THE ENERGY SPACES OF THE TANGENT POINT ENERGIES

Abstract

In this note, we will give a necessary and sufficient condition under which the tangent point energies introduced by von der Mosel and Strzelecki in [J. Geom. Anal., pp. 1–55 (2011), J. Knot Theory Ramifications21 (2012) 1250044] are bounded. We show that an admissible submanifold has bounded 𝔈 q -energy if and only if it is injective and locally agrees with the graph of functions that belong to Sobolev–Slobodeckij space [Formula: see text]. The known Morrey embedding theorems of von der Mosel and Strzelecki can then be interpreted as standard Morrey embedding theorems for these spaces. Especially, this shows that the Hölder exponents for the embeddings in [J. Geom. Anal., pp. 1–55 (2011)] are sharp.