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

David Berger; Rene Shilling; Eugene Shargorodsky; Teo Sharia

doi:10.4171/JST/509

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

David Berger, Rene Shilling, Eugene Shargorodsky, Teo Sharia

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Abstract

We study the equation $m(D)f=0$ in a large class of sub-exponentially growing functions. Under appropriate restrictions on $m \in C(\real^n)$ we show that every such solution can be analytically continued to a sub-exponentially growing entire function on $\comp^n$ if, and only if, $m(\xi) \neq 0$ for $\xi \neq 0$.

Original language	English
Pages (from-to)	665-695
Number of pages	31
Journal	Journal of Spectral Theory
Volume	14
Issue number	2
Early online date	30 May 2024
DOIs	https://doi.org/10.4171/JST/509
Publication status	Published - 13 Jun 2024

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10.4171/JST/509Licence: CC BY

Final published versionFinal published version, 382 KBLicence: CC BY

Cite this

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

abstract = "We study the equation $m(D)f=0$ in a large class of sub-exponentially growing functions. Under appropriate restrictions on $m \in C(\real^n)$ we show that every such solution can be analytically continued to a sub-exponentially growing entire function on $\comp^n$ if, and only if, $m(\xi) \neq 0$ for $\xi \neq 0$.",

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N2 - We study the equation $m(D)f=0$ in a large class of sub-exponentially growing functions. Under appropriate restrictions on $m \in C(\real^n)$ we show that every such solution can be analytically continued to a sub-exponentially growing entire function on $\comp^n$ if, and only if, $m(\xi) \neq 0$ for $\xi \neq 0$.

AB - We study the equation $m(D)f=0$ in a large class of sub-exponentially growing functions. Under appropriate restrictions on $m \in C(\real^n)$ we show that every such solution can be analytically continued to a sub-exponentially growing entire function on $\comp^n$ if, and only if, $m(\xi) \neq 0$ for $\xi \neq 0$.

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