On the growth of the Dirichlet integral for some function spaces (Q934559)

Let \(\mathbb D=\{z:| z| <1\}\) be the unit disk in the complex plane and denote by \(d\sigma_z\) the usual area measure on \(\mathbb D\). For \(z,a\in \mathbb D\) let \(g(z,a)\) be the Green function with pole at \(a\). For \(0<p<\infty\), we say that a function \(f\), analytic on \(\mathbb D\), belongs to \(Q_p\) if \[ \sup_{a\in \mathbb D} \int_\mathbb D | f'(z)| ^2 g(z,a)^p\,d\sigma_z<\infty. \] In this paper, the authors study the Dirichlet integral for functions belonging to \(Q_p\) spaces, they also prove the sharpness of these results. The main result is the following Theorem. For \(f\in Q_p\), \(0<p<\infty\), and any \(a\in \mathbb D\), \[ \lim_{t\to 0} t^p \int_{\mathbb D_t} | f'(z)| ^2 \,d\sigma_z=0. \]