# Proving that $u_0$ is a canonical solution

I've asked this question on stack exchange before but no one could help me so I wish I can get some help here. Let's first start with the definition of the canonical solution:

Consider $$\frac{\partial w}{\partial \bar{z}}=u$$. We say $$w_0$$ is a canonical solution if $$\frac{\partial w_0}{\partial \bar{z}}=u$$, and $$w_0 \perp \mathcal{H}(\mathbb{D})=\{ f \in \mathcal{O}(\mathbb{D}): f \in L^2(\mathbb{D}) \}$$, where $$\mathcal{O}(\mathbb{D})$$ is the space of holomorphic functions on the unit disk. And $$w_0 \perp \mathcal{H}(\mathbb{D}) \iff \int_{\mathbb{D}} w_0(\zeta)\cdot \overline{h(\zeta)} dV_{\zeta}=0$$, for any $$h \in \mathcal{H}(\mathbb{D})$$.

Here, we are finding $$w_0$$ as follows:

Let $$w_1(z)= \frac{1}{2\pi i} \int_{\mathbb{D}} \frac{u(\zeta)}{\zeta -z} d\overline{\zeta} \wedge d{\zeta}.$$ Define $$Pw_1(z):= \int_{\mathbb{D}} K(z,\zeta) w_1(\zeta) dV_{\zeta},$$ where $$K(z,\zeta)$$ is the Bergman kernel on the unit disk which's been calculated here. Now, we define $$w_0:= w_1 - Pw_1$$, and I want to prove this is a canonical solution.

What I've done so far is that if $$w_1$$ is holomorphic, then by the definition of the Bergman kernel, $$Pw_1(z)=w_1(z)$$ and hence, $$w_0=0$$ which is the trivial case.

So, suppose $$w_1$$ is not holomorphic. In this case, doing some calculations and using the Cauchy-Green's formula, I figured that we can actually write $$w_0=w_1-Pw_1= -\frac{1}{2\pi i} \int_{\mathbb{D}} \frac{(1-\lvert \zeta \rvert ^2) u(\zeta)}{(1-\overline{\zeta}z)(\zeta-z)} d\zeta \wedge d\overline{\zeta}.$$ So, I actually need to prove that this is a canonical solution but I don't know how to proceed from here. Any hint/help is appreciated.

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