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I'm studying $L^p$ space.

$1 \le p < r then $L^p \cap L^q \subset L^r$. More over $L^p \cap L^\infty \subset L^r$

I'm trying to prove that fact. Which theorem is useful for proving that?

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    Ah it's typo. I corrected $L^1$ to $L^q$2012-06-13

3 Answers 3

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Some hints:

$(1.)$ For $L^{p}\cap L^{q}\subset L^{r}$, take and $f\in L^{p}\cap L^{q}$ and divide the integration domain $X$ to $A:=\{x\in X:|f(x)|\leq 1\}$ and $A^{c}$. What can you say about $|f(x)|^{r}$ for $x\in A$ or $x\in A^{c}$? What can you conclude for $\|f\|_{r}$?

$(2.)$ Start showing that if $f\in L^{p}$ then $\mu(A^{c})<\infty$. Then use the fact that $|f(x)|\leq \|f\|_{\infty}$ for $\mu$-a.e. $x\in X$ and use the same logic as in the previous step to conclude $\|f\|_{r}<\infty$ if $f\in L^{p}\cap L^{\infty}$.

If you need some more hints or can't get started with these; we can discuss at the comment section below.

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(1.) For every positive real number $a$, one has $a^r\leqslant a^p+a^q$ (one can prove separately the cases $a\leqslant1$ and $a\gt1$). Thus, $|f(x)|^r\leqslant |f(x)|^p+|f(x)|^q$ for every $x$.

Hence $\|f\|_r^r\leqslant\|f\|_p^p+\|f\|_q^q$ and $\|f\|_r$ is finite for every $f$ in $L^p\cap L^q$. Thus, $L^p\cap L^q\subset L^r$.

(2.) For every positive real numbers $a$ and $b$ such that $a\leqslant b$, one has $a^r\leqslant a^pb^{r-p}$. Thus, $|f(x)|^r\leqslant |f(x)|^p\cdot\|f\|_\infty^{r-p}$ for every $x$.

Hence $\|f\|_r^r\leqslant\|f\|_p^p\cdot\|f\|_\infty^{r-p}$ and $\|f\|_r$ is finite for every $f$ in $L^p\cap L^\infty$. Thus, $L^p\cap L^\infty\subset L^r$.

Note: Although such uniform pointwise inequalities cannot yield optimal norm inequalities, they are (i) simple to prove, and (ii) sufficient to get the inclusions of spaces the OP is interested in.

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    For any $t\in\mathbb{R}$, $tf$ also satisfies your inequality. Therefore, $ \|f\|_r^r\le t^{p-r}\|f\|_p^p+t^{q-r}\|f\|_q^q $ Minimizing the rhs yields $ t=\left(\frac{r-p}{q-r} \frac{\|f\|_p^p}{\|f\|_q^q}\right)^{\Large\frac{1}{q-p}} $ plugging the latter into the former yields $ \|f\|_r\le\left(\left(\frac{q-r}{r-p}\right)^{\Large\frac{r-p}{q-p}} + \left(\frac{r-p}{q-r}\right)^{\Large\frac{q-r}{q-p}}\right)^{\Large\frac1r} \|f\|_p^{\Large\frac{p}{r}\frac{q-r}{q-p}} \|f\|_q^{\Large\frac{q}{r}\frac{r-p}{q-p}} $2012-06-14
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For $f(x)\ge0$, Jensen's Inequality yields $ \left(\frac{1}{\int_X f^p(x)\,\mathrm{d}x}\int_X f^{r-p}(x)f^p(x)\,\mathrm{d}x\right)^{\Large\frac{q-p}{r-p}}\le\frac{1}{\int_X f^p(x)\,\mathrm{d}x}\int_X f^{q-p}(x)f^p(x)\,\mathrm{d}x $ which becomes $ \left(\int_Xf^r(x)\,\mathrm{d}x\right)^{\Large\frac1r}\le\left(\int_Xf^p(x)\,\mathrm{d}x\right)^{\Large\frac1p\left(\frac{p}{r}\frac{q-r}{q-p}\right)}\left(\int_X f^q(x)\,\mathrm{d}x\right)^{\Large\frac1q\left(\frac{q}{r}\frac{r-p}{q-p}\right)} $ Thus for $f\in L^p\cap L^q$, $ \|f\|_r\le\|f\|_p^{\Large\frac{p}{r}\frac{q-r}{q-p}}\;\|f\|_q^{\Large\frac{q}{r}\frac{r-p}{q-p}}\tag{1} $ where $ \frac{p}{r}\frac{q-r}{q-p}+\frac{q}{r}\frac{r-p}{q-p}=1\tag{2} $ Note that when $q\to\infty$, $(1)$ becomes $ \|f\|_r\le\|f\|_p^{\Large\frac{p}{r}}\;\|f\|_\infty^{\Large1-\frac{p}{r}}\tag{3} $