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For α ∈ (1,2) we consider the equation ${\partial }_{t}u={\Delta }^{\alpha /2}u+b·\nabla u$, where b is a time-independent, divergence-free singular vector field of the Morrey class $M{₁}^{1-\alpha }$. We show that if the Morrey norm ${\left|\right|b\left|\right|}_{M{₁}^{1-\alpha }}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

We give a series representation of the logarithm of the bivariate Laplace exponent of -stable processes for almost all ∈ (0, 2].

On considère le noyau de Poisson du processus $\alpha$-stable symétrique pour un domaine conique. Puis on considère le problème d’intégrabilité du noyau de Poisson à la puissance $p$. On donne des conditions sur $q$ pour qu’il existe une solution au problème de Dirichlet pour les fonctions $\alpha$-harmoniques sur les domaines coniques, avec une condition au bord donnée par une fonction de $L^q$.

We construct the fundamental solution of ${\partial }_t-{\Delta }_y-q(t,y)$ for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We...