The Tangent-Point Energy from M-Dimensional Sets Revisited: Analogies to Pointwise Estimates for Sobolev Functions

The tangent-point energy from m-dimensional sets revisited: analogies to pointwise estimates for Sobolev functions.

Some 15 years ago, Heiko von der Mosel and the speaker have analyzed several nonlocal energies of non-smooth sets S, defined as multiple integrals of various geometrically defined quantities, depending on two or more points of S. One of the was the so-called tangent-point energy, equal to the double integral over S, with respect to the Hausdorff measure, of a function depending on two points:  (a power of) the radius of a sphere which is tangent to S at one point, and passes through another point of S. For this energy, optimal regularity results (for sets S having finite energy) are known; there is also a
full characterization (in terms of fractional Sobolev spaces) of those S that have finite energy, due to Simon Blatt.

I will present a survey of those results and indicate new proofs of optimal regularity and of Blatt's characterization which are simpler than those previously known.  They are both based on a lemma which holds for graphs with finite tangent-point energy and is reminiscent of Piotr Hajlasz's characterization of Sobolev spaces in terms of upper gradients.