Representations of Nevanlinna-type spaces by weighted Hardy spaces (Q1598354)

Let \(U\) denote the open unit disk and \(T\) the unit circle in the complex plane. The Nevanlinna class \(N\) consists of all holomorphic functions \(f\) in \(U\) which satisfy \[ \sup_{0<r<1}{\int_0^{2\pi} \log{(1+|f(re^{i\theta})|)} d\theta} < \infty . \] It is well-known that every \(f \in N\) has a nontangential limit \(f^*\) on \(T\). The Smirnov class \(N_*\) is the set of all \(f \in N\) such that \[ \lim_{r \to -1}{\int_0^{2\pi} \log{(1+|f(re^{i\theta})|)} d\theta} = \int_0^{2\pi} \log{(1+|f^*(e^{i\theta})|)} d\theta . \] Finally, for \(p>1\), the class \(N^p\) is the set of all holomorphic functions \(f\) in \(U\) which satisfy \[ \sup_{0<r<1}{\int_0^{2\pi} [\log{(1+|f(re^{i\theta})|)}]^p d\theta} < \infty . \] There holds \(H^q \subset N^p \subset N_* \subset N\) for \(0 < q \leq \infty\) and \(p>1\), where \(H^q\) denotes the usual Hardy space. The space \(N\) and its subspaces \(N_*\), \(N^p\) and \(H^q\) are called Nevanlinna-type spaces. Let \(w\) be a weight, that is a nonnegative \(L^1\)-function on \(T\), and denote by \(W_p\) the class of weights \(w\) satisfying \(\log{w} \in L^p(T)\) for \(1 \leq p < \infty\). For \(0<q<\infty\), let \(H^q(w)\) denote the closure of the polynomials in \(L^q(w d\theta)\). Then the author characterizes \(N^p\) as follows. Theorem. Let \(1 \leq p < \infty\) and \(0<q<\infty\). Then \[ N^p = \bigcup_{w \in W_p} H^q(w) . \] There is a natural topology \(\tau_p\) on \(N^p\), \(1 \leq p < \infty\) which is induced by the metric \[ d_p(f,g) = \left(\frac{1}{2\pi} \int_0^{2\pi} [\log{(1+|f^*(e^{i\theta})-g^*(e^{i\theta})|)}]^p d\theta\right)^{1/p} , \quad f,g \in N^p . \] Furthermore, by virtue of the above Theorem, it is possible to induce an inductive limit topology \(I_{p,q}\) on \(N^p\), \(0<q<\infty\). The author shows that the two topologies \(\tau_p\) and \(I_{p,q}\) on \(N^p\) are equivalent.