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It is an exercise in Wheeden Zygmund Measure and integration($9.2$(b)). That is I need to show if the conclusion of Young's inequality hold ($||f*g||_r\leq ||f||_p ||g||_q$), show that $$\frac{1}{p}+\frac{1}{q}-1=\frac{1}{r}$$

The hint tell me that apply the inequality to the convolution of $f(\lambda x)$ and $g(\lambda x)$ then vary $\lambda \geq 0$. But I have no idea how to use this hint.

I tried to apply the converse idea of proof of Young's convolution inequality. Of course it failed. I know the idea for the case of Young's product inequality, but I don't know how to find the relation of $p,q,r$ in this case

Can someone help me? Thanks!

aouii
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    What have you tried? –  Nov 03 '22 at 05:06
  • @PseudoLooped I try to apply the converse idea of proof of Young's convolution inequality. Of course it fails. I know the idea for the case of young's product inequality, but I don't know how to find the relation of p,q,r in this case – aouii Nov 03 '22 at 05:47
  • The ideas in this post may be helpful. – Feng Nov 03 '22 at 05:53

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