9
$\begingroup$

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial steps so that I could get to a specific calculation that my supervisor wanted me to look at. Now, looking back at the setup, I realise that I don't understand all of the details.

In Chapter 4, Hörmander introduces linear, closed, densely defined operators $T : L^2_{(p,q)}(\Omega, \varphi_1) \to L^2_{(p,q+1)}(\Omega, \varphi_2)$ and $S : L^2_{(p,q+1)}(\Omega, \varphi_2) \to L^2_{(p,q+2)}(\Omega, \varphi_3)$ which are defined by $\overline{\partial}$.

Some functional analysis shows that it is enough to prove that there is a positive constant $C$ such that $\|f\|^2 \leq C(\|T^*f\|^2 + \|Sf\|^2)$ for all $f \in D_{T^*}\cap D_S$. An argument is then given to show that $D_{(p,q+1)}(\Omega)$ is dense in $D_{T^*}\cap D_S$ with respect to the graph norm $f \mapsto \|f\| + \|T^*f\| + \|Sf\|$, where $D_{(p,q+1)}(\Omega)$ denotes the smooth compactly supported $(p,q+1)$ forms. The proof of this fact is where my troubles begin.

Hörmander shows that for suitable weights, and a sequence of compactly supported functions $(\eta_{\nu})_{\nu \in \mathbb{N}}$ with $0 \leq \eta_{\nu} \leq 1$ and $\eta_{\nu} = 1$ on any compact subset of $\Omega$ when $\nu$ is large (which satisfy an appropriate bound on $|\bar{\partial}\eta_{\nu}|$), we have $\|\eta_{\nu}f - f\|_{\varphi_{2}} \to 0$, $\|S(\eta_{\nu}f) - \eta_{\nu}Sf\|_{\varphi_{3}} \to 0$, and $\|T^*(\eta_{\nu}f) - \eta_{\nu}T^*f\|_{\varphi_{1}} \to 0$. I can understand why the first two are true, but not the third.

Hörmander shows that $\eta_{\nu}f \in D_{T^*}$. From there I can see how he gets, for $u \in D_T$, $|(T^*(\eta_{\nu}f) - \eta_{\nu}T^*f, u)_{\varphi_{1}}| \leq \int|f|e^{-\varphi_{2}/2}|u|e^{-\varphi_{1}/2}d\lambda$, but after this inequality, he states

$\dots$ which implies the bound $|T^*(\eta_{\nu}f) - \eta_{\nu}T^*f|^2e^{-\varphi_{1}} \leq |f|^2e^{-\varphi_{2}}$.

I don't see how this follows. How does Hörmander obtain this (pointwise) estimate? At the moment, the best I’ve got is a messy measure theoretic argument that I’m not even sure is correct. Any help would be much appreciated.

$\endgroup$
3
  • $\begingroup$ I'm sure I haven't quite processed everything but it seems to follow formally, by taking $u$ with appropriately chosen $C_0^\infty$ coefficients with tiny support around a given point, right? And such $u$ lie in $D_T$, if I'm not mistaken. $\endgroup$
    – Mike Hall
    Jun 5, 2012 at 9:13
  • $\begingroup$ You are correct, such $u$ are in $D_T$, but I haven't been able to explicitly determine what the 'appropriately chosen' coefficients should be. The idea is that if there was a point $z \in \Omega$ such that $(|T^*(\eta_{\nu}f) - \eta_{\nu}T^*f|^2e^{\varphi_1})(z) > (|f|^2e^{-\varphi_2})(z)$, then by taking a sequence of smooth $u$ with shrinking supports containing $z$, then the global inequality $|(T^*(\eta_{\nu}f) - \eta_{\nu}T^*f, u)_{\varphi_1}| \leq \int |f|e^{-\varphi_2/2}|u|e^{-\varphi_1/2}d\lambda$ would be violated for some particular $u$. $\endgroup$ Jun 7, 2012 at 15:31
  • $\begingroup$ Shouldn't it just be like if you were proving Cauchy-Schwarz? Choose $|u(z)|=1$ so that the (pointwise) inner product with $(T^*(\eta_{\nu}f) - \eta_{\nu}T^*f)(z)e^{-\varphi_1(z)}$ is equal to $|(T^*(\eta_{\nu}f) - \eta_{\nu}T^*f)(z)|e^{-\varphi_1(z)}$ and then extend to a tiny bump function. $\endgroup$
    – Mike Hall
    Jun 7, 2012 at 18:04

1 Answer 1

4
$\begingroup$

I also do not see how the desired pointwise bound $|T^*(\eta_{\nu}f) - \eta_{\nu}T^*f|^2e^{-\varphi_1} \leq |f|^2e^{-\varphi_2}$ follows from the $L^2$ estimate. The latter certainly implies by Cauchy-Schwarz that the operator norms of the commutators $[T^*,\eta_{\nu}]$ are uniformly bounded by unity, but this is not what is needed. Instead, I propose to argue as follows. The commutator $[T^*,\eta_{\nu}]$ is the adjoint of the multiplication operator $[\overline{\eta_{\nu}},T]=-\overline{\partial}\overline{\eta_{\nu}}\wedge$. Using only pointwise estimates the desired bound follows.

Let me add two observations: (1) Two pages later Hörmander computes $T^*$ and states that this gives another proof of his formula (4.1.8) which is what needs to be proved. Actually the argument I gave above is this alternative proof without computing the commutator very explicitly. (2) In his 1965 Acta. Math. paper on the $\overline{\partial}$ operator, Hörmander does prove the analog of (4.1.8) via an operator norm estimate of the commutator. However, in the proof of Proposition 2.1.1 of that paper he has a setup which gives that the commutators converge to zero in norm, in contrast to only strong convergence as in Lemma 4.1.3 of the book.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.