Pointwise multipliers from the Hardy space to the Bergman space (Q1300123)

The author studies the problem of characterizing regions \(G\) with the property that the Hardy space \(H^2(G)\) is contained in the Bergman space \(L^2_a(G)\). For example, he shows the following. When \(G\) is a simply connected region, \(D\) is an open unit disc and \(\tau:D\to G\) is a Riemann map, he shows that \(H^2(G)\subseteq L^2_a(G)\) if and only if \(\tau\) is Lipschitz of order \(1/2\). When \(\varphi\) is analytic on \(D\) and \(G=\varphi(D)\), \(H^2(G) \subseteq L^2_a(G)\) if \(\varphi'\) is a multiplier. The author also gives examples of multipliers and raises three questions.