THE SETS OF CONVERGENCE IN MEASURE OF MULTIPLE ORTHOGONAL FOURIER SERIES

Abstract

Let $\{ \vf_k(x),\,k=1,2,\ldots\}$ be an arbitrary orthonormal system on [0,1] that is uniformly bounded by a constant $M$. Let $T$ be a subset of [0,1]$^2$ such that the Fourier series of all Lebesgue integrable functions on [0,1]$^2$ with respect to the product system $\{ \vf_k(x) \vf_l(y),\,k, l=1,2,\ldots\}$ converge in measure by squares on $T$. The following problem is studied. How large may the measure of $T$ be?

A theorem is proved that implies that for each such system, there is $\mu_2T \leq 1-M^{-4} $ (for the $d$-fold product systems, $\mu_dT \,{\leq}\, 1-M^{-2d}$, d\,{\geq}\, 2$). This estimate is sharp in the class of all such product systems.