# Thread: Lp space problem, possibly using Holder inequality

1. ## Lp space problem, possibly using Holder inequality

Here's the problem statement:

If $\displaystyle f\in L_p$ for some $\displaystyle 0<p<\infty$, and every set of positive measure in $\displaystyle X$ has measure at least $\displaystyle m$, show that $\displaystyle f\in L_q$ for all $\displaystyle p<q\leq\infty$, with $\displaystyle \lVert f\rVert_q\leq m^{\frac{1}{q}-\frac{1}{p}}\lVert f\rVert_p$.
I have no problem showing that $\displaystyle f\in L_q$. It can also be shown that $\displaystyle f\in L_{\infty}$, in case that helps. However, I cannot show that the inequality

$\displaystyle \lVert f\rVert_q\leq m^{\frac{1}{q}-\frac{1}{p}}\lVert f\rVert_p$

holds. Any help would therefore be much appreciated !

(This is exercise 1.3.12 from this book, p41.)

2. ## Re: Lp space problem, possibly using Holder inequality

Let $\displaystyle X_a=\{ x\in X : |f(x)|>a\}$. If you've proven that $\displaystyle f\in L^\infty$ then let $\displaystyle b=\| f\|_{\infty}$ then we have

$\displaystyle \int_X |f|^p \geq \int_{X_b} |f|^p \geq b^pm$

(Edit: To be formal you'd have to take $\displaystyle X_{b-\varepsilon}$ but the bound on the left is independent of it and on the right it would look something like $\displaystyle (b-\varepsilon)^pm \to b^pm$)

ie.$\displaystyle \| f\|_{\infty} \leq m^{-\frac{1}{p}} \| f\|_p$. It's easy to prove by Hölder's inequality that $\displaystyle f\in L^q$ and

$\displaystyle \| f\|_q \leq \| f\|^{\theta}_p \| f\|^{1-\theta}_{\infty} \qquad \frac{1}{q}=\frac{\theta}{p}$

now substitute the estimate for $\displaystyle \| \cdot \|_\infty$ in this one and you're done.

3. ## Re: Lp space problem, possibly using Holder inequality

Very clever ! Thanks a bunch !