Hilbert-type operator induced by radial weight (Q1995750)

Let \({\mathcal H}(\mathbb{D})\) be the space of analytic functions on the open unit disc \(\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}\), and let \(H^p\) denote the Hardy space on \(\mathbb{D}\) with \(p\in[1,\infty]\). A nonnegative function \(\omega\in L^1(\mathbb{D})\) is called a radial weight if \(\omega(z)=\omega(|z|)\) for all \(z\in\mathbb{D}\). For \(p\in(1,\infty)\) and such an \(\omega\), the weighted Lebesgue space \(L^p_\omega\) consists of all complex-valued measurable functions \(f\) on \(\mathbb{D}\) such that \[ \big\|f\big\|^p_{L^p_\omega}=\int_{\mathbb{D}}|f(z)|^p\omega(z)dA(z)<\infty, \] where \(dA(z)=\frac{1}{\pi}dxdy\) is the normalized area measure on \(\mathbb{D}\). Then \(A^p_\omega=L^p_\omega\cap{\mathcal H}(\mathbb{D})\) is the corresponding weighted Bergman space. The classical weighted Bergman spaces \(A^p_\alpha\) are induced by the weights \(\omega(z)=(\alpha+1)(1-|z|^2)^\alpha\), \(\alpha>-1\). The Bloch space \({\mathcal B}\), the Dirichlet-type space \(D^p_{p-1}\) and the Hardy-Littlewood space \(HL(p)\) for \(p\in(1,\infty)\) consist of the functions \(f\in{\mathcal H}(\mathbb{D})\) such that, respectively, \begin{align*} \|f\|_{\mathcal B}&=|f(0)|+\sup_{z\in\mathbb{D}}(1-|z|^2)|f'(z)|<\infty,\\ \big\|f\big\|^p_{D^p_{p-1}}&=|f(0)|^p+\big\|f'\big\|^p_{A^p_{p-1}}<\infty,\\ \big\|f\big\|^p_{HL(p)}&=\sum_{n=0}^\infty|\widehat{f}_n|^p (n+1)^{p-2}<\infty\ \mbox{ for }\ f(z)=\sum_{n=0}^\infty\widehat{f}_n z^n. \end{align*} For radial weights \(\omega\) there exist Bergman reproducing kernels \(B^\omega_z \in A^2_\omega\), \[ f(z)=\int_{\mathbb{D}}f(\zeta){B^\omega_z(\zeta)}\omega(\zeta)dA(\zeta), \quad f\in A^2_\omega,\quad z\in\mathbb{D}. \] Then the Hilbert-type operator \(H_\omega\) is defined by \[ (H_\omega f)(z)=\int_0^1 f(t)\bigg(\frac{1}{z}\int_0^z B^\omega_t(\zeta)d\zeta \bigg)\omega(t)dt. \] A radial weight \(\omega\) belongs to the class \(\widehat{\mathcal D}\) if \(\widehat{\omega}(z)=\int_{|z|}^1\omega(s)ds>0\) for \(z\in\mathbb{D}\) and \(\widehat{\omega}\) possesses the doubling property \(\widehat{\omega}(r)\le C\widehat{\omega}\big(\frac{1+r}{2}\big)\) for a constant \(C=C(\omega)\ge 1\) and all \(r\in[0,1)\). For a radial weight \(\omega\), it is proved that the operator \(H_\omega:H^\infty\to{\mathcal B}\) is bounded if and only if \(\omega\in\widehat{\mathcal D}\). For \(\omega\in\widehat{\mathcal D}\), it is also proved that the boundedness of \(H_\omega\) on the Hardy space \(H^1\) is equivalent to the property \[ \sup_{a\in[0,1)}\frac{1}{1-a}\int_a^1\omega(t)\bigg(1+\int_0^t\frac{ds}{\widehat{\omega}(s)}\bigg)dt. \] If \(\omega\in\widehat{\mathcal D}\) and \(p\in(1,\infty)\), then the equivalence of the following statements is proved: \begin{itemize} \item[(i)] \(H_\omega:L^p[0,1)\to H^p\) is bounded; \item[(ii)] \(H_\omega:L^p[0,1)\to D^p_{p-1}\) is bounded; \item[(iii)] \(H_\omega:L^p[0,1)\to HL(p)\) is bounded; \item[(iv)] \(\omega\) satisfies the condition \end{itemize} \[\sup_{r\in(0,1)}\bigg(1+\int_0^r\widehat{\omega}(t)^{-p}dt\bigg)^{1/p} \bigg(\int_r^1\omega(t)^{p'}dt\bigg)^{1/p'}<\infty. \tag{1}\] In particular, if (1) holds, then the operator \(H_\omega:H^p\to H^p\) is bounded. If \(\omega\in\widehat{\mathcal D}\), \(p\in(1,\infty)\) and \(\nu\) is a radial weight on \(\mathbb{D}\), then a boundedness condition is also obtained for the operator \(H_\omega:A^p_\nu\to A^p_\nu\).