On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates
Liding Yao, Yuan Zhang
TL;DR
The paper addresses solving the ∂̄ equation on product domains Ω=∏_{j=1}^m Ω_j with optimal Sobolev regularity across all k∈Z and 1<p<∞. It builds product-form Nijenhuis–Woolf homotopy formulae by combining factorwise NW operators using a Fubini-based decomposition of Sobolev spaces, producing global P^Ω and H_q^Ω that satisfy f = ∂̄H_q^Ω f + H_{q+1}^Ω ∂̄ f and f = P^Ω f + H_1^Ω ∂̄ f with sharp Sobolev bounds. The main contributions include unconditional W^{k,p} solvability for all (0,q)-forms on product domains with Lipschitz factors (and stronger type domains for each factor), explicit product NW formulae and expansions, and the demonstration that the Sobolev regularity is sharp via Kerzman-type examples; the work extends NW-type integral representations to higher-dimensional, non-planar factors and provides a unified framework for ∂̄-solvability on complex product domains. These results have implications for several complex variables and PDEs, offering robust, scale-invariant Sobolev control in settings beyond classical pseudoconvex or finite-type domains.
Abstract
We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $Ω_1\times\dots\timesΩ_m$, where each $Ω_j$ is either a bounded Lipschitz domain in $\mathbb C^1$, a bounded strongly pseudoconvex domain with $C^2$ boundary, or a smooth convex domain of finite type. Such homotopy operators $\mathcal H_q$ yield solutions to the $\overline\partial$ equation with optimal Sobolev regularity $W^{k,p}\to W^{k,p}$ simultaneously for all $k\in\mathbb Z$ and $1<p<\infty$.