Essential norms of integration operators on weighted Bergman spaces
Let \(\omega\) be a radial weight on the open unit disc \(\mathbb{D}\subset \mathbb{C}\) satisfying the doubling property \(\omega\left([r,1)\right)\leq C\omega\left([(1+r)/2,1)\right)\), \(0\leq r<1\). Let \(A^p_\omega\) be its induced weighted Bergman space. The authors obtain operator norm and essential operator norm estimates for the Volterra integral operator \(T_g: A^p_\omega\to A^q_\omega\), \(0<p\leq q<\infty\), defined by \(T_g(f)(z)=\int_0^z f(\zeta)g'(\zeta)d\zeta\). The main results prove that if \(0\leq \frac 1p-\frac 1q<1\) then these operators satisfy \[ \|T_g\|^2\simeq \sup_{a\in \mathbb{D}} \frac{ \int_{S_a}|g'|^2\omega^*} {(\int_{S_a}\omega)^{2(1/p-1/q)+1}} \qquad \text{and} \qquad \|T_g\|_e^2\simeq\limsup_{|a|\to 1^-} \frac{ \int_{S_a}|g'|^2\omega^*} {(\int_{S_a}\omega)^{2(1/p-1/q)+1}}, \] where \(\omega^*(z)=\int_{|z|}^1 s\omega(s)\log\frac{s}{|z|}ds\), \(z\neq 0\), and \(S_a\) is a Carleson square with a basis point \(a\in\mathbb{D}\). If, in addition, \(p<q\), then \[ \|T_g\|\simeq \sup_{a\in \mathbb{D}} \frac{|g'(a)|(1-|a|)}{\omega^*(a)^{1/p-1/q}} \quad\text{and}\quad \|T_g\|_e\simeq \limsup_{|a|\to 1^-} \frac{|g'(a)|(1-|a|)}{\omega^*(a)^{1/p-1/q}}. \]
scientific article; zbMATH DE number 6684635
scientific article