2026/07/30 by Yuting Guo, Pengcheng Tang
Mathematics · #math.FA
arxiv created 2026/07/30 · arxiv updated 2026/07/31
Let g∈ H(\mathbb D), the generalized Hilbert operator \mathcal Hg is defined by \mathcal Hg(f)(z)=∫01 f(t)g'(tz)dt, z∈ \mathbb D f ∈ H(\mathbb D). Let \mathcal Rp=\mathcal H(Hp) be the range of the classical Hilbert operator on Hardy space, equipped with the pullback norm, and let (\mathcal Rp,Hp) denote the Hadamard multiplier space. For 1<p<∞, we prove the exact multiplier characterization \mathcal Hg:Hp\longrightarrow Hp is bounded \Longleftrightarrow g'∈(\mathcal Rp,Hp), and an equivalent Hilbert-matrix bilinear criterion \mathfrak Bp(g)<∞. We identify the multiplier space completely when 1<p≤2: (\mathcal Rp,Hp)=H(p,∞,\frac1p'). For p>2, we prove that the multiplier space (\mathcal Rp,Hp) is strictly contained in H(p,∞,\frac1p'). This shows that \(g∈ Λ(p,1/p)\) does not imply that \mathcal Hg is bounded on \(Hp\), 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 Hg on Hp for g ∈ H(\mathbb D) with nonnegative decreasing Taylor coefficients. We then study the structure of (\mathcal Rp,Hp). % It turns out that (Rp,Hp) contains all polynomials as well as Cauchy transforms. We show that the multiplier spaces (Rp,Hp) form a strictly increasing family with respect to the exponent p.