Abstract
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.
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