Biholomorphic invariance of capacity and the capacity of an annulus
Bivariate Revuz measures and the Feynman-Kac formula
BMO and Lipschitz approximation by solutions of elliptic equations
We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.
Boundary behaviour of eigenfunctions of the Laplace operator on trees
Boundary behaviour of eigenfunctions of the Laplacian in a bi-tree.
Boundary estimates for derivatives of harmonic functions
Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications.
Boundary potential theory for stable Lévy processes
We investigate properties of harmonic functions of the symmetric stable Lévy process on without the assumption that the process is rotation invariant. Our main goal is to prove the boundary Harnack principle for Lipschitz domains. To this end we improve the estimates for the Poisson kernel obtained in a previous work. We also investigate properties of harmonic functions of Feynman-Kac semigroups based on the stable process. In particular, we prove the continuity and the Harnack inequality for...
Boundary value problems and layer potentials on manifolds with cylindrical ends
We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the...
Bounded Analytic Functions in the Dirichlet Space.
Bounded holomorphic functions with multiple sheeted pluripolar hulls
We describe compact subsets K of ∂𝔻 and ℝ admitting holomorphic functions f with the domains of existence equal to ℂ∖K and such that the pluripolar hulls of their graphs are infinitely sheeted. The paper is motivated by a recent paper of Poletsky and Wiegerinck.
Brownian motion on fractals and function spaces.
Capacitary integrals in Drichlet spaces.
Capacitary strong type estimates in semilinear problems
We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures for which the Sobolev space can be imbedded into . The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...
Capacités de Choquet finies et profinies
On définit les capacités de Choquet dans le cas fini en utilisant une forme bilinéaire non dégénérée associée à la base de Choquet. On montre que, dans le cas fini, une capacité de Choquet est la donnée d’un convexe de mesure qu’on caractérise. Le cas profini, issu des arbres, est obtenu par passage à la limite projective du cas fini. Sur les capacités profinies, on définit une forme bilinéaire dont le rapport avec l’intégration, dans des cas simples, est étudié.
Capacités gaussiennes
On étudie les espaces de Sobolev construits sur un espace localement convexe muni d’une mesure gaussienne centree . Si est de Radon, on démontre que les capacités naturelles sont tendues sur les compacts. Cela résulte d’un principe général relatif aux quasi-normes.On s’intéresse également aux fonctions quasi-continues a valeurs banachiques, ce qui est utile pour les propriétés de Nikodym, et à des applications à la continuité des trajectoires des intégrales stochastiques.
Capacités invariantes par translation
Capacities and Geometric Transformations of Subsets.
Capacity associated to a positive closed current. (Capacité associée à un courant positif fermé.)
Cauchy-Poisson transform and polynomial inequalities
We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in is Hölder continuous then E admits a Szegö type inequality with weight function with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.