We consider compact pseudodifferential operators with symbols whose decaying order with respect to the variable ξ depends on the space variable. We obtain the estimates for singular values, as well as validity conditions of the Weyl asymptotics. The results are formulated in terms of whether the symbol belongs to the classes of multipliers of integral operators. We give applications of the results to the L2-small ball deviation asymptotics for Gaussian processes with variable Hurst parameter. Bibliography: 21 titles.