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.
CITATION STYLE
Martín, J., & Milman, M. (2014). Towards a unified theory of Sobolev inequalities. In Springer Proceedings in Mathematics and Statistics (Vol. 108, pp. 163–201). Springer New York LLC. https://doi.org/10.1007/978-3-319-10545-1_13
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