Boundness of Fourier thansform

Asked Mar 01 '19 at 08:48

Active Mar 01 '19 at 13:25

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Given $1\leq p\leq\infty$, prove that if it exists a constant C s.t. $$\|\hat{f}\|_{p'}\leq C \|f\|_p$$here $1/p+1/p'=1;$ then $1\leq p\leq 2$

I know that if $1\leq p\leq 2$, then it exists a constant C s.t. $$\|\hat{f}\|_{p'}\leq C \|f\|_p$$ and I have no idea how to prove this result.

Any idea will be helpful. Thanks.

asked Mar 01 '19 at 08:48

yufeng lu

1

See https://math.stackexchange.com/questions/146067/fourier-transform-in-lp – Kavi Rama Murthy Mar 01 '19 at 09:02
If not than the Fourier transform is a linear homeomorphism between $L^p $and $L^p'$ which is a contradiction， – yufeng lu Apr 01 '19 at 13:14

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