Bilateral estimates of some pseudo-derivatives of the transition probability density of an isotropic $α$-stable stochastic process - Osypchuk, M.M. (Carpathian Mathematical Publications, 2023)
In the paper, the transition probability density of an isotropic $α$-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 $|λ|^\varkappa$ and $λ|λ|^{\varkappa-1}$, where $\varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.