$\lim_{t \to 0} \frac{1}{t^{1/q}}\int_{[0,t]}|f|dm=0$ where $f \in L^{p}$ and $q=p^*$

Question

Consider the measure space $([0,1], \mathcal {B}_{[0,1]}, m)$ where $m$ is Lebesgue measure, and let $f \in L^{p}(m)$ for some $p>1$.

Let $q=p^*$ (that is, $1/p+1/q=1$).

Prove or disprove:

$$\lim_{t \to 0} \frac{1}{t^{1/q}}\int_{[0,t]}|f|dm=0$$

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Tried to use Hölder's inequality:

$\int_{[0,t]}|f|dm=\|f\chi_{[0,t]}\|_{1} \le \|f\|_{p}\|\chi_{[0,t]}\|_{q}= \|f\|_{p} \cdot t^{1/q}$

So:

$\lim_{t \to 0} \frac{1}{t^{1/q}}\int_{[0,t]}|f|dm \le \lim_{t \to 0} \frac{1}{t^{1/q}}\cdot\|f\|_{p} \cdot t^{1/q}= \|f\|_{p} < \infty$

Which does not prove the desirable outcome...

Thanks!

Answer 1

The trick is to write $f\chi_{[0,t]}$ as $(f\chi_{[0,t]})\chi_{[0,t]}$ and note that $\|f\chi_{[0,t]}\|_{p} \to 0$.

$t^{-1/q}\int_{[0,t]}|f|dm=t^{-1/q}\|f\chi_{[0,t]}\|_{1} \le t^{-1/q} \|f\chi_{[0,t]}\|_{p}\|\chi_{[0,t]}\|_{q}= \|f\chi_{[0,t]}\|_{p} $ and $\|f\chi_{[0,t]}\|_{p} \to 0$ as $ t \to 0$.