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On the\(\bar \partial \) equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

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Abstract

For a large class of subharmonicφ, the equation\(\bar \partial u = f\) is studied in\(\mathcal{H} = L^2 (\mathbb{C}^1 ,e^{ - \varphi } )\). Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL p norms are derived as a corollary. A class of counterexamples, related to the failure of\(\bar \partial _b \) to be analytic-hypoelliptic on certain CR manifolds, is discussed.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    Michael Christ

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  1. Michael Christ
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Additional information

Communicated by Steven Krantz

Research supported by the Alfred P. Sloan Foundation, Institut des Hautes Etudes Scientifiques, and National Science Foundation.

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Christ, M. On the\(\bar \partial \) equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1 . J Geom Anal 1, 193–230 (1991). https://doi.org/10.1007/BF02921303

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  • DOI: https://doi.org/10.1007/BF02921303

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