Weakly Canceling Operators and Singular Integrals

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 $\|f\|_{L_\infty } \lesssim \|Af\|_{L_1}$ if $A$ is a weakly canceling operator of order $d$ and the inequality $\|f\|_{L_2} \lesssim \|Af\|_{L_1}$ if $A$ is a canceling operator of order $d/2$, provided $f$ is a function of $d$ variables.