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

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

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 $m(ξ) \not= 0$ for $ξ\not= 0$.

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

Abstract

We study the equation in a large class of sub-exponentially growing functions. Under appropriate restrictions on , we show that every such solution can be analytically continued to a sub-exponentially growing entire function on if and only if for .
Paper Structure (5 sections, 8 theorems, 138 equations)

This paper contains 5 sections, 8 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Submultiplicative functions and the Beurling--Domar condition
  3. Estimates for entire functions
  4. Main results
  5. Concluding remarks

Key Result

Lemma 2.1

Let $g : \mathbb{R}^n\to [1, \infty)$ be a locally bounded, measurable submultiplicative function satisfying the Beurling--Domar condition B-D. Then for every $\varepsilon > 0$, there exists $R_\varepsilon \in (0, \infty)$ such that

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 4.1
  • ...and 12 more