications of general radial weights, under certain conditions. Finally, we apply our techniques to the weight |x|α(log 1/|x|)β. We find that the weight belongs to Ap(μ) if α ∈ (-q, (p-1)q), where q = sup Q(μ) is a constant associated with the dimensionality of μ. The Ap-conditions in this thesis are found to be sharp.