Let $\mu$ be positive Borel measure on unit disk D. We say that $\mu$ is a Carleson measure on the Bergman space if there is a constant $C> 0$ such that $f\in A^2(D,dA)$,
$$\left\|f\right\|_\mu^2 := \int_D|f(z)|^2d\mu\leq C \left\|f\right\|^2$$.
Let $I$ be an arc in the unit circle $\partial D$. Let $S(I)$ be the Carleson square defined by
$$S(I)=\left\{z\in D: 1-|I|\leq |z|<1, |\theta_0-arg z|\leq |I|\right\}.$$
Problem 1: Let $\mu$ be positive Borel measure on unit disk D. Then the following statements are equivalent
i) There is a constant $C_1 > 0$ such that $f\in A^2 (D,dA)$, $$\left\|f\right\|_\mu^2 :=\int_D|f(z)|^2d\mu\leq C_1 \left\|f\right\|^2$$. ii) There is a constant $C_2 > 0$ such that, for every $a \in D$,
$$\int_D\left(\dfrac{1-|a|^2}{|1-\overline {a}z|^2}\right)^2d\mu\leq C_2.$$
Proof of me: i) => ii): Set $f(z)=\dfrac{|1-|a|^2|}{|1-\overline {a}z|^2}$.Then, we get the inequality that needs to be proven.
ii) => i) I'm having trouble here. I use the Minkowski's integral inequality as follows $$\left\|f\right\|_\mu\leq\left(\int_D\left(\int_D\dfrac{|f(a)|}{|1-\overline{a}z|^2}dA(a)\right)^2 d\mu\right)^{\frac{1}{2}}\leq\int_D\left(\int_D\dfrac{|f(a)|^2}{|1-\overline{a}z|^4}d\mu(z)\right)^\frac{1}{2}dA(a)=\int_{D}\dfrac{|f(a)|^2}{(1-|a|^2)^2}\left(\int_D\dfrac{(1-|a|^2)^2}{|1-\overline {a}z|^4}d\mu (z)\right)^{\frac{1}{2}}dA(a)\leq C_2\int_D\dfrac{|f(a)|^2}{(1-|a|^2)^2}dA(a).$$
I'm stuck here.
However, I plan to do the same as the article in the link: https://soar.suny.edu/handle/20.500.12648/2607. (Theorem 2.2). But I don't know how to evaluate $|f(z)|^2$ so that $\left(\dfrac{1-|a|^2}{|1-\overline {a}z|^2}\right)^2$ appears.
Problem 2: Let $\mu$ be positive Borel measure on unit disk D. Then the following statements are equivalent
i) There is a constant $C_1>0$ such that $f\in A^2 (D,dA)$,
$$\left\|f\right\|_\mu^2 :=\int_D|f(z)|^2d\mu\leq C_1 \left\|f\right\|^2$$. ii) There is a constant $C_2> 0$ such that for any arc $I\in\partial D$, $$\mu (S(I))\leq C_2 |I|^2$$ iii) There is a constant $C_3 > 0$ such that, for every $a \in D$,
$$\int_D\left(\dfrac{1-|a|^2}{|1-\overline {a}z|^2}\right)^2d\mu\leq C_3$$
Proof of me: I have shown that (i) is equivalent to (ii) and (iii) => (ii). However, I'm stuck (ii) => (iii). I read the proof of Lemma 2.1 in the paper "Some Subclasses of BMOA and their Characterization in Terms of Carleson Measures, R. Aulaskari, David A. Stegenga, Jie Xiao". I don't understand where they have $\mu (\Delta)\leq C_\mu$.
