Dirichlet finite biharmonic functions on the Poincaré N-ball.
Let be a locally compact group and a compact subgroup such that the algebra of biinvariant integrable functions is commutative. We characterize the -invariant Dirichlet forms on the homogeneous space using harmonic analysis of . This extends results from Ch. Berg, Séminaire Brelot-Choquet-Deny, Paris, 13e année 1969/70 and J. Deny, Potential theory (C.I.M.E., I ciclo, Stresa), Ed. Cremonese, Rome, 1970. Every non-zero -invariant Dirichlet form on a symmetric space of non compact type...
Let be a sub-laplacian on a stratified Lie group . In this paper we study the Dirichlet problem for with -boundary data, on domains which are contractible with respect to the natural dilations of . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.
We prove good- inequalities for the area integral, the nontangential maximal function, and the maximal density of the area integral. This answers a question raised by R. F. Gundy. We also prove a Kesten type law of the iterated logarithm for harmonic functions. Our Theorems 1 and 2 are for Lipschitz domains. However, all our results are new even in the case of .
Given an irreducible algebraic curves in , let be the dimension of the complex vector space of all holomorphic polynomials of degree at most restricted to . Let be a nonpolar compact subset of , and for each choose points in . Finally, let be the -th Lebesgue constant of the array ; i.e., is the operator norm of the Lagrange interpolation operator acting on , where is the Lagrange interpolating polynomial for of degree at the points . Using techniques of pluripotential...
L’object de ce travail est l’etude des fonctions fonctions localement sommable sur , vérifiant (où est Laplacien pris au sens des distributions) et que se comportent à l’infini comme des fonctions sousharmoniques. En parculier, nous caractérisons les fonctious qui sont à la fois bi-sousharmoniques et sousharmoniques.