Asymptotic Hölder absolute values (Q1607913)

Write \(R\) for a commutative ring with an identity, \(\|\cdot\|\) and \(|\cdot |\) for mappings with source \(R\) and target \(\mathbb R\). Definitions: 1) Let \(1\leqslant c_1,c_2\in\mathbb R\). The mapping \(\|\cdot\|\) is said to be a \((c_1,c_2)\)-Hölder absolute value provided (HAV1) \(\forall x\in R, \qquad\|x\|=0\iff x=0\); (HAV2) \(\forall x\in R,\quad \forall y\in R, \qquad \|x+y\|\leqslant c_2(\|x\|+\|y\|)\); (HAV3) \(\forall x\in R,\quad \forall y\in R, \qquad c_1^{-1}\|x\|\cdot\|y\|\leqslant \|x\cdot y\|\leqslant c_1\|x\|\cdot\|y\|\). 2) Let \(\mathcal C_1 : ]1,+\infty [\times\mathbb N\to ]1,+\infty [\) be a mapping such that \(\exists L(\mathcal C_1)\in \mathbb R\) \(\forall \gamma >1\), \(L(\mathcal C_1)= \overline{\lim}_{n\to\infty} \frac{\log_2 \mathcal C_1(\gamma ,n)}n \), and \(c_2\in\mathbb R\), \(c_2\geqslant 1\). The mapping \(|\cdot |\) is said to be a \((\mathcal C_1,c_2)\)-asymptotic Hölder absolute value provided (HAV1) and (HAV2) hold for it and \( \forall\gamma >1\) \(\forall n\geqslant 2\) \(\forall(x_1,\dots ,x_n)\in R^n\) \[ \mathcal C_1(\gamma ,n)^{-1}\cdot\prod_{i=1}^n |x_i|^{{\gamma_i}^{-1}}\leqslant \Biggl|\prod_{i=1}^n x_i\Biggr|\leqslant \mathcal C_1(\gamma ,n)\cdot\prod_{i=1}^n |x_i|^{\gamma_i}. \] 3) Let \(p,q\in\mathbb R\). Then Hölder absolute values \(\|\cdot\|\) and \(|\cdot|\) are said to be \((p,q)\)-Hölder equivalent provided \(\forall x\in R\), \[ p^{-1}|x|^q\leqslant\|x\|\leqslant p|x|^q. \] Results obtained: 1) If \(\|\cdot\|\) is a \((c_1,c_2)\)-Hölder absolute value, \(\alpha=\log_2(2c_2)\) and \(\forall x\in R\), \(|x|=\lim_{n\to\infty}\|x\|^{\frac 1{n\alpha}}\) then \(|\cdot |\) is an absolute value and is \((c_1^\alpha,\alpha)\)-Hölder equivalent to \(\|\cdot\|\). 2) If \(\|\cdot\|\) is a \((\mathcal C_1,c_2)\)-asymptotic Hölder absolute value, and \(\beta =\frac 1{\log_2(2c_2)}\) then \((e^{L(\mathcal C_1)\beta},\beta)\) is Hölder equivalent to a certain absolute value provided \(L(\mathcal C_1)\neq 0\) and \(\|\cdot\|^{\beta}\) is an absolute value provided \(L(\mathcal C_1)=0\).