### The atomic decomposition of harmonic functions satisfying certain conditions of integrability.

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We study nonnegative functions which are harmonic on a Lipschitz domain with respect to symmetric stable processes. We prove that if two such functions vanish continuously outside the domain near a part of its boundary, then their ratio is bounded near this part of the boundary.

The purpose of the paper is to extend results of the potential theory of the classical Schrödinger operator to the α-stable case. To obtain this we analyze a weak version of the Schrödinger operator based on the fractional Laplacian and we prove the Conditional Gauge Theorem.

We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone.

For ${C}^{1,1}$ domains we give exact asymptotics near the domain’s boundary for the Green function and Martin kernel of the rotation invariant α-stable Lévy process. We also obtain a relative Fatou theorem for harmonic functions of the stable process.

We characterize those homogeneous translation invariant symmetric non-local operators with positive maximum principle whose harmonic functions satisfy Harnack's inequality. We also estimate the corresponding semigroup and the potential kernel.

We propose a new general method of estimating Schrödinger perturbations of transition densities using an auxiliary transition density as a majorant of the perturbation series. We present applications to Gaussian bounds by proving an optimal inequality involving four Gaussian kernels, which we call the 4G Theorem. The applications come with honest control of constants in estimates of Schrödinger perturbations of Gaussian-type heat kernels and also allow for specific non-Kato perturbations.

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 ${L}^{q}$.

The 2018 Forum of Mathematician was organized with cooperation of Italian Mathematical Union (Union e Matematica Italiana) and the Italian Society of Industrial and Applied Mathematics (Società Italiana di Matematica Applicata e Industriale). The meeting aims at continuation of the tradition of bilateral meetings held in the last years by the Polish Mathematical Society together with other national societies. This forum does not exclude the participation of mathematicians from other countries. The...

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...

We investigate properties of functions which are harmonic with respect to α-stable processes on d-sets such as the Sierpiński gasket or carpet. We prove the Harnack inequality for such functions. For every process we estimate its transition density and harmonic measure of the ball. We prove continuity of the density of the harmonic measure. We also give some results on the decay rate of harmonic functions on regular subsets of the d-set. In the case of the Sierpiński gasket we even obtain the Boundary...

We study Fourier multipliers resulting from martingale transforms of general Lévy processes.

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