What is the geometric intuition of $n/p$?

Asked May 21 '19 at 13:25

Active May 21 '19 at 13:25

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(volume for ball): Let B$_{r}^{p}(0):=\{x\in\mathbb R^n; \|x\|_p\leq r\}$. Then the volume of B$_{r}^{p}(0)$ is \begin{align} {\rm V}_r^{p}=2^n\cdot\frac{\left\{\Gamma(\frac{1}{p}+1)\right\}^n}{\Gamma\left(\frac{n}{p}+1\right)}\cdot r^n. \quad\text{(calculation of multi-integral)} \end{align}

(Sobolev embedding): Sobolev space $W^{k,p}(n)(1\leq p<\infty)\rightarrow \frac{n}{p}-k:=i$.

If $0<i$, then $W^{k,p}\hookrightarrow W^{\ell,q}$, where $\frac{n}{q}-\ell=i, k>\ell$.

If $0>i$, then $W^{k,p}\hookrightarrow C^{r,\alpha}$, where $-(r+\alpha)=i, 0<\alpha\leq1$.

Notice that there is a well-marked factor \begin{align} \frac{n}{p}=\frac{\text{dimension of space}}{\text{norm index of space}}.\end{align} What is the significance or geometric intuition of $\frac{n}{p}$, which I think is essential and fundamental? (Maybe I think too much, thanks!)

asked May 21 '19 at 13:25

1

What happens for $x^a = \prod_{j=1}^n x_j^{a_j}$ ? – reuns May 21 '19 at 14:10
Sorry, I don't know. – May 22 '19 at 09:47

What is the geometric intuition of $n/p$?

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