Gain of Regularity in Extension Problem on Convex Domains

M Jasiczak

doi:10.1155/2015/295759

Abstract

We investigate the extension problem from higher codimensional linear subvarieties on convex domains of finite type. We prove that there exists a constantdsuch that on Bergman spacesHp(D)with1≤p<dthere appears the so-called “gain regularity.” The constantddepends on the minimum of the dimension and the codimension of the subvariety. This means that the space of functions which admit an extension to a function in the Bergman spaceHp(D)is strictly larger thanHp(D∩A), whereAis a subvariety.

Highlights

Let D be a bounded pseudoconvex domain, ψ : D → R ∪ {−∞} a plurisubharmonic function, and A ⊂ Cn a complex linear hyperplane
The first one is to study the extension problem on different function spaces, for instance other Lp spaces, with the prominent case of bounded extensions of bounded holomorphic functions
Comparing with the Ohsawa-Takegoshi Theorem for ψ = 0, the result of Cumenge says that there is a strictly larger class of functions than L2(D ∩ A) ∩ H(D) which admit an extension to a holomorphic function in L2(D)

Summary

Introduction

Let D be a bounded pseudoconvex domain, ψ : D → R ∪ {−∞} a plurisubharmonic function, and A ⊂ Cn a complex linear hyperplane. Cumenge in [8] considered the extension problem in case of strictly pseudoconvex domains and subvarieties A of codimension k ≥ 1 which are nonsingular and cut bD transversally. She proved that in this case any holomorphic function in Lp((−r)sdVD∩A), s ≥ k, 1 ≤ p < ∞, admits an extension to a holomorphic function which belongs to Lp((−r)s−kdV). Comparing with the Ohsawa-Takegoshi Theorem for ψ = 0, the result of Cumenge says that there is a strictly larger class of functions than L2(D ∩ A) ∩ H(D) which admit an extension to a holomorphic function in L2(D). Such phenomena have been central in the whole ∂-problem theory and PDE’s in general (a natural example here is subelliptic estimates [9,10,11,12] on finite type domains)

Journal of Function Spaces

The last estimate follows from the fact that