William Beckner

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OBJECTIVE To evaluate the quality of reports of complementary and alternative medicine (CAM) randomized controlled trials (RCTs) in the pediatric population. We also examined whether there was a change in the quality of reporting over time. METHODS We used a systematic sample of 251 reports of RCTs that used a CAM intervention. The quality of each report(More)
Sharp L p extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. Optimal constants are obtained for the full Stein-Weiss potential as a map from L p to itself which in turn yield semi-classical Rellich inequalities on R n. Additional results are obtained for Stein-Weiss(More)
Sharp extensions of Pitt's inequality and bounds for Stein-Weiss fractional integrals are obtained that incorporate gradient forms and vector-valued operators. Such results include Hardy-Rellich inequalities. Weighted inequalities provide quantitative information to characterize integrability for differential and integral operators and intrinsically are(More)
Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss(More)
I am indebted for many years of support and guidance, and also for leading me into the area of partial differential equations. Also, I am very grateful to Professor Rafael de la Llave for his encouragement and useful disscutions during my thesis work. I also express my gratitude to Professor Lawrence C. Evans for many interesting discussions, and for(More)
Dedicated, like every good thing, to my parents. Acknowledgments My path in life to a PhD in mathematics has been longer than most; so too will be my acknowledgements. My parents I must thank for everything, but most particularly for demonstrating to me by the example of their lives that knowledge and rationality are virtues, while ignorance is not. I thank(More)
Linear interpolation inequalities that combine Hardy's inequality with sharp Sobolev embedding are obtained using classical arguments of Hardy and Littlewood (Bliss lemma). Such results are equivalent to Caffarelli-Kohn-Nirenberg inequalities with sharp constants. A one-dimensional convolution inequality for the exponential density is derived as an(More)
Although the listing that follows represents the majority of documents cited in NRC publications , it is not Intended to be exhaustive. Referenced documents available for Inspection and copying for a fee from the NRC Public Document Room include NRC correspondence and internal NRC memoranda; NRC Office of The following documents in the NUREG series are(More)