The Bohr-Pál theorem and the Sobolev space \(W_2^{1/2}\) (Q2787153)

The classical result of Pál and Bohr asserts that for every continuous real-valued function~\(f\) on the unit circle~\(\mathbb{T}\) there is a change of variable, that is, a homeomorphism~\(h\) of \(\mathbb{T}\) into itself, such that the Fourier series of the composition \(f\circ h\) converges uniformly. An improvement of this result, due to A. A. Saakyan, says that in fact \(h\) can be chosen in such a way that \(f\circ h\) belongs to the Sobolev space~\(W_{2}^{1/2}(\mathbb{T})\). It turns out that this fails for complex-valued functions.NEWLINENEWLINEIn this paper, the author shows that there exist smooth complex-valued functions \(f\) for which \(f\circ h\notin W_{2}^{1/2}(\mathbb{T})\) for every homeomorphism~\(h\) of \(\mathbb{T}\). The function~\(f\) can be chosen in the class~\(\mathrm {Lip}_{\alpha}(\mathbb{T})\) for \(\alpha<1/2\).NEWLINENEWLINELet \(\omega\) be a modulus of continuity, i.\,e. a nondecreasing continuous function on \([0,\infty)\) such that \(\omega(0)=0\) and \(\omega (x+y)\leq\omega (x)+\omega(y)\). Denote by \(\mathrm {Lip}_{\omega}(\mathbb{T})\) the class of all complex-valued functions \(f\) on \(\mathbb{T}\) with \(\omega(f,\delta)=O(\omega(\delta))\), \(\delta\to +0\), where NEWLINE\[NEWLINE \omega (f,\delta)=\sup_{|t_{1}-t_{2}|\leq\delta}|f(t_{1})-f(t_{2})|,\quad \delta\geq0, NEWLINE\]NEWLINE is the modulus of continuity of \(f\). For \(\omega(\delta)=\delta^{\alpha}\) one writes \(\mathrm {Lip}_{\alpha}\) instead of \(\mathrm {Lip}_{\omega}\).NEWLINENEWLINEThe result proved here is the following one:NEWLINENEWLINE\(\bullet\) Suppose that \(\limsup_{\delta\to +0}\omega(\delta)/\sqrt{\delta}=\infty\). Then there exists a complex-valued function \(f\in\mathrm {Lip}_{\omega}(\mathbb{T})\) such that \(f\circ h\notin W_{2}^{1/2}(\mathbb{T})\) for every homeomorphism~\(h\) of the circle~\(\mathbb{T}\) onto itself. In particular, if \(\alpha<1/2\), then there exists a function of class~\(\mathrm {Lip}_{\alpha}(\mathbb{T})\) with this property.