Function Space and Hölder Inequality

Question

Let the measure space $([0,1],\mathbb{B}([0,1]),\lambda)$ be given ($\lambda$ is the Lebesgue measure restricted to $[0,1]$). Given $f\in L^p$ (so $\int |f|^p d\lambda < \infty$), show that if $\frac{1}{p}+\frac{1}{q}=1$, then $$\lim_{n\to \infty} n^{1/q} \int_{[0,\frac{1}{n}]} |f| d\lambda = 0 $$ My progress so far: $\lim_{n\to \infty} n^{1/q} \int_{[0,\frac{1}{n}]} |f| d\lambda \leq ||f||_p$ and thus bounded. I can show this by the Holder Inequality, which I think must be a part of the solution. But of course is boundedness not nearly enough. Could someone give a hint (not a solution!) how I need to proceed/approach this question?

Answer 1

In the computation you made in your coment, use $\chi_{[0,1/n]}$ twice instead of once:

$$\int_0^{1/n} \lvert f\rvert\,d\lambda = \int_0^1 \chi_{[0,1/n]}\cdot \lvert f\cdot \chi_{[0,1/n]}\rvert\,d\lambda \leqslant \lVert \chi_{[0,1/n]}\rVert_q \lVert f\cdot \chi_{[0,1/n]}\rVert_p = n^{-1/q}\lVert f\cdot\chi_{[0,1/n]}\rVert_p.$$

By the dominated convergence theorem, $f\cdot\chi_{[0,1/n]} \to 0$ in $L^p$.