Generalized Hilbert operators on Hardy spaces

Abstract

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 $1<p\le 2$: \[ (\mathcal R_p,H^p)=H\left(p,\infty,\frac1{p'}\right). \] For $p>2$, we prove that the multiplier space $({\mathcal{R}}_p,H^p)$ is strictly contained in $H\left(p,\infty ,\frac{1}{p^{\prime }}\right)$. 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 $p$.

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