Weighted fractional differentiation composition operators from mixed-norm spaces to weighted-type spaces (Q1637824)

Summary: Let \(\mathbb D\) be an open unit disc in the complex plane \(\mathbb C\) and let \(\varphi: \mathbb D\to\mathbb D\) as well as \(u: \mathbb D\to\mathbb C\) be analytic maps. For an analytic function \(f(z)=\sum^\infty_{n=0}a_n z^n\) on \(\mathbb D\) the weighted fractional differentiation composition operator is defined as \((D^\beta_{\varphi,u}f)(z)=u(z)f^{[\beta]}(\varphi(z))\), where \(\beta\geq 0\), \(f^{[\beta]}(z)=\sum^\infty_{n=0}(\Gamma(n+1+\beta)/\Gamma(n+1))a_n z^n\), and \(f^{[0]}(z)=f(z)\). In this paper, we obtain a characterization of boundedness and compactness of weighted fractional differentiation composition operator from mixed-norm space \(H(p,q,\phi)\) to weighted-type space \(H^\infty_\mu\).