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On the Neumann-Poincaré operator

Josef Král, Dagmar Medková (1998)

Czechoslovak Mathematical Journal

Let Γ be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L C 2 ( Γ ) the space of all complex-valued functions on Γ which are square integrable w.r. to the arc-length on Γ . Let L 2 ( Γ ) stand for the space of all real-valued functions in L C 2 ( Γ ) and put L 0 2 ( Γ ) = { h L 2 ( Γ ) Γ h ( ζ ) | d ζ | = 0 } . Since the Cauchy singular operator is bounded on L C 2 ( Γ ) , the Neumann-Poincaré operator C 1 Γ sending each h L 2 ( Γ ) into C 1 Γ h ( ζ 0 ) : = ( π i ) - 1 P . V . Γ h ( ζ ) ζ - ζ 0 d ζ , ζ 0 Γ , is bounded on L 2 ( Γ ) . We show that the inclusion C 1 Γ ( L 0 2 ( Γ ) ) L 0 2 ( Γ ) characterizes the circle in the class of all...

On the spectral Nevanlinna-Pick problem

Constantin Costara (2005)

Studia Mathematica

We give several characterizations of the symmetrized n-disc Gₙ which generalize to the case n ≥ 3 the characterizations of the symmetrized bidisc that were used in order to solve the two-point spectral Nevanlinna-Pick problem in ℳ ₂(ℂ). Using these characterizations of the symmetrized n-disc, which give necessary and sufficient conditions for an element to belong to Gₙ, we obtain necessary conditions of interpolation for the general spectral Nevanlinna-Pick problem. They also allow us to give a...

On the stability by convolution product of a resurgent algebra

Yafei Ou (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We consider the space of holomorphic functions at the origin which extend analytically on the universal covering of ω , ω . We show that this space is stable by convolution product, thus is a resurgent algebra.

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