In my vs. post a few weeks ago, I sketched a more or less standard proof of the similarity principle, and then wrote:
I defy the reader to come up with any alternative version of the above proof that does not use properties of the operator for some .
Two readers responded to this challenge: they were Jean-Claude Sikorav and Patrick Massot, and in this post I’m going to explain (as I did last week on the topic of regularity and bubbling) my reinterpretation of the proof that they sent me. It should be said that after I’d managed to understand this proof, I still felt rather surprised that it works, and while I can’t speak for anyone else, it strikes me as something that I would never come up with if I had not first seen the standard proof in .
Recall the statement: the most useful version of the similarity principle can be viewed as saying that if is a real-linear Cauchy-Riemann type operator on a smooth complex vector bundle over a Riemann surface , and satisfies and , then on some neighborhood of in , admits a continuous trivialization that identifies with a holomorphic function. This is useful because it implies a unique continuation result: either or it has an isolated zero at (which is also of positive order if is a line bundle).
As I outlined in or not , that is the question, the similarity principle is a corollary of the following local existence result for solutions of linear Cauchy-Riemann type equations. Let’s fix the usual notation: will denote the open disks of radius and respectively, write the standard coordinate on as , and the standard Cauchy-Riemann operator as .
Lemma 1. Suppose , , and is a function of class . Then for any sufficiently small, there exists a linear map associating to each a function that satisfies in the sense of distributions and .
One remark before we get into the proof. The regularity assumption on the zeroth order term may strike you as absurdly weak — normally geometers are only interested in smooth Cauchy-Riemann type operators. Recall however that the first step in proving the similarity principle is to replace a real-linear Cauchy-Riemann operator with one that is complex linear but still annihilates the given section : this can always be done by changing the zeroth order term, but since we do not know a priori what the zero set of looks like, the price we pay is that can at best be assumed to be of class after this change. The above statement weakens the hypothesis to for just because that will turn out to be what we need in the proof. The similarity principle then follows because we can use the local existence result to construct each column of a continuous matrix-valued function on that satisfies and , and since is now complex linear, the Leibniz rule implies that if and , then .
There’s a fairly straightforward way to prove Lemma 1 if you’re willing to accept the fact — essentially equivalent to the Calderón-Zygmund inequality — that
has a bounded right inverse for . The idea is to look for solutions to the equation , where is defined to match on and to vanish everywhere else. By the Sobolev embedding theorem, we have
hence in the space of bounded linear operators as . Now consider the continuous family of bounded linear maps
for , and notice that since constant functions are holomorphic, one can use the bounded right inverse of to construct a bounded right inverse for . The existence of bounded right inverses is an open condition, so it follows that also admits such an inverse for all sufficiently small; call it . After embedding into , the linear map promised by Lemma 1 can now be written as
and this completes the proof.
We used the assumption quite a few times in the above argument: without it, the map would not be continuous because is not continuous, and even if we could obtain solutions of class in the end, they might not be in . For these reasons, I previously could not imagine how it might be possible to prove such a local existence result without relying on the elliptic estimates for with .
But it is possible.
Here’s a slightly different kind of local existence result.
Lemma 2. Suppose , , is a function of class , and is a holomorphic function on a disk of some radius . Then for any number , there exists a number and a continuous function such that and in the sense of distributions.
In this lemma we’ve dropped the requirement that our solution take a prescribed value at the origin, instead just asking for it to be -close to some prescribed function. Nonetheless, it’s not too hard to see that Lemma 2 implies Lemma 1: one can use Lemma 2 to construct the columns of a continuous matrix-valued function that satisfies and is -close to the identity, hence everywhere invertible. Solutions with prescribed values at 0 can then be constructed in the form where is constant.
So how can we prove Lemma 2 using only estimates? We shall again look for continuous functions satisfying . Notice that it is still true that
in the space of bounded linear operators , though for different reasons than in the case: is a Sobolev borderline case and embeds continuously into for every , so picking such that and using Hölder’s inequality, we have
It follows that since has a bounded right inverse, so does for sufficiently small: denote this right inverse by
It should now at least seem plausible that any solution to the equation admits a -close perturbation satisfying : indeed, the latter is equivalent to the equation
so an obvious solution presents itself in the form
Since is -small for small and is close to (the right inverse of ) in the operator norm, our solution is evidently -small. This is nice, of course, but it’s not good enough. We also need to be continuous, and -small. Is it?
A nice little fact about the right inverse of
As it turns out, yes: the solution we just found is continuous and -small. This is easy to see if you don’t mind using Calderón-Zygmund, because is also small in and the right inverse of restricts to as a continuous operator , which is then continuous into by the Sobolev embedding theorem. But actually, this -bound on admits a much more direct proof that is orders of magnitude easier than either Calderón-Zygmund or the Sobolev embedding theorem.
Proposition. The standard Cauchy-Riemann operator admits a bounded right inverse such that for each , restricts to as a bounded linear operator .
In fact, with a little bit more effort one can prove that maps continuously to the Hölder space , but the -bound will be plenty sufficient for our purposes. To see why it is true, let us quickly recall how is constructed (cf. Section 2.6 in the current version of my book in progress on holomorphic curves). The Cauchy-Riemann operator has a fundamental solution , defined by
Being a fundamental solution means that satisfies in the sense of distributions, so the equation can be solved for sufficiently nice functions by writing as the convolution
where denotes the Lebesgue measure for functions of . This is well defined in particular whenever is smooth with compact support in , and in this case one can prove a straightforward variation on Young’s inequality to bound in terms of for any , so extends to bounded linear map . Since , one obtains a -bound on if one can also bound in terms of for all . This is always possible if , and in the case this is the essence of the Calderón-Zygmund inequality. But for there is an easy proof by Fourier transforms: observe that the Fourier transform of the relation gives
where denotes the Fourier transform of in the sense of tempered distributions. Similarly, taking the Fourier transform of the equation gives , hence , and we can now use Plancherel’s theorem to compute
This proves that the map extends to a bounded linear operator , and we define to be this extension. With this explicit formula in hand, the proof of the proposition is very quick: notice in particular that for every , so if and , then for every and every ,
where is the supremum of all -norms of restricted to disks of unit radius in . Since the convolution maps smooth functions to smooth functions and the -closure of is , this proves the proposition.
There remains just one niggling detail: we’ve shown that maps to , but in our proof of Lemma 2, we need to know that this is also true for , the right inverse of the perturbed operator . To see this, it will help to give a slightly more precise definition of . Notice that
is a bounded linear operator on and is close to the identity in the operator norm since is continuous and is small. But for slightly different reasons, this operator is also close to the identity in the space of bounded linear operators on : indeed, we showed above that is continuous, and is also small since
Thus if we choose small enough, defines an isomorphism on both and , so defining
gives a right inverse of that is continuous both from to and from to . The proof of Lemma 2 is now complete, and though we appealed to Calderón-Zygmund once or twice for intuition, we never actually used it.
This will be my last post on the vs. debate for a while, as I’m sure it’s clear to everyone by now that I’ve been thinking about this far too much lately. The evidence currently available to me suggests that it might very well be possible to develop the entire theory of pseudoholomorphic curves using only estimates — useful perhaps if you want to feel honest without taking the time to read the proof of Calderón-Zygmund, or if you’re one of those strange people with an aversion to the axiom of choice (it’s needed for the Hahn-Banach theorem, which is needed for regularity and transversality arguments in for , but you can avoid it if you only work with Hilbert spaces).
But just as proving the estimates requires effort, avoiding them also requires effort, and some of the resulting proofs become arguably less straightforward and less elegant. In the end, it’s a matter of taste.