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Phragmén-Lindelöf conditions for graph varieties

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Abstract

For Pm ∈ ₵[z1, …, zn], homogeneous of degree m we investigate when the graph of Pm in ₵n+1 satisfies the Phragmén-Lindelöf condition PL(ℝn+1, log), or equivalently, when the operator \(i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)\) admits a continuous solution operator on C∞(ℝn+1). This is shown to happen if the varieties V+- ≔ {z ∈ ₵n: Pm(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V+- and u(x) ≤ 0 on V+- ∩ ℝn also satisfies u(z) A¦ Im z¦ on V+-. Necessary as well as sufficient conditions for V+- to satisfy (SPL) are derived and several examples are given.

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Authors and Affiliations

  1. Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Düsseldorf

    Reinhold Meise

  2. Department of Mathematics, University of Michigan, Ann Arbor, USA

    B. A. Taylor

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  1. Reinhold Meise
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  2. B. A. Taylor
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Meise, R., Taylor, B.A. Phragmén-Lindelöf conditions for graph varieties. Results. Math. 36, 121–148 (1999). https://doi.org/10.1007/BF03322107

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