Extrapolation is one of the most signi?cant and powerful properties of the weighted theory. It basically states that an estimate on a weighted Lpo space for a single exponent po ? 1 and all weights in the Muckenhoupt class Apo implies a corresponding Lp estimate for all p, 1 < p < ?, and all weights in Ap . Sharp Extrapolation Theorems track down the dependence on the Ap characteristic of the weight. In this dissertation we generalize the Sharp xtrapolation Theorem to the case where the underlying measure is d? = uo dx, and uo is an A? weight. We also use it to extend Lerner's extrapolation techniques. Such Theorems can then be used to extrapolate some known initial weighted estimates in L2 (wd? ). In addition, for some operators this approach allows us to specify the weights w?1 = uo and to use known weighted results in Lp (wd? ) to obtain some estimates on the unweighted space. This work was inspired by the paper [Per1] where the L2 weighted estimates for the dyadic square function were considered to obtain the sharp estimates for the so-called Haar Multiplier in L2.

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