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.

  • $\begingroup$ 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 '12 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 $u$. $\endgroup$ – Michael Albanese Jun 7 '12 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 '12 at 18:04

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.


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