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On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates (2502.00925v1)

Published 2 Feb 2025 in math.CV

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 $\Omega_1\times\dots\times\Omega_m$, where each $\Omega_j$ is either a bounded Lipschitz domain in $\mathbb C1$, a bounded strongly pseudoconvex domain with $C2$ 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 $1<p<\infty$.

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