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 what the canonical shape in a given knot type is, 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 paper, we calculate the first and second variational formulae of the [Formula: see text]-O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the [Formula: see text]-O’Hara energies, their techniques do not seem to be applicable to the case [Formula: see text]. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the [Formula: see text]-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.

XL.—On Fourier's Repeated Integral

§1. Pringsheim has recently reopened the question as to the circumstances under which Fourier's repeated integral exists and represents the function to which it corresponds. In its simplest form the theorem concerning this integral asserts that, with provisos to be specified—ƒ( + 0) being the unique limit, supposed to exist, of ƒ(u) as u approaches the value zero. From this equation, indeed, the remainder of the theory follows immediately. If it is to be true, certain conditions must be satisfied at the origin, at infinity, and in the finite part of the range of values of the independent variable. Till Pringsheim's paper appeared, the only condition known to be sufficient at infinity was of a very special character, requiring no less than the absolute integrability of the function at infinity.