Compactness Estimates for the $$ \bar \partial $$ -Neumann Problem in Weighted L 2-spaces

Abstract

In this paper we discuss compactness estimates for the ∂-Neumann problem in the setting of weighted L 2 -spaces on C n . For this purpose we use a version of the Rellich -Lemma for weighted Sobolev spaces.2000 Mathematics Subject Classification. Primary 32W05; Secondary 32A36, 35J10.

Compactness for the $${\overline{\partial}}$$ -Neumann problem: a functional analysis approach

“…We are now able to give a short proof of the main result in [9] or [8] Proposition 2.5 Let ϕ be a plurisubharmonic C 2 -weight function. If the lowest eigenvalue μ ϕ (z) of the Levi -matrix M ϕ satisfies ( * * ), then N ϕ is compact.…”

“…. , n where χ R is a C ∞ cutoff function which is identically 1 on B 2R and zero outside B 3R and by Gårding's inequality for B 3R (see [5,8,15])…”

“…and that forms with coefficients in [8]). Now we suppose that the lowest eigenvalue μ ϕ of the Levi-matrix…”

“…We will give a new proof of the main result in [9] using a direct approach, see [1], Theorem 2.32 or [3], Corollaire IV.26, where two conditions are given which imply that a subset of an L 2 -space is relatively compact. The first of these conditions will correspond to Gårding's inequality (see for instance [5,8]) and the second condition corresponds to our assumption on the lowest eigenvalue of the Levi matrix M ϕ .…”

Closed range of∂¯in L2-Sobolev spaces on unbounded domains inCnHarrington1, Raich2 2018Journal of Mathematical Analysis and Applications7010View full textAdd to dashboardCite

“…This means that great care is required when defining Sobolev spaces. In [HR14], the authors developed the theory of weighted Sobolev spaces on unbounded domains building on ideas in [GH10] and [Gan]. Boundary smoothness also requires greater care, since derivatives of defining functions may still be unbounded even when the domain itself is smooth.…”