arXiv 1901.063342019-01-18math.FA

（-1，1）上有限希尔伯特变换的逆与延拓

Inversion and extension of the finite Hilbert transform on (-1,1)

Guillermo P. Curbera, Susumu Okada, Werner J. Ricker

摘要：

优化不等式的原理，或其等价的算子理论公式，在分析中得到了很好的建立。对于操作者，这对应于将其动作扩展到更大的域，希望扩展到最大可能的这样的域（即，其\textit{optimal domain}）。一些经典的算子已经被最优地定义（例如，$L^p(\mathbb{R})$、$1<p<\infty$中的希尔伯特变换）和其它变换不是（例如，$L^p(\mathbb{T})$、$1<p<2$中Hausdorff-Young不等式或各种空间中的Sobolev不等式）。本文对作用于重排不变空间$X$上的有限希尔伯特变换$T$进行了详细的研究。对于一大类这样的空间$X$它表明，$T$已经最佳定义$X$（这是已知的$L^p(-1,1)$的所有$1<p<\infty$，除了$p=2$）。$p=2$的情况是显著不同的，因为$T$的范围是$L^2(-1,1)$的真稠密子空间。然而，通过完全不同的方法，建立了$T$也在$L^2(-1,1)$上最佳地定义。我们的方法也被用来表明翼型方程的解，这是众所周知的空间$L^p(-1,1)$每当$p\not=2$（由于某些性质的$T$），也可以扩展到类的r. i。空间$X$考虑本文。

英文摘要：

The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible such domain (i.e, its \textit{optimal domain}). Some classical operators are already optimally defined (e.g., the Hilbert transform in $L^p(\mathbb{R})$, $1<p<\infty$) and others are not (e.g., the Hausdorff-Young inequality in $L^p(\mathbb{T})$, $1<p<2$, or Sobolev's inequality in various spaces). In this paper a detailed investigation is undertaken of the finite Hilbert transform $T$ acting on rearrangement invariant spaces $X$ on $(-1,1)$, an operator whose singular kernel is neither positive nor does it possess any monotonicity properties. For a large class of such spaces $X$ it is shown that $T$ is already optimally defined on $X$ (this is known for $L^p(-1,1)$ for all $1<p<\infty$, except $p=2$). The case $p=2$ is significantly different because the range of $T$ is a proper dense subspace of $L^2(-1,1)$. Nevertheless, by a completely different approach, it is established that $T$ is also optimally defined on $L^2(-1,1)$. Our methods are also used to show that the solution of the airfoil equation, which is well known for the spaces $L^p(-1,1)$ whenever $p\not=2$ (due to certain properties of $T$), can also be extended to the class of r.i.\ spaces $X$ considered in this paper.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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