Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.
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)
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 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)
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)
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)
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)