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Another prelim question... Suppose $0<q<p<\infty$, and $E\subseteq \mathbb{R}^n$ has finite measure. Suppose $f$ is in weak $L^p$, i.e. $\lambda(|f| > t) \leq N/t^p$. Show $f \in L^q(E)$ and moreover $\int_E |f|^q \leq C_{n,p,q} N^{q/p}\lambda(E)^{1-q/p}$.

First let's show $f \in L^q(E)$. For $g$ a strictly increasing function with $g(0)=0$ $$ \int_E |f|^q = \sum_{k} \int_E |f|^q 1_{g(k) < f \leq g(k+1)} \leq \sum_k g^q(k+1) (\lambda(f > g(k)) \wedge \lambda(E)) $$ $$ \leq\sum_k g^q(k+1) (\frac{N}{g^p(k)} \wedge \lambda(E)) $$ so it suffices to choose $g$ such that $\sum_{k\geq M} \frac{g^q(k+1)}{g^p(k)} < \infty$ for some $M$. Taking $g(k) = 2^k$ (except $g(0)=0$) and $M>1$ this turns into $\sum_{k\geq M} 2^q (2^{(q-p)})^n < \infty$ which is of course true. Okay so we have $f \in L^q(E)$. Now what can be done to show the bound?

I've tried writng $|f|^q = (|f|^p)^{q/p}$ and using Fubini and Holder $$ \int_E |f|^q = \int_0^\infty \int \frac{p}{q}t^{p/q-1} 1_{E} 1_{0\leq |f|^p \leq t}dx dt \leq \int_0^\infty\frac{p}{q}t^{p/q-1} \lambda(E)^{1-q/p} \lambda(f>t^{1/p})^{q/p}dt $$ $$ = \frac{p}{q}\lambda(E)^{1-q/p} \int_0^\infty t^{p/q-1} \lambda(f>t^{1/p})^{q/p} dt $$ but I can't get an estiamte like this to turn out to be finite.

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The usual way to estimate $L^p$ norms using weak estimates is by using the Layer-Cake formula (see $L^p$-norm of a non-negative measurable function)

$$ \int |f|^q \, d\mu = \int_0^\infty q \cdot \lambda^{q-1} \cdot \mu(\{x \mid |f(x)| \geq \lambda\}) \, d\lambda. $$

Now, we split the integral into two parts: \begin{eqnarray*} \int_{0}^{\infty}\lambda^{q-1}\mu\left(\left\{ x\mid\left|f\left(x\right)\right|\geq\lambda\right\} \right)\,{\rm d}\lambda & = & \int_{0}^{t}\lambda^{q-1}\mu\left(\left\{ x\mid\left|f\left(x\right)\right|\geq\lambda\right\} \right)\,{\rm d}\lambda+\int_{t}^{\infty}\lambda^{q-1}\mu\left(\left\{ x\mid\left|f\left(x\right)\right|\geq\lambda\right\} \right)\,{\rm d}\lambda\\ & \leq & \int_{0}^{t}\lambda^{q-1}\mu\left(E\right)\,{\rm d}\lambda+N\cdot\int_{t}^{\infty}\lambda^{q-1}\lambda^{-p}\,{\rm d}\lambda\\ & = & \mu\left(E\right)\cdot\frac{\lambda^{q}}{q}\bigg|_{0}^{t}+N\cdot\frac{\lambda^{q-p}}{q-p}\bigg|_{t}^{\infty}\\ & \overset{q<p}{=} & \mu\left(E\right)\cdot\frac{t^{q}}{q}+N\cdot\frac{t^{q-p}}{p-q}. \end{eqnarray*}

I will leave the rest of the argument (optimize the value of $t> 0$) to you.

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PhoemueX
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