Normed inequalities for fractional derivatives
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The objective of this thesis is to obtain weighted norm estimates for the homogeneous and inhomogeneous fractional derivatives. The primary focus is on a detailed exploration of the weighted fractional Leibniz rule and the weighted fractional chain rule. This work seeks to augment the current body of knowledge by establishing appropriate inequalities for more generalized weight classes and extending the range of indices for which the inequalities are valid. In particular the fractional Liebniz rule is extended to the product of m factors and the weight class is improved from the product of Muckenhoupt weight classes to the more natural multiple weight class denoted Ap. Furthermore, a fractional chain rule is extended from Lebesgue spaces for p [greater than] 1 to weighted Triebel-Lizorkin spaces with 0 [less than] p [less than] [infinity], where an explicit relationship holds between the permissible Muckenhoupt weights, the smoothness index, and the integrability index.
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Ph. D.