Convergence and divergence of series conjugate to a convergent multiple Fourier series

Abstract:

In this note we consider to what extent the classical theorems of Plessner and Kuttner comparing the set of convergence of a trigonometric series with that of the conjugate trigonometric series can be generalized to higher dimensions. We show that if a function belongs to ${L^p},p > 1$, of the $2$-torus, then the convergence (= unrestricted rectangular convergence) of the Fourier series on a set implies its three conjugate functions converge almost everywhere on that set. That this theorem approaches the best possible may be seen from two examples which show that the dimension may not be increased to 3, nor the required power of integrability be decreased to 1. We also construct a continuous function having a boundedly divergent Fourier series of power series type and an a.e. circularly convergent double Fourier series whose $y$-conjugate diverges circularly a.e. Our ${L^p}$ result depends on a theorem of L. Gogöladze (our proof is included for the reader’s convenience), work of J. M. Ash and G. Welland on $(C,1,0)$ summability, and on a result deducing the boundedness of certain partial linear means from convergence of those partial means. The construction of the counterexamples utilizes examples given by C. Fefferman, J. Marcinkiewicz, A. Zygmund, D. Menšov, and the present authors’ earlier work.