On the Fourier coefficients of homeomorphisms of the circle (Q1271376)

Let \(f(e^{is})\) be a sense preserving homeomorphism of the circle. In order to study the Fourier coefficients of \(f\) the author considers \(2\pi\)-periodic functions \(e^{i\omega(s)}\) with nondecreasing \(\omega\), and the Fourier series \(e^{i\omega(s)}\sim \sum^\infty_{n= -\infty} c_ne^{ins}\). Extremals for Fourier coefficient estimates do not always exist in the class of homeomorphisms, but are limits of the form \(e^{i\omega(s)}\) where \(\omega\) may have discontinuities at a countable set. The author shows that if \(\omega\) is a nondecreasing function such that \(e^{i\omega(s)}\) is \(2\pi\)-periodic, \(\omega(2\pi)= \omega(0)+ 2\pi\), then the coefficients \(c_0\), \(c_1\) in the Fourier expansion satisfy \(|c_0|+ |c_1|\geq 2/\pi\). This inequality is sharp. The result generalizes a similar inequality known to hold for a certain subclass of functions \(\omega\), and improves from \(1/2\) to \(2/\pi\) the lower bound of an analogous inequality known to hold for all \(\omega\). The proof uses variational arguments.