\overline∂_{𝑏}-equations on certain unbounded weakly pseudo-convex domains

Kang, Hyeonbae

doi:10.1090/S0002-9947-1989-0989577-5

$\overline \partial _ b$-equations on certain unbounded weakly pseudo-convex domains
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Trans. Amer. Math. Soc. 315 (1989), 389-413 Request permission

Abstract:

We found an explicit closed formula for the relative fundamental solution of ${\bar \partial _b}$ on the surface ${H_k} = \{ ({z_1},{z_2}) \in {\mathbb {C}^2}:\operatorname {Im} {z_2} = |{z_1}{|^{2k}}\}$ . We then make estimates of the relative fundamental solution in terms of the nonisotropic metric associated with the surface. The estimates lead us to the regularity results. We also study the problem of finding weights $\omega$ so that ${\bar \partial _b}$ as an operator from $L_\omega ^2$ to ${L^2}$ has a closed range. We find the best possible weight among radial weights.

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