Abstract. The integrability classes for even trigonometric and Walsh systems, such as Fomin's classes ${\cal F}_p$, $p>1$, and their enlargements $dv^2$, $\overline{cv}^2$ and $cv^2$, are not necessarily integrability classes for general Vilenkin systems. Recently Aubertin and J.F. Fourier have resolved the problem for the class $dv^2$ proving that for $(c_k) \in dv^2$, (*) $\sum |c_k-c_{\tilde k}|/k< \infty $ is a necessary and sufficient condition in order that the sum of the Vilenkin series $\sum c_k \chi_k$ be an integrable function. Here $\tilde k$ is that index for which $\chi_{\tilde k} = \overline{\chi }_k$, and hence depends on the characteristic sequence of primes $p=(p_{j+1})$. We improve and extend this result from $dv^2$ to new larger classes $lv^2(p)$, $\overline{mv}^2(p)$ and $mv^2(p)$, where $dv^2 \subset lv^2(p) \subset\overline {mv}^2(p)\cap bv$ and $\overline{cv}^2 \subset\overline {mv}^2(p) \subset mv^2(p)$. We prove that for $(c_k)\in mv^2(p)\cap bv$, (*) is also a necessary and sufficient condition for the integrability of $\sum c_k \chi_k $ and derive new equivalent forms of (*). Applications of these results yield several known theorems on integrability of Vilenkin series.

AMS Subject Classification (1991): 42C10, 43A55

Received April 17, 2002, and in revised form June 21, 2002. (Registered under 2926/2009.)