Abstract. Alexits' theorem is generalized to the case of $L^\infty $ space on the real line. Known results were obtained under additional restrictive assumptions. We avoid these by introducing a new analog of conjugate function.

AMS Subject Classification (1991): 42A38

Keyword(s): Alexits' theorem, Fourier transform, Hilbert transform, conjugate function, Lipschitz classes

Received July 17, 1997 and in revised form November 12, 1997. (Registered under 2654/2009.)