Riemann-Stieltjes operator between iterated logarithmic Bloch space and mixed-norm space on the unit ball (Q2912671)

In this paper, the authors study the boundedness and the compactness of the Riemann-Stieljes operator NEWLINE\[NEWLINE I_h(f)(z):=\int_0^1 Rf(tz)h(tz) \frac{dt}{t},\qquad (h\in H), NEWLINE\]NEWLINE acting between the iterated logarithmic Bloch spaces NEWLINE\[NEWLINE \mathcal{B}_{\log_k}:=\left\{f\in H:\sup_{z\in\mathbb{B}}(1-|z|^2)\omega_{\log_k} (|z|) |Rf(z)|<\infty\right\} NEWLINE\]NEWLINE and the mixed-norm spaces NEWLINE\[NEWLINE H(p,q,\phi):=\left\{f\in H:\,f(r\zeta)\in L^p\left(L^q(d\sigma(\zeta));\frac{\phi^p(r)}{1-r}dr\right)\right\}, \quad 0<p,q<\infty. NEWLINE\]NEWLINE Here, \(H\) denotes the space of holomorphic functions on the unit ball \(\mathbb{B}\) of \(\mathbb{C}^n\), \(R\) is the radial derivative, NEWLINE\[NEWLINE \omega_{\log_k}(r):=\prod_{j=1}^k\log ..^{(j)}..\log\frac{e^{[k]}}{1-r^2}, \quad \log..^{(k)}..\log e^{[k]}=1, NEWLINE\]NEWLINE \(d\sigma\) is the surface measure on the unit sphere and \(\phi\) is a positive continuous function on \([0,1)\) satisfying \((1-r)^{-a}\phi(r)\to 0\) and \((1-r)^{-b}\phi(r)\to \infty\) as \(r\to 1\) for some \(0<a<b\).NEWLINENEWLINEObserve that, if \(\phi(r)=(1-r^2)^s\), \(s>0\), then \(H(p,p,\phi)\) is the weighted Bergman space \(H\cap L^p((1-|z|^2)^{sp-1}dm(z))\). If \(k=1\), then \(\mathcal{B}_{\log_1}\) is the logarithmic Bloch space \(\mathcal{B}_{\log}\).NEWLINENEWLINETwo of the main results proved by the authors are:NEWLINENEWLINE(1) For \(I_h:\mathcal{B}_{\log_k}\to H(p,q,\phi)\), the following statements are equivalent: (i) \(I_h\) is bounded. (ii) \(I_h\) is compact. (iii) \(h\in H(p,q,\phi/\omega_{\log_k})\).NEWLINENEWLINE(2) \(I_h:H(p,q,\phi)\to \mathcal{B}_{\log_k}\) is bounded (respectively, compact) if and only if the function NEWLINE\[NEWLINE \frac{\omega_{\log_k}(|z|)}{\phi(|z|)(1-|z|^2)^{n/q}}|h(z)| NEWLINE\]NEWLINE is bounded on \(\mathbb{B}\) (respectively, tends to 0 as \(|z|\to 1\)).NEWLINENEWLINEThe paper also includes a characterization of the boundedness and compactness of \(I_h\) as an operator from \(H(p,q,\phi)\) to the little iterated logarithmic Bloch space \(\mathcal{B}_{\log_k,0}\).