Variational formulae and estimates of O'Hara's knot energies

O'Hara's energies, introduced by Jun O'Hara, were proposed to answer the question of what is the canonical shape in a given knot type, and were configured so that the less the energy value of a knot is, the "better" its shape is. The existence and regularity of minimizers has been well studied. In this article, we calculate the first and second variational formulae of the $(\alpha,p)$-O'Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the $(\alpha,1)$-O'Hara energies, their techniques do not seem to be applicable to the case $p>1$. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the $(\alpha,p)$-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also estimates in several function spaces.