If $f\in L^{p_0}[0,1]$, $p_0$ is a given number big enough. Prove: $\lim_{p \to 0+}\Vert f\Vert_p=e^{\int_0^1 \ln\vert f(x)\vert dx}.$

Asked May 22 '20 at 14:24

Active May 23 '20 at 02:44

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Problem: if $f\in L^{p_0}[0,1]$, $p_0$ is a given number big enough. Prove: $$\lim_{p \to 0+}\Vert f\Vert_p=e^{\int_0^1 \ln\vert f(x)\vert dx}~.$$

I've proved one side of the equation by using Jensen's Inequality: we have $$-\ln (\int_0^1 \vert f\vert^p dx)\le\int_0^1-\ln\vert f\vert^pdx,$$ then we have $$\Vert f\Vert_p\ge e^{\int_0^1 \ln\vert f(x)\vert dx}.$$ Now I have trouble sovling the other side of the euqation. Could anyone help me solve the problem? Thanks a lot!

edited May 23 '20 at 02:44

asked May 22 '20 at 14:24

LonnerT

link could also solve the problem! – LonnerT May 23 '20 at 03:15

If $f\in L^{p_0}[0,1]$, $p_0$ is a given number big enough. Prove: $\lim_{p \to 0+}\Vert f\Vert_p=e^{\int_0^1 \ln\vert f(x)\vert dx}.$

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