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

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

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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

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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 $m \in C(\mathbb{R}^n)$, we show that every such solution can be analytically continued to a sub-exponentially growing entire function on $\mathbb{C}^n$ if and only if $m(\xi) \not= 0$ for $\xi \not= 0$.

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David Berger, René L. Schilling, Eugene Shargorodsky, Teo Sharia. 2024-01-23. An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions. https://arxiv.org/abs/2401.12876

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