Daniel Spector

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This work presents some new characterizations of Sobolev spaces and the space of functions of Bounded Variation. Additionally it gives new proofs of continuity and lower semicontinuity theorems due to Reshetnyak. In the recent paper [12], Bourgain, Brezis, and Mironescu studied the limiting behavior of the semi-norm |f | W s,p (Ω) := Ω Ω |f (x) − f (y)| p(More)
for their help during my stay here at Pittsburgh. Finally, I would like to thank the Department of Mathematical Sciences at Carnegie Mellon University for creating a comfortable environment of study. In particular, I want to thank Professor William J. Hrusa for being such an understanding and generous graduate advisor. Special thanks go to my family for(More)
Acknowledgments There are many people that must be thanked for helping me get to where I am today. I am the most grateful to my adviser Steve Shreve, for the numerous hours of discussions, answers and encouragement. He has taught me more about mathematics and how to think like a mathematician, than anyone else. My thesis committee: Dmitry Kramkov, Kasper(More)
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