AI 中文总结
该研究将半直线上分数阶离散拉普拉斯算子的Toeplitz实现视为全空间算子的压缩，证明了强最优的分数阶Hardy不等式，且等号无法由非零元达到，进而得到离散Birman不等式的分数阶推广。
AI 中文摘要
我们将半直线$\n\mathbb{N}$上的分数阶离散拉普拉斯算子$(-\Delta)^\alpha$的Toeplitz实现，视为全直线分数阶离散拉普拉斯算子到$\ell^{2}(\mathbb{N})$的压缩。对所有$\alpha>0$，我们证明分数阶Hardy不等式$$(-\Delta)^\alpha\geq\frac{4^\alpha\Gamma^2(\alpha+1/2)}\pi\frac{\Gamma(2\\,\cdot\\,-1)}{\Gamma(2\\,\cdot\\,-1+2\alpha)}$$在$\ell^{2}(\mathbb{N})$上成立，且在强意义下是最优的。特别地，我们表明该不等式无法改进，且$\ell^{2}(\mathbb{N})$中任何非零元都无法达到等号。作为推论，我们推导出离散Birman不等式的一个分数阶推广。
英文摘要
We consider a Toeplitz realisation of the fractional discrete Laplacian $(-Δ)^α$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $α>0$, we prove that the fractional Hardy inequality $$(-Δ)^α\geq\frac{4^αΓ^2(α+1/2)}π\frac{Γ(2\,\cdot\,-1)}{Γ(2\,\cdot\,-1+2α)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.
