[Paper Review] Closed range of $\bar\partial$ in $L^2$-Sobolev spaces on unbounded domains in $\mathbb{C}^n$

Phillip S. Harrington, Andrew Raich|arXiv (Cornell University)|Apr 25, 2017

Advanced Harmonic Analysis Research6 references3 citations

TL;DR

This paper establishes a sufficient condition—weak $Z(q)$—for the closed range of the $ar heta$ operator on $(0,q)$-forms in weighted $L^2$-Sobolev spaces over unbounded domains in $\mathbb{C}^n$, generalizing classical results to non-pseudoconvex and unbounded settings. The key contribution is proving that weighted Sobolev spaces are essential for closed range in such domains, with the weak $Z(q)$ condition ensuring regularity and solvability of the $ar heta$-Neumann problem.

ABSTRACT

Let $Ω\subset\mathbb{C}^n$ be a domain and $1 \leq q \leq n-1$ fixed. Our purpose in this article is to establish a general sufficient condition for the closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$-Sobolev spaces on $(0,q)$-forms. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and, even more critically, in $L^2$-Sobolev spaces.

Motivation & Objective

• To establish a general sufficient condition for the closed range of the $\bar\partial$ operator on $(0,q)$-forms in weighted $L^2$-Sobolev spaces over unbounded domains in $\mathbb{C}^n$.
• To generalize the classical $Z(q)$ condition to a weaker curvature condition—weak $Z(q)$—that applies to non-pseudoconvex and unbounded domains.
• To demonstrate that weighted $L^2$-Sobolev spaces are necessary for closed range estimates in unbounded domains, where unweighted spaces fail due to geometric and analytic obstructions.
• To extend the $\bar\partial$-Neumann theory to unbounded domains by constructing weighted Sobolev estimates and proving regularity of solutions.

Proposed method

• Introduce the weak $Z(q)$ condition as a curvature condition on the Levi form that generalizes the classical $Z(q)$ condition and applies to unbounded domains.
• Use weighted $L^2$-Sobolev norms with weight $e^{-t|z|^2}$ to stabilize the $\bar\partial$-Neumann problem on unbounded domains.
• Apply integration by parts and commutator estimates to control higher-order derivatives of $\bar\partial$ and its adjoint $\bar\partial_t^*$ in weighted Sobolev norms.
• Leverage the tangentiality of vector fields near the boundary to control commutators and absorb lower-order terms via small-constant/large-constant arguments.
• Prove weighted $H^k$ estimates for the $\bar\partial$-Neumann operator $N_{q,t}$, showing $\|u\|_{t,k,\Omega}^2 \leq C_k(\|\bar\partial u\|_{t,k,\Omega}^2 + \|\bar\partial_t^* u\|_{t,k,\Omega}^2) + C_{k,t}\|u\|_{t,k-1,\Omega}^2$.
• Use the canonical solution operators and continuity of the $\bar\partial$-Neumann operator to establish solvability in $C^\infty$ and regularity up to the boundary.

Experimental results

Research questions

• RQ1 Can the closed range of $\bar\partial$ on $(0,q)$-forms be established in $L^2$-Sobolev spaces on unbounded domains without assuming pseudoconvexity or boundedness?
• RQ2 What is the weakest curvature condition on the Levi form that ensures closed range of $\bar\partial$ in weighted $L^2$-Sobolev spaces on unbounded domains?
• RQ3 Why are weighted $L^2$-Sobolev spaces necessary for closed range estimates in unbounded domains, and what geometric obstructions prevent unweighted estimates?
• RQ4 How does the weak $Z(q)$ condition generalize classical $Z(q)$ and related conditions in the literature, and what is its role in ensuring regularity of the $\bar\partial$-Neumann operator?
• RQ5 Can the $\bar\partial$-Neumann problem be solved in $C^\infty$ on unbounded domains using weighted Sobolev estimates, and what are the structural requirements on the weight and domain?

Key findings

• The weak $Z(q)$ condition is a sufficient curvature condition for the closed range of $\bar\partial$ on $(0,q)$-forms in weighted $L^2$-Sobolev spaces on unbounded domains in $\mathbb{C}^n$, even when the domain is not pseudoconvex.
• Weighted $L^2$-Sobolev spaces are essential for closed range estimates in unbounded domains; unweighted estimates fail due to the presence of arbitrarily large balls, as shown by a scaling argument with $u_R$.
• The $\bar\partial$-Neumann operator $N_{q,t}$ satisfies weighted $H^k$ estimates: $\|u\|_{t,k,\Omega}^2 \leq C_k(\|\bar\partial u\|_{t,k,\Omega}^2 + \|\bar\partial_t^* u\|_{t,k,\Omega}^2) + C_{k,t}\|u\|_{t,k-1,\Omega}^2$ for $t$ large enough.
• The theory extends to $C^\infty$-smooth solutions via standard arguments, showing that the $\bar\partial$-Neumann problem is solvable in $C^\infty$ on unbounded domains under the weak $Z(q)$ condition.
• The use of tangential vector fields in the proof allows control of commutators and absorption of lower-order terms, enabling the derivation of the key $H^k$ estimates.
• The failure of the Rellich lemma in unbounded domains with infinitely many disjoint balls of fixed radius implies that $H^1(\Omega)$ is not compact in $L^2(\Omega)$, justifying the need for weighted spaces in Sobolev theory.