Problem: For $1\le p\le +\infty$, find the values of the parameter $\lambda$ for which $\mathop{lim}\limits_{\epsilon\to 0+} \frac1{\epsilon^\lambda} \int_0^\epsilon f = 0$ for all $f \in L^p[0, 1]$.
I have tried to use $H\ddot{o}lder$'s inequality to prove that every $\lambda<1/q,\ q$ is the conjugate of $p$, satisfies the equation:
$$\forall f \in L^p[0,1], \int_0^\epsilon f dm=\int_0^1 f\chi_{[0,\epsilon]}dm\le\Vert f\Vert_p\epsilon^{1/q}.$$
As for $\lambda>1/q$, let $f(x)=x^{-\frac1 {kp}},k>1$, it is easy to find that $\forall\lambda>1/q,$ there exists a $k_\star$ s.t. $\mathop{lim}\limits_{\epsilon\to 0+} \frac1{\epsilon^\lambda} \int_0^\epsilon f = +\infty.$
My problem is how to solve the condition of $\lambda=1/q.$ I 've thought it for several hours but have no idea. Could anyone help out? Thanks a lot in advance!