An extension of a theorem of Hadamard and domination in the Bergman space (Q2368160)

The Bergman space \(A_ p\) consists of holomorphic functions \(f(z)\) in the unit disk \(\Delta = \{z \in \mathbb{C} : | z | < 1\}\) which satisfy \[ \| f \|_{A_ p} = \left( \int_ \Delta \bigl | f(z) \bigr |^ pd A(z) \right)^{1/p} < \infty, \] where \(dA(z)\) is the Lebesgue measure on \(\Delta\). The paper deals with certain extensions of the factorization theory of \(A_ p\) developed by Hedenmalm \((p = 2)\) and later Duren-Khavinson-Shapiro-Sundberg \((1 \leq p < \infty)\). Specifically, the main result of this theory is that a function \(G \in A_ p\) satisfying \[ \int_ \Delta \biggl( \bigl | G(z) \bigr |^ p - 1 \biggr) h(z) dA(z) = 0 \tag{1} \] for all \(h \in H_ \infty\) is a contractive zero-divisor, that is \[ \left \| {f \over G} \right \|_{A_ p} \leq \| f \|_{A_ p} \tag{2} \] for all functions \(f\) from the invariant subspace of \(A_ p\) generated by \(G\). Moreover, for every discrete set \(D \subset \Delta\) which is a zero set for an \(A_ p\)-function, there is a function \(G_ D \in A_ p\), unique up to a unimodular scalar factor, satisfying (1) and vanishing precisely at \(D\). The second part of the factorization theorem is shown to hold in the case of finite \(D\) and weighted spaces \((p=2)\) with \(dA(z)\) replaced by \(| f(z) |^ 2 dA(z)\), \(f\) a polynomial of prescribed degree sufficiently close to 1 in \(\Delta\). The theorem is a consequence of an extension of a theorem of Hadamard about the Green function of weighted biharmonic operators proved by the author.