A family of sharp Sobolev inequalities is established by averaging the length of -dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them-the affine Sobolev inequality of Lutwak, Yang, and Zhang. When , 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.
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| http://dx.doi.org/10.1007/s12220-020-00544-6 | DOI Listing |
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