This thesis studies several problems dealing with weighted inequalities and vector-valued operators. A weight is a nonnegative locally integrable function, and weighted inequalities refers to studying a given operator's continuity from one weighted Lebesgue space to another. The case where the underlying measure of both Lebesgue spaces is given by the same weight is known as a one weight inequality and the case where the weights are different is called a two weight inequality. These types of inequalities appear naturally in harmonic analysis from attempts to extend classical results to function spaces where the underlying measure is not necessarily Lebesgue measure. For most operators from harmonic analysis, Muckenhoupt weights represent the class of weights for which a one weight inequality holds. Chapters II and III study questions involving these weights. In particular, Chapter II focuses on determining the sharp dependence of a vector-valued singular integral operator's norm on a Muckenhoupt weight's characteristic; we determine that the vector-valued operator recovers the scalar dependence. Chapter III presents material from a joint work with M. Lacey. Specifically, in this chapter we estimate the weak-type norms of a simple class of vector-valued operators, but are unable to obtain a sharp result. The final two chapters consider two weight inequalities. Chapter IV characterizes the two weight inequality for a subset of the vector-valued operators considered in Chapter III. The final chapter presents examples to argue there is no relationship between the Hilbert transform and the Hardy-Littlewood maximal operator in the two weight setting; the material is taken from a joint work with M. Reguera.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

D. Tech. Chemistry / The avocado oil contains compounds with potency against ailments caused by some bacteria, fungi, viruses as well as some insects and parasites. Present in the oil are long aliphatic hydrocarbons (acetogenins) documented to have a wide range of biological activities such as anticancer, antibacterial, antifungal, insecticidal and nematocidal properties and as a result the oil is fast gaining popularity amongst consumers. The demand and popularity of the oil makes it a target for contamination and adulteration of with cheaper oils. Chromatographic techniques such as Gas chromatography (GC) and High performance liquid chromatography (HPLC) are the frequently used methods of oil analysis to safe guard against adulteration. Derivatization of acetogenins uniquely found in avocado oil would provide derivatives which can be used as markers for authentication of the oil, an alternative method to fatty acid methyl ester analysis.

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published_or_final_version / Statistics / Master / Master of Philosophy

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