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| ON SOME CONSTANTS OF QUASICONFORMAL DEFORMATION AND ZYGMUND CLASS |
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Citation： |
Chen Jixiu,Wei Hanbai.ON SOME CONSTANTS OF QUASICONFORMAL DEFORMATION AND ZYGMUND CLASS[J].Chinese Annals of Mathematics B,1995,16(3):325~330 |
| Page view： 1022
Net amount： 765 |
Authors： |
Chen Jixiu; Wei Hanbai |
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| Abstract： |
A real-valued function $f(x)$ on $ \Re$ belongs
to Zygmund class $\Lambda_{*}(\Re)$ if its Zygmund norm
$\|f\|_z=\underset{x,t}\to{\inf} \Bigl|\frac {f(x+t)-2f(x)+f(x-t)}t\Bigr|$ is
finite. It is proved that when
$f\in\Lambda_{*}(\Re)$, there exists an extension $F(z)$ of $f$ to $
H=\{\text{Im}z>0\}$ such
that
$$\aligned
\|\overline {\partial}F\|_{\infty}\le\frac {\sqrt {1+53^2}}{72}
\|f\|_z.\endaligned
$$
It is also proved that if $f(0)=f(1)=0$, then
$$\aligned
\max_{x\in [0,1]}|f(x)|\le\!\frac 13\|f\|_z.\endaligned
$$ |
Keywords： |
Quasiconformal deformation, Zygmund class,
Beurling-Ahlfors extension |
Classification： |
30C62 |
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