论文标题
Zygmund太空Llogl作用的有限希尔伯特变换
The finite Hilbert transform acting in the Zygmund space LlogL
论文作者
论文摘要
有限的Hilbert Transform T是一个单数积分运算符，它映射Zygmund Space $ Llogl：= llogl（-1,1）$将$连续成$ l^1：= l^1（-1,1,1）$。通过将Parseval和Poincaré-Bertrand公式扩展到此设置，可以在$ l^1 $（显示为包含$ lllogl $的$ l^1 $中）建立求解airfoil equation $ t（f）= g $所需的反转结果。到目前为止，这仅以$ g $属于所有$ l^p $ spaces的结合，$ p> 1 $。 （由于Stein的结果）确定了$ t $不能扩展到$ llogl $以外的任何域空间，而仍然以$ l^1 $（即$ t：llogl \ t：llogl \ t：llogl \ to llogl \ t：l^1 $）的值进行了最佳定义。
The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincaré-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation $T(f)=g$ whenever the data function $g$ lies in the range of $T$ within $L^1$ (shown to contain $LlogL$). Until now this was only known for $g$ belonging to the union of all $L^p$ spaces with $p>1$. It is established (due to a result of Stein) that $T$ cannot be extended to any domain space beyond $LlogL$ whilst still taking its values in $L^1$, i.e., $T:LlogL\to L^1$ is optimally defined.
