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

Vladimir Lebedev
2016 Studia Mathematica  
The well-known Bohr--Pál theorem asserts that for every continuous real-valued function f on the circle T there exists a change of variable, i.e., a homeomorphism h of T onto itself, such that the Fourier series of the superposition f∘ 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( T). This refined version of the Bohr--Pál theorem does not extend to complex-valued functions. We show
more » ... if α<1/2, then there exists a complex-valued f that satisfies the Lipschitz condition of order α and at the same time has the property that f∘ h∉ W_2^1/2( T) for every homeomorphism h of T.
doi:10.4064/sm8438-1-2016 fatcat:qqukbug2tbdoxbrt4n4kt4jx2m