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As it is known that if 1 ≤ p < 2d/d+1, d ≥2, the Fourier transform of L^p($\mathbb{R^d}$) radial function is a continuous function away from origin. How do we prove that this range for p cannot be extended?

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For $p>\frac{2d}{d+1}$, the function $f(x)=\|x\|^{-d/2}J_{d/2}(\|x\|)$ is in $L^p(\mathbf R^d)$ by standard Bessel asymptotics, and its Fourier transform is the (obviously discontinuous) characteristic function of the unit ball.

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  • $\begingroup$ what about p=2d/d+1? $\endgroup$
    – OwenKING
    Apr 14 '17 at 0:07
  • $\begingroup$ @OwenKING Then this $f$ is barely not in $L^p$ ($\int_{\|x\|\leqslant r}|f|^p\simeq\log r$), so it's no longer a counterexample. $\endgroup$ Apr 14 '17 at 1:21

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