Reverse estimates in growth spaces
Recall that for for a bounded circular domain \(D\) which is strictly convex and has \(\mathcal C^{2}\) smooth boundary, the growth space \(A^{w}(D)\) is defined as follows. It consists of all holomorphic in \( D\) functions \(f\) such that \(|f(z)|<Cw(r(z))\) for all \(z\in D\). Here \(w\) is the so-called weight function which is a general non decreasing continuous and unbounded on the interval \([0,1)\) function and \(r(z)\) is the Minkowski (or gauge) function of \(D\). The main theorem is that when \(w\) is a doubling function then for some \(M\in\mathbb N\), which depends only on \(D\), one can find \(M+1\) functions from the space \(A^{w}(D)\) such that the sum of their absolute values at every point \(z\in D\) dominates the quantity \(w(r(z))\). The exact value of the smallest such \(M\) is unknown in general (and probably an interesting quantity to study). One obviously has that \(M>0\), which is true even without the assumption of \(w\) being doubling. The authors manage to prove that, when \(D\) is the unit disc in the complex plane, \(M\) is \(1\) (this generalizes a result of Ramey and Ullrich from 1991), but even for such particular domains as the unit balls in \(\mathbb C^n\), \(M\) is unknown (and far from clear). It is also interesting what happens when the property of \(w\) being doubling is replaced by some weaker assumption. The authors provide some corollaries to their result. In particular they apply it in the study of composition and multiplication operators in various spaces of holomorphic functions.
scientific article; zbMATH DE number 6050604
scientific article