Weighted L^2 Version of Mergelyan and Carleman Approximation

Séverine Biard,Jujie Wu,John Erik Fornæss

doi:10.1007/s12220-020-00418-x

Abstract

We study the density of polynomials in H^2(E,varphi ), the space of square integrable functions with respect to mathrm{e}^{-varphi }mathrm{d}m and holomorphic on the interior of E in {mathbb {C}}, where varphi is a subharmonic function and dm is a measure on E. We give a result where E is the union of a Lipschitz graph and a Carathéodory domain, which we state as a weighted L^2-version of the Mergelyan theorem. We also prove a weighted L^2-version of the Carleman theorem.

Highlights

Let E ⊂ C be a measurable set, m a measure on E and φ a measurable function, locally bounded above on E
We generalize the classical holomorphic approximation theorems to weighted L2-spaces
We prove the following weighted L2-version of the Weierstrass theorem: Theorem 1.1 Let γ be a Lipschitz graph over a bounded interval and φ a subharmonic function in a neighborhood of γ in C

Summary

Introduction

Let E ⊂ C be a measurable set, m a measure on E and φ a measurable function, locally bounded above on E. We prove the following weighted L2-version of the Weierstrass theorem: Theorem 1.1 Let γ be a Lipschitz graph over a bounded interval and φ a subharmonic function in a neighborhood of γ in C. Thanks to Theorem 1.1, we prove the following weighted L2-version of the Carleman theorem: Theorem 1.4 Let be the graph of a locally Lipschitz function over the real axis in C and φ a subharmonic function in a neighborhood of. We recall the following result which is a converse of the previous property in complex dimension one: Theorem 2.11 If φ ≡ −∞ is subharmonic and ν(φ)(z) < 2 for a point z, e−φ is locally integrable in a neighborhood of z. Theorem 2.12 Let γ be a Lipschitz graph and φ ≡ −∞ a subharmonic function on C. 1 2π φ(γ (t)) < 1 so this is a direct consequence of Corollary 2.10

Many authors refer to this result as “Skoda’s exponential integrability”

Rectifiable Non-Lipschitz Arcs