Solution of a problem concerning functions of harmonic bounded variation (Q1099269)

It is known [\textit{C. Goffmann}, Real. Anal. Exch. 3(1977), 7-23 (1978; Zbl 0372.26005)] that the space of all regulated functions for which the Fourier series converges for every change of variable (GW) contains the space of functions of harmonic bounded variation (HBV) and the space of all functions for which the Fourier series converges uniformly for every change of variable (UGW) contains the space of continuous functions of harmonic bounded variation \((HBV_ c)\). In \textit{C. Goffman}'s paper (loc. cit., p. 17) and by \textit{D. Waterman} [Proc. Am. Math. Soc. 80, 445-447 (1980; Zbl 0461.26007)] the question is raised whether \(GW=HBV\) and \(UGW=HBV_ c\). \textit{D. Waterman} [Real. Anal. Exch. 9(1983), 146-153 (1984)] also pointed out that his joint paper with \textit{A. Baernstein} [Indiana Univ. Math. J. 22, 569-576 (1972; Zbl 0235.42004)] contains the result \(HBV_ c\subseteq UGW\subsetneqq GW_ c,\) implying HBV\(\subsetneqq GW\). The purpose of the present note is to prove that \(HBV_ c\subsetneqq UGW\).