The purpose of the present paper is to discuss the integrability of the Fourier transform of \(L^\infty \)-functions subject to a very weak decay condition. This will include the negative power of the logarithm. For a start, the case when \(d=1\) is investigated by the use of the second mean value theorem. Based on the discussion of this case, a passage to the weighted Hankel transform is done, which covers the integrability of the d-dimensional Fourier transform of the radial functions.

中文翻译：

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加权Hankel变换及其在傅立叶变换中的应用。

本文的目的是讨论在非常弱的衰减条件下\（L ^ \ infty \）函数的傅立叶变换的可积性。这将包括对数的负幂。首先，使用第二平均值定理研究\（d = 1 \）的情况。基于对这种情况的讨论，完成了加权Hankel变换的遍历，它涵盖了径向函数的d维傅里叶变换的可积性。