  We consider the semiclassical asymptotic behaviour of the number of
eigenvalues smaller than $E$ for elliptic operators in $L\sp 2 ({\bf R}\sp d)$.
We describe a method of finding remainder estimates related to the volume of
the region of the phase space in which the principal symbol takes values
belonging to the interval $[E'-h; E'+h]$, where $E'$ is close to $E$. This
method allows to derive remainder estimates $O(h\sp {1-d})$ for a class of
symbols with critical points and non-smooth coefficients.
