### 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.