Bilateral estimates of some pseudo-derivatives of the transition probability density of an isotropic $\alpha$-stable stochastic process

In the paper, the transition probability density of isotropic $\alpha$-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols $\mathrm{\mid }\lambda {\mathrm{\mid }}^{\mathrm{\varkappa }}$ and $\lambda \mathrm{\mid }\lambda {\mathrm{\mid }}^{\mathrm{\varkappa }-1}$, where $\mathrm{\varkappa }$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.