论文标题
衍生物 - 希尔伯特操作员，作用于强壮的空间
A Derivative-Hilbert operator acting on Hardy spaces
论文作者
论文摘要
在间隔[0,1）上设置$μ$为阳性的borel量度。带有条目的hankel矩阵$ \ MATHCAL {h}_μ=（μ_{n，k}）_ {n，k \ geq0} $，带有$μ_{n，k} =μ__{n+k} $ $$ \ MATHCAL {DH}_μ（f）（z）= \ sum_ {n = 0}^\ infty（\ sum_ {K = 0}^\inftyμ_{n，k} a _k）（n+1）z^n $ $单位光盘$ \ mathbb {d} $。我们在$ [0,1）$上表征了这些积极的borel措施，以使$ \ nathcal {dh}_μ（f）（z）= \ int _ {[0,1）} \ frac {f（t）} {（t）} {（1-tz）^2}dμ（t） $ \ MATHCAL {DH}_μ$是$ h^p（0 <p <\ infty）$ to $ h^q（q> p $和$ q \ geq 1 $）的有限（sups。，compact）运算符。 我们还研究了Hardy空间中的类似问题$ h^p（1 \ leq p \ leq 2）$。
Let $μ$ be a positive Borel measure on the interval [0,1). The Hankel matrix $\mathcal{H}_μ= (μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}= μ_{n+k}$, where $μ_n=\int_{ [0,1)}t^ndμ(t)$, induces formally the operator $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^\infty (\sum_{k=0}^\infty μ_{n,k}a_k)(n+1)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. We characterize those positive Borel measures on $[0,1)$ such that $\mathcal{DH}_μ(f)(z)= \int_{[0,1)} \frac{f(t)}{(1-tz)^2} dμ(t)$ for all in Hardy spaces $H^p(0<p<\infty)$, and among them we describe those for which $\mathcal{DH}_μ$ is a bounded(resp.,compact) operator from $H^p(0<p <\infty)$ into $H^q(q > p$ and $q\geq 1$). We also study the analogous problem in Hardy spaces $H^p(1\leq p\leq 2)$.
