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Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, where in local coordinates $|\nabla^{p} \overline{\nabla^q} f|^2_g = g^{i_1\bar k_1} \ldots g^{l_q \bar j_q}\partial_{i_1} \ldots \partial_{i_p}\partial_{\bar j_1} \ldots \partial_{\bar j_q}f\partial_{l_1} \ldots \partial_{l_q}\partial_{\bar k_1} \ldots \partial_{\bar k_p}f$.

I was told that there is an inclusion from $\mathcal{C}^{k+1} \to \mathcal{C}^k$ and it is compact. I suppose that I first need to show that if the $\mathcal{C}^{k+1}$ norm of $f$ is bounded by a multiple(independent of $f$) of its $\mathcal{C^k}$ norm. I am not sure how I should do that. My guess is that some kind of integration by part is needed here to "decrease the partials".

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    $\begingroup$ Is this not (equivalent to) the standard $C^k$ norm (at least locally, in charts)? If so, this is just the Arzela-Ascoli theorem. $\endgroup$
    – Ryan Unger
    Commented Jul 17, 2019 at 17:00
  • $\begingroup$ May I ask what $C^k$ norm are you referring to？Perhaps any reference so I can look it up? $\endgroup$
    – archer
    Commented Jul 17, 2019 at 17:45
  • $\begingroup$ Here's a version that gives what I claim. Suppose $X$ is compact and let $\theta_1,\dotsc,\theta_N$ be a partition of unity relative to some system of charts that covers $X$. Then define $$\|u\|_{C^k}=\sum_{i=1}^N\sum_{|\alpha|\le k}\sup_X |D^\alpha(\theta_i u)|,$$ where $D^\alpha$ is the $\alpha$-th multiderivative in the chart associated to $\theta_i$. This norm should be equivalent (in the sense of norms) to what you wrote. It has the desired compactness property, so your norm does too. $\endgroup$
    – Ryan Unger
    Commented Jul 17, 2019 at 18:26
  • $\begingroup$ I see. Is there any reference on it? $\endgroup$
    – archer
    Commented Jul 17, 2019 at 18:35
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    $\begingroup$ This is a standard exercise in advanced calculus. You apply the Arzela-Ascoli theorem to the sequence and its derivatives. The main ideas are in Chapter 5 of Jost's postmodern analysis book. $\endgroup$
    – Ryan Unger
    Commented Jul 17, 2019 at 18:47

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