改进的离散对偶 $p$-Hardy 不等式与带权离散 $p$-Birman 不等式

中文摘要 AI 辅助

本文建立了带移位的一维广义离散对偶 $p$-Hardy 不等式的新版本。利用该广义离散对偶 $p$-Hardy 不等式，我们获得了两个离散对偶 $p$-Hardy 不等式的改进。具体而言，对于 $p>1$ 且满足 $A_{0}=A_{1}=0$ 的 $A\in C_c(\mathbb{N}_{0})$，我们首先改进了离散对偶 $p$-Hardy 不等式 \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^{p}| A_{n}-A_{n-1}|^{p}\geq\frac{1}{p^{p}}\displaystyle\sum_{n=2}^{\infty}|A_{n}|^{p}, \end{align*} 其中相关常数项是精确的。随后，我们改进了其幂型加权离散对偶 $p$-Hardy 扩展 \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^{\alpha}|A_{n}-A_{n-1}|^{p}\geq\Big(\frac{\alpha+1-p}{p}\Big)^{p} \displaystyle\sum_{n=2}^{\infty}\frac{|A_{n}|^{p}}{n^{p-\alpha}} \end{align*} 对于 $p-1<\alpha\leq p$，其中相关常数项也是精确的。我们还建立了带幂权的离散 $p$-Birman 不等式。此外，我们建立了具有精确常数的多变量对偶 $p$-Hardy 不等式。证明过程首先建立两个变量的不等式，然后将论证扩展到多个变量，同时保持常数的精确性。

英文摘要

In this paper, we establish a new version of one dimensional generalized discrete dual $p$-Hardy inequality with a shift. Using this generalized discrete dual p-Hardy inequality, we obtain improvements of two discrete dual $p$-Hardy inequalities. To be specific, for $p>1$ and $A\in C_c(\mathbb{N}_{0})$ satisfying $A_{0}=A_{1}=0$, we first improve the discrete dual p-Hardy inequality \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^{p}| A_{n}-A_{n-1}|^{p}\geq\frac{1}{p^{p}}\displaystyle\sum_{n=2}^{\infty}|A_{n}|^{p}, \end{align*} where the associate constant term is sharp. Subsequently, we improve its power-type weighted discrete dual p-Hardy extension \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^α|A_{n}-A_{n-1}|^{p}\geq\Big(\frac{α+1-p}{p}\Big)^{p} \displaystyle\sum_{n=2}^{\infty}\frac{|A_{n}|^{p}}{n^{p-α}} \end{align*} for $p-1<α\leq p$, where the associated constant term is also sharp. We also establish a discrete $p$- Birman inequality with power weights. Furthermore, we establish a multivariable dual $p$-Hardy inequality with a sharp constant. The proof proceeds by first establishing the inequality for two variables and then extending the argument to multiple variables, while preserving the sharpness of the constant.

发表机构

• Indian Institute of Science（印度科学学院）

机构由 AI 辅助整理，请以论文原文为准。