  We find the logarithmic $L_2$-small ball asymptotics for a class of zero mean
Gaussian fields with covariances having the structure of "tensor product". The
main condition imposed on marginal covariances is slow growth at the origin of
counting functions of their eigenvalues. That is valid for Gaussian functions
with smooth covariances. Another type of marginal functions considered as well
are classical Wiener process, Brownian bridge, Ornstein--Uhlenbeck process,
etc., in the case of special self-similar measure of integration. Our results
are based on new theorem on spectral asymptotics for the tensor products of
compact self-adjoint operators in Hilbert space which is of independent
interest. Thus, we continue to develop the approach proposed in the paper
\cite{KNN}, where the regular behavior at infinity of marginal eigenvalues was
assumed.
