On the strong logarithmic summability of the double trigonometric Fourier series

We prove the following theorem: Suppose that E subset of [0, 2 pi)(2) is any Lebesgue measurable set, mu E-2 > 0, and phi(u) is a nonnegative, continuous and nondecreasing function on [0,8) such that u phi(u) is a convex function on [0,8) and f(u) = o(ln u), u -> infinity. Then there exists a function g is an element of L-1([0, 2 pi)(2)) such that integral([0,2 pi)2) |g(x, y)|phi(|g(x, y)|)dxdy < infinity and the sequence of the strong logarithmic means by squares of the double trigonometric Fourier series of g, that is, the sequence {1/ln N Sigma(N)(k=1) |S-k,S-k(g;x,y)-g(x,y)|/k, N = 2.3...} is not bounded in measure on E.