Sharp Sobolev Inequalities via Projection Averages
Keyword(s):
Abstract A family of sharp $$L^p$$ L p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $$L^p$$ L p Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them—the affine $$L^p$$ L p Sobolev inequality of Lutwak, Yang, and Zhang. When $$p = 1$$ p = 1 , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.
2018 ◽
Vol 149
(04)
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pp. 979-994
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2006 ◽
Vol 136
(2)
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pp. 277-300
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2018 ◽
Vol 38
(8)
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pp. 3939-3953
2007 ◽
Vol 244
(1)
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pp. 315-341
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