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Dimension of the harmonic measure of non-homogeneous Cantor sets

Athanasios Batakis (2006)

Annales de l’institut Fourier

We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets is a continuous function of the parameters determining these sets. This results extends a previous one of the author and do not use ergotic theoretic tools, not applicables to our case.

Dirichlet forms on symmetric spaces

Christian Berg (1973)

Annales de l'institut Fourier

Let G be a locally compact group and K a compact subgroup such that the algebra L 1 ( G ) ♯ of biinvariant integrable functions is commutative. We characterize the G -invariant Dirichlet forms on the homogeneous space G / K using harmonic analysis of L 1 ( G ) ♯ . 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 G -invariant Dirichlet form on a symmetric space G / K of non compact type...

Dirichlet problem with L p -boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli, Ermanno Lanconelli (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let ℒ be a sub-laplacian on a stratified Lie group G . In this paper we study the Dirichlet problem for ℒ with L p -boundary data, on domains Ω which are contractible with respect to the natural dilations of G . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for ℒ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Distribution function inequalities for the density of the area integral

R. Banuelos, C. N. Moore (1991)

Annales de l'institut Fourier

We prove good- λ 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 R + 2 .

Distribution of nodes on algebraic curves in ℂ N

Thomas Bloom, Norman Levenberg (2003)

Annales de l’institut Fourier

Given an irreducible algebraic curves A in ℂ N , let m d be the dimension of the complex vector space of all holomorphic polynomials of degree at most d restricted to A . Let K be a nonpolar compact subset of A , and for each d = 1 , 2 , . . . , choose m d points { A d j } j = 1 , . . . , m d in K . Finally, let Λ d be the d -th Lebesgue constant of the array { A d j } ; i.e., Λ d is the operator norm of the Lagrange interpolation operator L d acting on C ( K ) , where L d ( f ) is the Lagrange interpolating polynomial for f of degree d at the points { A d j } j = 1 , . . . , m d . Using techniques of pluripotential...

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