The anomalous fluxes are evaluated in the simplest possible geometric situation: drift waves in a shearless slab geometry, in the presence of density and temperature gradients. It is shown that, within the strict quasilinear framework, the linear transport equations relating the fluxes to the thermodynamic forces have serious limitations. Such a linear relation does not even exist for the ion energy flux. For all the fluxes, the first ‘‘correction’’ has a singularity whose location depends on the relative value of the density gradient and of the ion temperature gradient: its existence seriously restricts the domain of validity of the quasilinear transport theory. The ‘‘semiempirical quasilinear’’ formulas used in the comparisons with experiments are also discussed.