Towards a Unified Theory of Sobolev Inequalities

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 108)

Summary

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated with a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.

Notes

Acknowledgements

We are grateful to E. Milman for a number of useful comments that helped improve the presentation.

The author J. Martín was Partially supported in part by Grants MTM2010-14946, MTM-2010-16232.

The author M. Milman was partially supported by a grant from the Simons Foundation (#207929 to Mario Milman).

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA

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