Let \(\mathbb{D}\) be the unit disc of the complex plane, \(dm= dxdy/\pi\) the normalized area measure on \(\mathbb{D}\). Let \(w(r)\) for \(0< r< 1\) be positive and integrable in \((0,1)\), and extend \(w\) to \(\mathbb{D}\) by \(w(z)= w(|z|)\) for \(z\in\mathbb{D}\). Then, for \(0< p<\infty\), the weighted Bergman space \(A^p(w)\) is the space of functions \(f\) analytic on \(\mathbb{D}\) for which
\[ \| f\|^p_{A^p(w)}= \int_{\mathbb{D}} |f(z)|^p w(z)\,dm(z)< \infty. \]
The authors study Carleson measures and certain integration operators on \(A^p(w)\) for weights \(w\) in a certain class \(W\) of weights. \(W\) is defined by several technical conditions which will not be repeated here, but the class includes the exponential type weights
\[ w_{\gamma,\alpha}(r)= (1-r)^\gamma\exp\Biggl\{{-c\over (1-r)^\alpha}\Biggr\} \]
for \(\gamma\geq 0\), \(\alpha> 0\), and \(c> 0\), as well as “double exponential” weights.
Let \(X\) be a space of analytic functions on \(\mathbb{D}\). A positive Borel measure \(\mu\) in \(\mathbb{D}\) is called \(q\)-Carleson for \(X\) if the embedding \(X\subset L^q(\mu)\), \(0< q<\infty\), is continuous. The authors obtain a complete description of the \(q\)-Carleson measures for \(A^p(w)\), \(0< p,q<\infty\), for weights \(w\in W\). In addition, the following problem is considered. Let \(g\) be a fixed function analytic on \(\mathbb{D}\), and define the operator \((T_g f)(z)= \int^z_0 f(w)g'(w)\,dw\). The authors completely describe the boundedness and compactness of \(T_g: A^p(w)\to A^q(w)\), \(0< p,q<\infty\), for weights \(w\in W\).
Finally, for \(0< p<\infty\), \(H\) a separable Hilbert space, let \(S_p(H)\) denote the Schatten \(p\)-class of operators on \(H\). This is the class of compact operators on \(H\) whose sequence of singular numbers \(\lambda_n\) belong to \(\ell^p\), the \(p\)-summable sequence space. The authors characterize the \(g\) analytic in \(\mathbb{D}\) for which \(T_g\in S_p(A^2(w))\), for \(w\in W\).