I am struggling to prove the following "pqr theorem":
Let $1 \leq p < q < r \leq \infty$. Let $(f_n)_{n\in \mathbb N}$ be a sequence in $L^p(\Omega) \cap L^r(\Omega)$. Assume $\exists \epsilon >0$ such that $$\lVert f_n\rVert_p \leq 1, \lVert f_n \rVert_r \leq 1, \lVert f_n\rVert_q \geq \epsilon.$$
Then there is $\delta >0$ such that
$$\liminf_{n\to \infty}\mu(\{x: \lvert f_n(x)\rvert > \delta\}) \geq \delta.$$
My attempt: I first proved $f_n\in L^q(\Omega)$ and $\lVert f_n \rVert_q \leq 1$ using the Hölder inequality. Now we can write $$\lVert f_n\rVert_q^q = \int_0^\infty\mu(\lvert f_n\rvert > t^{1/q}) \, dt.$$
Now I was trying to derive a contradiction from assuming there can't be such $\delta$, e.g. by somehow moving the $\liminf$ inside but wasn't successful so far. Any hints?
Best Answer
For the sake of having an answer, I'll post the solution to this myself:
Note that equivalently we may show that there exist $\delta, \eta > 0$ such that $$\liminf_{n\to \infty} \mu(\lvert f_n \rvert > \delta) \geq \eta.$$
For $\eta \geq \delta $ it is clear, if $\eta < \delta$, then $$\{ \lvert f_n \rvert > \delta \} \subseteq \{\lvert f_n \rvert > \eta \} \implies \eta \leq \mu(\lvert f_n \rvert > \delta ) \leq \mu(|f_n | > \eta).$$ Now by the Layer-cake representation we may write $$\epsilon \leq \lVert f \rVert _q = \int_0^\infty q t^{q-1} \mu(|f_n| > t)\,dt \\= \int_0^\delta q t^{q-1} \mu(|f_n| > t)\,dt + \int_\delta^M q t^{q-1} \mu(|f_n| > t)\,dt + \int_M^\infty q t^{q-1} \mu(|f_n| > t)\,dt \\ = \int_0^\delta \frac{q}{p} p t^{q-p} t^{p-1} \mu(|f_n| > t)\,dt + \int_\delta^M q t^{q-1} \mu(|f_n| > t)\,dt + \int_M^\infty \frac{q}{r}r t^{q-r} t^{r-1} \mu(|f_n| > t)\,dt \\ \leq \int_0^\delta \frac{q}{p} p \delta^{q-p} t^{p-1} \mu(|f_n| > t)\,dt + \int_\delta^M q t^{q-1} \mu(|f_n| > t)\,dt + \int_M^\infty \frac{q}{r}r M^{q-r} t^{r-1} \mu(|f_n| > t)\,dt \\ \leq \frac{q}{p} \delta^{q-p} \lVert f_n \rVert_p^p + \int_\delta^M q t^{q-1} \mu(|f_n| > t) \, dt + \frac{q}{r}M^{q-r} \lVert f_n \rVert _r^r$$ and thus, $$\int_\delta^M qt^{q-r} \mu(|f_n| > t)\, dt \geq \underbrace{\epsilon - \frac{q}{p}\delta^{q-p} - \frac{q}{r}M^{q-r}}_{=: \eta_1} >0,$$ if $\delta$ sufficiently small and $M$ sufficiently large. Now we have for $t \geq \delta$ that $\mu(|f_n| > \delta) \geq \mu(|f_n| > t)$, so $$\int_\delta^M q t^{q-1} \mu(|f_n| > t) \, dt \leq \int_\delta^M q t^{q-1} \mu(|f_n| > \delta) \, dt = \mu(|f_n| > \delta) (M^q - \delta^q),$$ so $\eta := \frac{\eta_1}{M^q - \delta^q}$ is the desired constant.