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Working paper

The Bohr--Pal Theorem and the Sobolev Space W_2^{1/2}

arxiv.org. math. Cornell University, 2015. No. 1508.07167.
The well-known Bohr--Pal 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 o 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--Pal theorem does not extend to complex-valued functions. We show that, in general, there is no homeomorphism which will bring a complex-valued function satisfying the Lipschitz condition of order <1/2 into W_2^{1/2}(T) .