AI 中文总结
该数学研究刻画了Hardy空间上广义Hilbert算子的有界性条件，否定了相关猜想，明确了乘子空间的包含关系与结构。
AI 中文摘要
设$g\in H(\mathbb D)$，广义Hilbert算子$\mathcal H_g$定义为：$\mathcal H_g(f)(z)=\int_0^1 f(t)g'(tz)dt$，其中$z\in \mathbb D$，$f \in H(\mathbb D)$。设$\mathcal R_p=\mathcal H(H^p)$为经典Hilbert算子在Hardy空间上的值域，配有拉回范数，$(\u0026#x1D4A6;_p,H^p)$表示Hadamard乘子空间。对$1\u0026lt;p\u0026lt;\infty$，证明了精确乘子刻画：$\mathcal H_g:H^{p}\longrightarrow H^{p}$有界当且仅当$g'\in(\u0026#x1D4A6;_p,H^p)$，且等价的Hilbert矩阵双线性判据$\mathfrak B_p(g)\u0026lt;\infty$。确定了$1\u0026lt;p\leq2$时的乘子空间：$(\u0026#x1D4A6;_p,H^p)=H\left(p,\infty,\frac1{p'}\right)$；对$p\u0026gt;2$，证明该乘子空间严格包含于$H\left(p,\infty,\frac1{p'}\right)$，表明$g\in \Lambda(p,1/p)$不蕴含$\mathcal H_g$在$H^p$上有界，否定了Galanopoulos、Girela、Peláez和Siskakis提出的猜想。此外，确定了两类已知的充分类包含于该乘子空间，由此对$g \in H(\mathbb D)$且具有非负递减Taylor系数的情况，得到$\mathcal H_g$在$H^{p}$上的完整系数刻画。还研究了$(\u0026#x1D4A6;_p,H^p)$的结构，发现其包含所有多项式及Cauchy变换，且该乘子空间随指数$p$构成严格递增族。
英文摘要
Let $g\in H(\mathbb D)$, the generalized Hilbert operator $\mathcal H_g$ is defined by \[ \mathcal H_g(f)(z)=\int_0^1 f(t)g'(tz)dt,\ \ z\in \mathbb D\, \ \ f \in H(\mathbb D). \] Let $\mathcal R_p=\mathcal H(H^p)$ be the range of the classical Hilbert operator on Hardy space, equipped with the pullback norm, and let $(\mathcal R_p,H^p)$ denote the Hadamard multiplier space. For $1<p<\infty$, we prove the exact multiplier characterization \[ \mathcal H_g:H^{p}\longrightarrow H^{p} \ \ \text{is bounded} \quad\Longleftrightarrow\quad g'\in(\mathcal R_p,H^p), \] and an equivalent Hilbert-matrix bilinear criterion $\mathfrak B_p(g)<\infty$. We identify the multiplier space completely when $1<p\le2$: \[ (\mathcal R_p,H^p)=H\left(p,\infty,\frac1{p'}\right). \] For $p>2$, we prove that the multiplier space $(\mathcal R_p,H^p)$ is strictly contained in $H\left(p,\infty,\frac1{p'}\right)$. This shows that \(g\in Λ(p,1/p)\) does not imply that $\mathcal H_g$ is bounded on \(H^p\), giving a negative answer to the conjecture posed by Galanopoulos, Girela, Peláez and Siskakis. In addition, we locate two previously known sufficient classes inside the multiplier space. This allow us obtain a complete coefficient characterization of $\mathcal H_g$ on $H^{p}$ for $g \in H(\mathbb D) $ with nonnegative decreasing Taylor coefficients. We then study the structure of $(\mathcal R_p,H^p)$. % It turns out that $(\mathcal{R}_p,H^p)$ contains all polynomials as well as Cauchy transforms. We show that the multiplier spaces $(\mathcal{R}_p,H^p)$ form a strictly increasing family with respect to the exponent $p$.
