I just stumbled across the John-Nirenberg inequality which bounds the growth of functions in the BMO space. So I got curious whether such a bound on the growth also exists for $L_p$ functions. Let me state the problem:
Let $f:[0,1)\to\mathbb{R}$ be a measurable function. Does it hold that $f\in L_p[0,1)$ if and only if there exists an $\varepsilon>0$ and a $C\ge 0$ such that \begin{equation}\label{eqn} \lambda(\{|f|\ge x\})\le Cx^{-(p+\varepsilon)} \end{equation}
where $\lambda$ is the Lebesgue measure?
One direction is easy. Assume that the inequality holds. Then
\begin{eqnarray*} \|f\|_p^p &=& \int_0^1f(x)^pdx=p\int_0^\infty\lambda(\{|f|\ge u\})u^{p-1}du\\ &=& p\int_0^1\lambda(\{|f|\ge u\})u^{p-1}du+p\int_1^\infty\lambda(\{|f|\ge u\})u^{p-1}du\\ &\le& 1+ p\int_1^\infty Cu^{-(1+\varepsilon)}du<\infty. \end{eqnarray*}
How does one show the other direction of the equivalence? Or, if the equivalence is false, is there a similar statement bounding the growth of functions in $L_p[0,1)$?