Some questions on \(L^1\)-approximation in Bergman spaces (Q2040234)

Let $A^1(\mathbb D)=L^1(\mathbb D, dA/\pi)\cap \operatorname{Hol}(\mathbb D)$ be the usual Bergman space and $B_{H^\infty}$ the unit ball in the space $H^\infty\subseteq \operatorname{Hol}(\mathbb D)$ of bounded holomorphic functions. The following $A^1(\mathbb D)$-approximation results are shown: (1) Given a continuous function $f$ on $\overline{ \mathbb D}$ with $||f||_\infty=1$ and $\epsilon>0$, there exists a function $\varphi:[0,1]\to \mathbb R^+$ continuous at $0$ with $\varphi(0)=0$, such that $\mathrm{dist}_{L^1(\mathbb D)}(f, B_{H^\infty})\leq \varphi(\epsilon)$ whenever $\mathrm{dist}_{L^1(\mathbb D)}(f, A^1(\mathbb D))\leq\epsilon$. (2) For all $\epsilon$ sufficiently small there exists functions $q_\epsilon\in C(\overline{\mathbb D})$ with $||q_\epsilon||_\infty=1$, such that $\mathrm{dist}_{L^1(\mathbb D)}(q_\epsilon, A^1(\mathbb D))\leq\epsilon$, but $$\frac{1}{\epsilon} \mathrm{dist}_{L^1(\mathbb D)}(q_\epsilon, B_{H^\infty})\to\infty\text{ as }\epsilon\to 0.$$ We note that the situation is quite different in the Hardy-space setting; see [the author et al., Constr. Approx. 14, No. 3, 401--410 (1998; Zbl 0916.46040); \textit{V. Totik}, Anal. Math. Physics 12:4 (2022)].