On Inequalities of Weak Type

Citation

Sawyer, Stanley Arthur (1964) On Inequalities of Weak Type. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2P68-HS42. https://resolver.caltech.edu/CaltechETD:etd-10072002-100405

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let {T[subscript n]} be a sequence of continuous linear transformations on L[superscript p](X) for finite measure space X and 1<=p<=2. Assume further that lim T[subscript n]f(x) exists a.e. for all f(x) in L[superscript p](X). Then, under the added assumptions that X is a compact group or homogeneous space and that each operator T[subscript n] commutes with translations on X, E.M.Stein was able to prove the existence of a constant [...] such that [...] for all f(x) in L[superscript p](X) and A > 0. The first result of this paper is to prove (1) from convergence under the weaker assumption that the sequence {T[subscript n]} commutes with each member of a family of measure-preserving transformations on X, a family which is large enough to have only trivial fixed sets. This result contains Stein's theorem, concludes maximal ergodic theorems from individual ergodic theorems, and applies in situations arising in probability theory. The conditions above are then weakened so that the domain of {T[subscript n]} becomes an F-space of functions satisfying a certain concordance condition on its topology, and the operators {T[subscript n]} become continuous in measure with range in the space of measurable functions on X. Then, under the assumption that {T[subscript n]} commutes with enough measure-preserving transformations as above, a slightly weaker version of (1) is concluded. Now, assume that {T[subscript n]} is a sequence of continuous-in-measure linear transformations of an abstract F-space [...] into measurable functions on finite measure space X, and that [...] for every f in a dense subset of E. A decomposition of the measure space X is then obtained, such that [...] on one of the sets for all f in E, and such that for all f in the complement of a set of the first category in E, [...] a.e. on the other set of the decomposition. A, theorem of Banach then applies on the first set to give a result which can be viewed as similar to (1). The decomposition is then applied to the preceeding results to prove new theorems

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