论文标题
希尔伯特操作员和衍生产品的范围，以$ h^1 $为代表
The range of Hilbert operator and Derivative-Hilbert operator acting on $H^1$
论文作者
论文摘要
令$μ$为间隔$ [0,1）$的积极borel量度。 hankel矩阵$ \ Mathcal {h}_μ=（μ_{n，k}）_ {n，k \ geq0} $带有条目$μ_{n，k} =μ__{n+k} $对于$ f（z）= \ sum_ {n = 0}^{\ infty} a_nz^n $是$ \ mathbb {d} $中的分析函数，希尔伯特操作员定义了$$ \ MATHCAL {h}_μ（f）（z）= \ sum_ {n = 0}^{\ infty} \ bigG（\ sum_ {k = 0}^{\ infty}^{\ infty}μ__{操作员的定义为$ \ Mathcal {dh}_μ（f）（z）= \ sum_ {n = 0}^{\ infty} \ bigG（\ sum_ {k = 0}^{\ infty} {\ infty}^{ \ Mathbb {d}。$$在本文中，我们确定了希尔伯特操作员和衍生物 - 希尔伯特操作员的范围，该操作员在$ h^{\ infty} $上作用。
Let $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_μ=(μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}=μ_{n+k}$, where $μ_n=\int_{[0,1)}t^{n}dμ(t)$. For $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$, the Hilbert operator is defined by $$\mathcal{H}_μ(f)(z)=\sum_{n=0}^{\infty}\Bigg(\sum_{k=0}^{\infty}μ_{n,k}a_k\Bigg)z^n, \quad z\in \mathbb{D}.$$ The Derivative-Hilbert operator is defined as $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^{\infty}\Bigg(\sum_{k=0}^{\infty}μ_{n,k}a_k\Bigg)(n+1)z^n, \quad z\in \mathbb{D}.$$ In this paper, we determine the range of the Hilbert operator and Derivative-Hilbert operator acting on $H^{\infty}$.
