Abstract. Let $d\in{\msbm N}$, $\psi :[0,\infty )\to[0,\infty )$ be a nondecreasing function and $$\psi(u)=o\left(u(\log u)^{d-1}\log\log u\right ) (u\to\infty ).$$ Then there exists a function $f_1$ integrable on ${\msbm T}^d$ such that $$\int_{{\msbm T}^d}| \psi(f_1(x))| dx< \infty,$$ and the trigonometric Fourier series of the function $f$ over cubes unboundedly diverges everywhere. There exists also a function $f_2$ integrable on ${\msbm T}^d$ such that $$\int_{{\msbm T}^d}| f_2(x+h)-f_2(x)| dx= o(| h| ^{-1}/\psi(| h| ^{-1}))\qquad (h\to0),$$ and the trigonometric Fourier series of the function $f_2$ over cubes unboundedly diverges almost everywhere.

AMS Subject Classification (1991): 42B05

Received October 12, 1994, and in revised form June 16, 1995. (Registered under 5688/2009.)