The uncharted territory of W^α,1 Sobolev spaces

Prof. Augusto Ponce

Abstract: I will address properties of the fractional Sobolev space W^α,1(R^N) which cannot be answered by classical representation formulas from Harmonic Analysis, for any exponent α > 0. They can be handled instead in terms of a strong capacitary inequality which is based itself on a new geometric boxing inequality that connects the Hausdorff content of dimension N − α and the fractional perimeter of order 0<α<1. These results have been obtained in collaboration with D. Spector (Okinawa, Japan).