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

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

doi:10.48550/arXiv.2401.12876

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

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

Research output: Working paper/Preprint › Preprint

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Abstract

We study the equation m(D)ƒ=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(ξ)≠0 for ξ≠0.

Original language	English
Publisher	arXiv
Number of pages	25
DOIs	https://doi.org/10.48550/arXiv.2401.12876
Publication status	Published - 23 Jan 2024

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math.AP

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abstract = "We study the equation m(D)ƒ=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(ξ)≠0 for ξ≠0.",

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

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