Vol. 60, No. 1, 1975

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Plessner’s theorem for Riesz conjugates

Gerald E. Peterson and Grant Welland

Vol. 60 (1975), No. 1, 307–317

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.

Mathematical Subject Classification
Primary: 42A92, 42A92
Secondary: 42A40
Received: 26 September 1974
Published: 1 September 1975
Gerald E. Peterson
Grant Welland