Plessner’s theorem states
that if a trigonometric series converges everywhere in a set E of positive measure,
then its conjugate series converges almost everywhere in E. Recently, Ash and Gluck
have shown that this theorem is false in two dimensions by exhibiting a Fourier series
of an L1 function which converges almost everywhere, but each of its conjugates is
divergent almost everywhere. We show that if instead of the usual conjugates in two
dimensions, one uses Riesz conjugates, then Plessner’s theorem remains true
provided the conjugates are required only to be restrictedly convergent almost
everywhere in E. The techniques used to obtain this result are similar to
those used in the one-dimensional case and involve the notions of stable
convergence, nontangential convergence, the theory of Riesz conjugates as
developed by E. M. Stein and G. Weiss, and a Tauberian theorem for Abel
summability.
