TITLE: An interpolation inequality
QUESTION [2 upvotes]: I am interested in the following statement. Let $q>p$. Then there are positive numbers $\alpha$ and $\beta$ so that for all $f\in C^1(\mathbb{R}^n)$, one has
$$ \left(\int|\nabla f|^p dx\right)^\frac{1}{p} \leq C \left(\int|f|^p dx\right)^\frac{\alpha}{p} \left(\int|\nabla f|^q dx\right)^\frac{\beta}{q}. $$
Does someone have an idea or a reference?

REPLY [3 votes]: Such an inequality is false. Take $n=1$, $\phi$ a standard cut-off function supported in $[0,1]$ and $f=\sum_n c_n \phi (\frac{x-n}{r_n})$. Then $\|f\|_p \approx \sum_n |c_n|^p r_n$, $\|f'\|_p \approx \sum_n \frac{|c_n|^p}{r_n^{p-1}}$, $\|f'\|_q \approx \sum_n \frac{|c_n|^p}{r_n^{q-1}}$. If $p=1$, $q=2$, $c_n=1/n$ and $r_n=(\log n)^{-2}$ then $f\in L^1$, $f' \in L^2$, $f' \not \in L^1$.
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