An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions

Eugene Shargorodsky; David Berger; René L. Schilling; Teo Sharia

doi:10.4171/JST/509

An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions

Eugene Shargorodsky, David Berger, René L. Schilling, Teo Sharia

Research output: Contribution to journal › Article › peer-review

Abstract

We study the equation m(D)f D 0 in a large class of sub-exponentially growing functions. Under appropriate restrictions on m ∈ C(Rn), we show that every such solution can be analytically continued to a sub-exponentially growing entire function on Cn if, and only if, m(ζ) ≠ 0 for ζ ≠ 0.

Original language	English
Pages (from-to)	665-695
Number of pages	31
Journal	Journal of Spectral Theory
Volume	14
Issue number	2
DOIs	https://doi.org/10.4171/JST/509
Publication status	Published - 13 Jun 2024

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title = "An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions",

abstract = "We study the equation m(D)f D 0 in a large class of sub-exponentially growing functions. Under appropriate restrictions on m ∈ C(Rn), we show that every such solution can be analytically continued to a sub-exponentially growing entire function on Cn if, and only if, m(ζ) ≠ 0 for ζ ≠ 0.",

author = "Eugene Shargorodsky and David Berger and Schilling, {Ren{\'e} L.} and Teo Sharia",

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AB - We study the equation m(D)f D 0 in a large class of sub-exponentially growing functions. Under appropriate restrictions on m ∈ C(Rn), we show that every such solution can be analytically continued to a sub-exponentially growing entire function on Cn if, and only if, m(ζ) ≠ 0 for ζ ≠ 0.

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An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions

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