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Let $a\in (0,1)$. Prove $$\|u\|_{L^r(\mathbb R^d)}\leq C\|\nabla u\|_{L^p(\mathbb R^d)}^a\|u\|_{L^q(\mathbb R^d)}^{1-a},\quad u\in \mathcal C_c^1(\mathbb R^d)$$ if $\frac{1}{r}=\frac{1-a}{q}$ and $\frac{1}{p}-\frac{1}{d}=0$ with $C$ independent of $u$.

Context

I had to prove this inequality when $$\frac{1}{r}=a\left(\frac{1}{p}-\frac{1}{d}\right)+\frac{1-a}{q}.$$ I proved it when $\frac{1}{p}-\frac{1}{d}>0$ and $\frac{1}{p}-\frac{1}{d}<0$. I want now to prove it when $\frac{1}{p}-\frac{1}{d}=0$, but I don't know how to proceed. Notice that for $\frac{1}{p}-\frac{1}{d}>0$ we used Sobolev inequality and for $\frac{1}{p}-\frac{1}{d}<0$ we use the fact that $W^{1,p}(\mathbb R^d)\hookrightarrow L^\infty (\mathbb R^d)$ continuously. But when $\frac{1}{p}-\frac{1}{d}=0$ is there such a trick ?

idm
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  • It would help if you sketched the proof for $p\neq d$. When $p=d$, $W^{1,d}$ does not embed into $L^\infty$, but it does embed into $L^q$ for arbitrarily large $q$, which might be useful to you. – Jeff Jan 13 '18 at 21:17
  • Are you aware that you are trying to prove the Gagliardo-Nirenberg interpolation inequality? Maybe you find a similar proof strategy as yours if you search for this inequality in some books. – Cahn Jan 14 '18 at 13:22
  • For $p=d$ we have $| u|{BMO}\leq C| \nabla u|{L^d}$. To see this note that by Poincare's inequality and Holder's inequality we have $$ \int_B |u-u_B|, dx \leq C|B|^{1/d} \int_B |\nabla u|, dx \leq C|B|| \nabla u|{L^d}. $$ Therefore the desired result would follow if in the usual $L^p$ interpolation inequalities we can substitute $BMO$ for $L^\infty$, i.e. if it holds $$ | u|{L^p} \leq C| u|{BMO}^\theta | u|{L^q}^{1-\theta}, $$ where $1/p=(1-\theta)/q$. I'm pretty sure this is true but I can't find a reference for it. – Jose27 Jan 15 '18 at 04:00
  • @Jose27: nice idea :-) – idm Jan 19 '18 at 14:31

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