We suggest an elementary harmonic analysis approach to canceling and weakly canceling differential operators, which allows us to extend these notions to the anisotropic setting and replace differential operators with Fourier multiplies with mild smoothness regularity. In this more general setting of anisotropic Fourier multipliers, we prove the inequality parallel to f parallel to(L infinity) less than or similar to parallel to Af parallel to(L1) if A is a weakly canceling operator of order d and the inequality parallel to f parallel to(L2) less than or similar to parallel to Af parallel to(L1) if A is a canceling operator of order d/2, provided f is a function of d variables.