Abstract or Additional Information
In the study of the physical universe, PDE and the calculus
of variations have emerged as useful frameworks for describing the
interconnectedness of various phenomena. The appropriate function
spaces for these models are typically Sobolev spaces, each of which is
equipped with a capacity that encodes the behavior of the fine
properties of its elements. In particular, if one is interested in
modeling the physical universe with mathematics, capacities emerge
naturally as objects of intrinsic interest. In this talk we introduce
some results related to one of the most basic examples of a capacity,
the Hausdorff content. This includes the Sobolev inequality of N.
Meyers and W.P. Ziemer, D. R. Adams' functional analysis observation
concerning this inequality and maximal estimates, J. Orobitg and J.
Verdera's extension of this result, and concluding with a recent
result of the speaker, Benson Chen and K.H. Ooi which clarifies the
exponents in Orobitg-Verdera's result.