Riesz transforms and partial derivatives

New estimates are given in the two-dimensional case for special operators that are linear combinations of Riesz transforms. They are used to investigate the distances between partial derivatives $\dfrac{\partial^nf}{\partial x_1^k\partial z_2^l}$, $k+l=n$, on the class
$$ K_n=\biggl\{f\colon\biggl\|\frac{\partial^nf}{\partial{x_1^n}}\biggr\|_p\leqslant 1,\ \biggl\|\frac{\partial^nf}{\partial{x_1^n}}\biggr\|_p\leqslant 1\biggr\}, \qquad 1<p<\infty. $$