Abstract
Let Ω be a pseudoconvex domain in \(\mathbb {C}^{n}\) with smooth boundary bΩ. We define general estimates \((f\text {-}\mathcal {M})^{k}_{\Omega }\) and \((f\text {-}\mathcal {M})^{k}_{b{\Omega }}\) on k-forms for the complex Laplacian □ on Ω and the Kohn–Laplacian □ b on bΩ. For 1 ≤ k ≤ n−2, we show that \((f\text {-}\mathcal {M})^{k}_{b{\Omega }}\) holds if and only if \((f\text {-}\mathcal {M})^{k}_{\Omega }\) and \((f\text {-}\mathcal {M})^{n-k-1}_{\Omega }\) hold. Our proof relies on Kohn’s method in Kohn (Ann. Math. 156(2), 213–248, 2002).
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Acknowledgments
This research is partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2012.16. The author gratefully acknowledges the careful reading by the referee. The exposition and rigor of the paper were improved by the close reading.
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This author was a Plenary speaker at the Vietnam Congress of Mathematicians 2013.
Dedicated to Professor Eberhard Zeidler on the occasion of his 75th birthday.
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Khanh, T.V. Equivalence of Estimates on a Domain and Its Boundary. Vietnam J. Math. 44, 29–48 (2016). https://doi.org/10.1007/s10013-015-0160-0
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DOI: https://doi.org/10.1007/s10013-015-0160-0