Journal
AMERICAN JOURNAL OF MATHEMATICS
Volume 144, Issue 4, Pages 1115-1157Publisher
JOHNS HOPKINS UNIV PRESS
DOI: 10.1353/ajm.2022.0025
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Funding
- NSF [DMS-1448846]
- National Science Foundation Mathematical Science Postdoctoral Research Fellowship [DMS-1606260]
- Royal Swedish Academy of Sciences from Stiftelsen GS Magnusons fond [MG2018-0092]
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This study examines the boundary behavior of rational inner functions (RIFs) in higher dimensions from both analytic and geometric perspectives. The results highlight the loss of certain favorable behavior observed in the two-variable case.
We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree (m, n, 1) RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of nontrivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.
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