Quantitative differentiation

As we learn in Calculus, differentiation is about approximating a given function by an affine one in infinitesimal balls. In quantitative differentiation, we would like to do the same in "macroscopic" balls of quantified size. There are now three approaches to the problem: "geometric", "analytic", and "dynamic". I will concentrate on the latter two, which I have considered in joint work with Assaf Naor (both) and Sean Li (the analytic approach). A key to both is a quantitative elaboration of Dorronsoro's classical embedding theorem of a Sobolev space into a certain local approximation space; in the "dynamic" version, this is achieved with the help of the heat flow.

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