Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D}$ ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:38:27Z http://mathoverflow.net/feeds/question/71455 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71455/regularity-properties-of-the-derivatives-of-a-particular-function-on-d-times-d Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D}$ ? Analysis Now 2011-07-28T02:48:24Z 2011-07-29T03:36:33Z <p>This question might sound a little less rigorously formulated, but I hope the question still makes sense.</p> <p>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 :</p> <p>$G (z,w) = \int_{S^1} \frac{h(t)-w}{1-\bar{w}h(t)} p(z,t) |dt|$. 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.</p> <p>Let $1 \le k &lt; \infty .$ 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 </p> <p>1) bounded away from zero ( for fixed $z$, and for fixed $w$ ), </p> <p>2) the $k$ th order derivatives are in $L^p(D\times D), 1\le p \le \infty$ ? </p> <p>3) the $k$ th order derivatives are in $\mathcal{C}^1(\bar{D}\times\bar{D})$ ?</p> <p>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 ?</p> <p>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 !</p>