Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has uniformly bounded derivatives in $L^p (\mathcal{M})$ for each $p \geq 1$ ?
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asked Feb 6, 2012 at 19:35
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$\begingroup$ Have you tried to figure this out yourself? What difficulty did you run into? Also, note that you need to clarify what you mean by "uniformly bounded derivatives in $L^p(M)$". $\endgroup$– Deane YangCommented Feb 6, 2012 at 19:46
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$\begingroup$ I mean that $|| \nabla ^k \phi _j ||_{ L^p (\mathcal{M})} \leq K$ for each $k \geq 0$, where $K$ can depend on $j$ but is independent of $k$. Here, $\phi _j$ is one function in the partition of unity. In terms of trying it for myself, I have given it some thought but was hoping I was missing something trivial. $\endgroup$– Michael CoffeyCommented Feb 6, 2012 at 20:19
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$\begingroup$ @Michael, something is not competely clear to me: you want a bound on the $L^p$ norms of the derivatives, which is uniform on which of the following: $p\ge1$; any partition of unity subordinated to the subcover $\mathcal{U}$; at least some partition of unity subordinated to the subcover $\mathcal{U}$; the subcover $\mathcal{U}$ itself ? Thanks. $\endgroup$– Pietro MajerCommented Feb 6, 2012 at 20:24
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$\begingroup$ It appears that he wants it to be uniform in the number of derivatives. I don't have an answer to this, but I think it suffices to find a single compactly supported smooth function on $R^n$ with a uniformly bound for any number of derivatives in $L^p$. $\endgroup$– Deane YangCommented Feb 6, 2012 at 20:44
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$\begingroup$ I doubt it. Here's why: I would guess that you can reduce this to a partition of unity for a closed interval. Now the derivatives of a given element of the "standard explicit" partition (see, e.g., Dubrovin, Fomenko, and Novikov's construction) are going to oscillate more as the order of the derivative increases. Looking at tanh (which is similarish) I don't see why the extrema of these derivatives would be bounded (though I don't have ready access to software to check this numerically at the moment). $\endgroup$– Steve HuntsmanCommented Feb 6, 2012 at 21:02
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1 Answer
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A counterexample: Let $M$ be the unit circle. Let the two charts be the arcs $A = (0,2\pi)$ and $B = (\pi/2, 5\pi/2)$. For each $n$, consider the partition of unity subordinate to $\{A,B\}$ given by
$$ \psi_{A,n} = \sin^2 ( (2n+1) \theta ) $$
and
$$ \psi_{B,n} = \cos^2 ( (2n+1) \theta ) $$
Check that $\psi_{A,n} (0) = 0 = \psi_{B,n} (\pi/2)$, and clearly $\psi_{A,n} + \psi_{B,n} = 1$.
By the scaling property it is easy to check that
$$ \| \partial^k\psi_{A,n} \| = (2n+1)^k \| \partial^k\psi_{A,1} \| $$
And using that
$$ \sin^2(\theta) = \frac12 (1-\cos 2\theta) $$
you see immediately that your desired uniform bound is impossible.
answered Mar 1, 2012 at 9:51