Abstract. We prove the following theorem: {\it Suppose that $E\subset[0,2\pi )^2$ is any Lebesgue measurable set, $\mu_{2}E >0,$ and $\phi(u)$ is a nonnegative, continuous and nondecreasing function on $[0,\infty )$ such that $u\phi(u)$ is a convex function on $[0,\infty )$ and $ \phi(u) = o(\ln u), u \to\infty. $ Then there exists a function $g \in L_1([0,2\pi )^2)$ such that $ \int_{[0,2\pi )^2} | g(x,y) |\phi(| g(x,y) |)dx dy < \infty_{\strut }^{\strut } $ and the sequence of the strong logarithmic means by squares of the double trigonometric Fourier series of $g$, that is, the sequence $ \left\{{1\over\ln N}\sum_{k=1}^N {| S_{k,k}(g;x,y) - g (x,y)| \over k}, N=2,3,\ldots\right \} _{\strut }^{\strut } $ is not bounded in measure on $E$.}

AMS Subject Classification (1991): 42C15, 42C10

Keyword(s): double Fourier series, strong logarithmic means, bounded in measure

Received April 7, 2010. (Registered under 23/2010.)