The Bohr–Pál theorem and the Sobolev space $W_2^{1/2}$

Vladimir Lebedev ¹

¹ National Research University Higher School of Economics 34 Tallinskaya St. Moscow, 123458, Russia

Studia Mathematica, Tome 231 (2015) no. 1, pp. 73-81

Résumé

The well-known Bohr–Pál theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of the superposition $f\circ h$ converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings $f$ into the Sobolev space $W_2^{1/2}(\mathbb T)$. This refined version of the Bohr–Pál theorem does not extend to complex-valued functions. We show that if $\alpha \lt 1/2$, then there exists a complex-valued $f$ that satisfies the Lipschitz condition of order $\alpha $ and at the same time has the property that $f\circ h\notin W_2^{1/2}(\mathbb T)$ for every homeomorphism $h$ of $\mathbb T$.

Détail

DOI : 10.4064/sm8438-1-2016

Mots-clés : well known bohr theorem asserts every continuous real valued function circle mathbb there exists change variable homeomorphism mathbb itself fourier series superposition circ converges uniformly subsequent improvements result imply actually there exists homeomorphism brings sobolev space mathbb refined version bohr theorem does extend complex valued functions alpha there exists complex valued satisfies lipschitz condition order alpha time has property circ notin mathbb every homeomorphism mathbb