Repulsive knot energies and pseudodifferential calculus : rigorous analysis and regularity theory for O'Hara's knot energy family E (alpha), alpha in [2,3)

In this thesis, we consider J. O'Hara's knot functionals E^(alpha), $alphain[2,3)$, proving Fréchet differentiability and $C^infty$ regularity of critical points. Using some ideas of Z.-X. He and filling major gaps in his investigation of the Möbius Energy E^(2), we furnish a rigorous proof of an even more general statement. We start with proving continuity of E^(alpha) on injective and regular H^2 curves, moreover we establish Fréchet differentiability of E^(alpha). Among other things, the proof draws on the fact that reparametrization of a sequence of curves to arc-length preserves H^2 convergence. Additionally, we derive several formulae of the first variation. In the second part, we consider the rescaled functional $ilde E = ext{length}^{alpha-2}E$ establishing a bootstrap argument, which gives $C^infty$ regularity for critical points in $H^alphacap H^{2,3}$ being injective and parametrized by arc-length. The major technique is to introduce fractional Sobolev spaces on a periodic interval and to study bilinear Fourier multipliers