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Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of E with pole at infinity and M ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces

Marco Biroli, Umberto Mosco (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove local embeddings of Sobolev and Morrey type for Dirichlet forms on spaces of homogeneous type. Our results apply to some general classes of selfadjoint subelliptic operators as well as to Dirichlet operators on certain self-similar fractals, like the Sierpinski gasket. We also define intrinsic BV spaces and perimeters and prove related isoperimetric inequalities.

Solutions positives et mesure harmonique pour des opérateurs paraboliques dans des ouverts «lipschitziens»

Yanick Heurteaux (1991)

Annales de l'institut Fourier

Soit L un opérateur parabolique sur R n + 1 écrit sous forme divergence et à coefficients lipschitziens relativement à une métrique adaptée. Nous cherchons à comparer près de la frontière le comportement relatif des L -solutions positives sur un domaine “lipschitzien”. Dans un premier temps, nous démontrons un principe de Harnack uniforme pour certaines L -solutions positives. Ce principe nous permet alors de démontrer une inégalité de Harnack forte à la frontière pour certains couples de L -solutions positives....

Some applications of the trace condition for pluriharmonic functions in Cn.

Alessandro Perotti (2000)

Publicacions Matemàtiques

In this paper we investigate some applications of the trace condition for pluriharmonic functions on a smooth, bounded domain in Cn. This condition, related to the normal component on ∂D of the ∂-operator, permits us to study the Neumann problem for pluriharmonic functions and the ∂-problem for (0,1)-forms on D with solutions having assigned real part on the boundary.

Some Dirichlet spaces obtained by subordinate reflected diffusions.

Niels Jacob, René L. Schilling (1999)

Revista Matemática Iberoamericana

In this paper we want to show how well-known results from the theory of (regular) elliptic boundary value problems, function spaces and interpolation, subordination in the sense of Bochner and Dirichlet forms can be combined and how one can thus get some new aspects in each of these fields.

Some non-linear function theoretic properties of Riemannian manifolds.

Stefano Pigola, Marco Rigoli, Alberto G. Setti (2006)

Revista Matemática Iberoamericana

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.

Some properties of α-harmonic measure

Dimitrios Betsakos (2008)

Colloquium Mathematicae

The α-harmonic measure is the hitting distribution of symmetric α-stable processes upon exiting an open set in ℝⁿ (0 < α < 2, n ≥ 2). It can also be defined in the context of Riesz potential theory and the fractional Laplacian. We prove some geometric estimates for α-harmonic measure.

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