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This question might sound a little less rigorously formulated, but I hope the question still makes sense.

Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = \frac{1}{2\pi}\frac{1-|z|^2}{|z-t|^2} $denote the Poisson kernel, $D$ denote the open unit disk in $\mathbb{C} $. Let us consider the function $ G: D\times D \to \bar{D} $ given by :

$ G (z,w) = \int_{S^1} \frac{h(t)-w}{1-\bar{w}h(t)} p(z,t) |dt| $. Note that for fixed $w$,the map $z\to G(z,w) $ is harmonic, because so is the Poisson kernel.$|dt|$ denoted the standard Lebesgue measure on the circle.

Let $1 \le k < \infty .$ I want to know the properties of the $k$ th order ( mostly upto $k \le 2 $ and if possible , higher ) order partial ( $ G_z, G_w, G_{zz}, G_{zw}, G_ {z \bar{z}}$ etc. ) derivatives of $G$ defined on $D$. In particular, my question is : is there a regularity [ smooth, real-analytic etc. ] condition on $h$ such that the $k$ th order derivatives of $G$ are

1) bounded away from zero ( for fixed $z$, and for fixed $w$ ),

2) the $k$ th order derivatives are in $L^p(D\times D), 1\le p \le \infty $ ?

3) the $k$ th order derivatives are in $ \mathcal{C}^1(\bar{D}\times\bar{D}) $ ?

4) For fixed $z$, can the map $w \to G(z,w)$ be quasiconformal at all, if $h$ has nice properties, like quasi-symmetric, smooth, bi-Lipchitz, real-analytic etc ?

I guess the question could be partially or fully answered using harmonic analysis or analysis. I would appreciate if you answer, or refer a book/paper discussing similar results. Any partial answer would be greatly appreciated as well !

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