Why bootstrapping for $J$-holomorphic curves fails in $C^k$

We present a simple example for the failure of the Calder\'on-Zygmund estimate for the $\bar{\partial}$-operator when the Sobolev $(k,p)$-norms are replaced by the $C^k$-norms. This example is discussed in the context of elliptic bootstrapping, Fredholm theory, and the regularity of $J$-holomorphic curves.


Introduction
In the theory of elliptic partial differential equations, with the Laplace equation as the prototype, it is well known (see [4,6], for instance) that regularity results can be established for solutions lying in Sobolev or Hölder spaces, with the help of Calderón-Zygmund or Schauder estimates, respectively.
In symplectic topology, moduli spaces of J-holomorphic curves are described as solution sets of a nonlinear Cauchy-Riemann equation. Since the implicit function theorem fails in the Fréchet space of smooth maps, one needs to work with a Banach space of maps having lower regularity. A Calderón-Zygmund estimate for the ∂operator then is essential for two purposes: (i) regularity results for J-holomorphic curves, in the sense that solutions of the nonlinear Cauchy-Riemann equation in the Sobolev space W 1,p actually turn out to be of class C ∞ ; (ii) existence of C k -bounds for all k that guarantee compactness of the relevant moduli space of J-holomorphic curves.
The Calderón-Zygmund estimate allows one to bootstrap from W k,p to W k+1,p ; smoothness and C k -bounds (for p > 2) then follow from the Sobolev embedding theorem and the corresponding Sobolev inequality. Roughly speaking, the estimates say that a (weak) solution u ∈ W k,p of the inhomogeneous Laplace equation ∆u = f is two derivatives more regular than f . The literature abounds with examples that this statement fails in the smooth theory, that is, from ∆u, understood in the distributional sense, of class C k one cannot, in general, conclude that u ∈ C k+2 . This means that a C k -analogue of these estimates cannot be formulated in a sensible way, because u C k+2 may not be defined.
However, we have not seen it emphasised that even when the correct order of differentiability is assumed a priori, the Calderón-Zygmund (or Schauder) estimates fail when the Sobolev (or Hölder) norms are replaced by C k -norms. We allow that this may be fairly apparent to the more analytically inclined.
In this expository note we adapt an example of Sikorav [9] (where the C k -estimate cannot be formulated) to define an explicit family of solutions of the inhomogeneous Cauchy-Riemann equation, having the correct regularity (in the C k -theory), but violating the Calderón-Zygmund estimate. We also place this in the context of the Fredholm property of the Cauchy-Riemann operator, which is essential for showing that the relevant moduli space of J-holomorphic curves is a smooth manifold.
As we shall explain, in the theory of J-holomorphic curves one often deals with these estimates in a setting where the maps are known to be smooth. As a consequence, the consideration of Sobolev norms on such maps, or the introduction of Sobolev spaces of J-holomorphic curves, may seem to lack motivation. Our example clarifies why one has to work with Sobolev completions.

The Calderón-Zygmund estimate
In this section we formulate the Calderón-Zygmund estimate for the inhomogeneous Cauchy-Riemann equation and briefly discuss its relevance for the bootstrapping of J-holomorphic curves.
Let B R ⊂ C be the open disc of radius R centred at 0. We write C ∞ c (B R , C n ) for the space of compactly supported smooth maps B R → C n , and W k+1,p 0 (B R , C n ) for its closure in the Sobolev space of k + 1 times weakly differentiable maps of finite Sobolev (k + 1, p)-norm. The Cauchy-Riemann operator is Likewise, we are going to set ∂ := 1 2 (∂ x − i∂ y ). For a proof of the following estimate see [5].
Proposition 1. For any k ∈ N 0 and real numbers p > 1 and R > 0, there is a positive constant c = c(k, p, R) such that u k+1,p ≤ c ∂u k,p for all u ∈ W k+1,p 0 (B R , C n ).
Without the assumption on compact support, one needs to add the term c u k,p on the right-hand side, as can be seen by a partition of unity argument. This is the more common formulation of the Calderón-Zygmund estimate (and sometimes referred to as a semi-Fredholm estimate, cf. [5]).
One first has to prove the proposition for u ∈ C ∞ c (B R , C n ), the stated version then follows by writing u ∈ W k+1,p 0 (B R , C n ) as a limit of compactly supported smooth maps.
Remark. For p = 2, the proof of Proposition 1 simplifies considerably, see [1,Section 4.2] or [5, Section III.1.2], but for the subsequent application of the Sobolev embedding theorem one needs p > 2.
We are going to show by an example that there is no such uniform estimate u C k+1 ≤ c ∂u C k .
From Proposition 1, in [5] the regularity of J-holomorphic discs (with Lagrangian boundary condition) is established by a localisation argument and the difference quotient technique of Abbas and Hofer [1]. That difference quotient technique allows one to bootstrap from W k,p to W k+1,p , but not from C k to C k+1 , so a Calderón-Zygmund estimate in C k would not be of help.
A different approach to the regularity of J-holomorphic curves can be found in [8,Section B.4]. Here the nonlinear Cauchy-Riemann equation u x + J(u)u y = 0 is reformulated as an inhomogeneous linear equation, and then one directly uses the regularity theory for the ∂-operator. This approach would stumble at the first hurdle in the C k -theory by Sikorav's example.
Much of the compactness theory of J-holomorphic curves as in [5] would go through in the smooth theory if one had a Calderón-Zygmund estimate (for smooth maps) in the C k -norms. Our example shows why this hope is in vain.

The example
3.1. Sikorav's example. We begin with an example of a function f : B 1/2 → C that is not of class C 1 , even though ∂f (in the distributional sense) is of class C 0 . This is a slight modification (and correction) of an example presented by Sikorav [9], which is closely related to the standard example illustrating the corresponding phenomenon for the Laplace operator, see [4]. Set This extends continuously (with value 0) into z = 0, and this continuous extension is the distributional derivative ∂f on B 1/2 (see the discussion in Section 4).
On the other hand, we have ∂f = log log |z| −2 − 1 log |z| −2 for z = 0, which does not extend continuously into z = 0.

Proof of Proposition 2.
Here is the example for the failure of the Calderón-Zygmund estimate in the C k -theory. Choose a smooth function ψ : Since |z| 1/ν log log |z| −2 → 0 as z → 0, the function f ν is complex differentiable in z = 0 with ∂f ν (0) equal to 0, and hence differentiable with ∂f ν (0) likewise equal to 0.
Writing |z| 1/ν as (zz) 1/2ν we see that We then compute Since both ∂f ν (z) and ∂f ν (z) go to 0 as z → 0, we conclude that f ν (and hence u ν ) is of class C 1 . The second summand in ∂f ν is bounded uniformly in ν on B 1/2 \ {0}. Writing |z| = r ν for z = 0 with 0 < r < 2 −1/ν < 1, we see that the first summand in ∂f ν is likewise bounded uniformly in ν, since which extends continuously into r = 0. Clearly, these bounds also take care of the second summand in ∂f ν . On the other hand, the first summand of ∂f ν (z) evaluated at z = 2 −ν yields 1 2 log log 2 2ν , which goes to infinity as ν → ∞. It follows that ∂f ν C 0 is bounded uniformly in ν, whereas f ν C 1 goes to infinity as ν → ∞.
The same is true for the compactly supported functions u ν , since f ν C 0 is bounded uniformly in ν, and with a similar expression for ∂u ν , which means that the limiting behaviour of ∂u ν and ∂u ν equals that of ∂f ν and ∂f ν , respectively.

The Fredholm property of ∂
The discussion so far shows that one cannot forgo Sobolev norms for bootstrapping arguments, but it still seems to leave room for the possibility to stay within the framework of C k -maps. In the compactness theory of J-holomorphic curves one cannot simply work in a space of C k -maps for some fixed k, since one typically relies on the Arzelà-Ascoli theorem and C k+1 -bounds to guarantee convergence in C k . However, one might want to start with J-holomorphic curves of class C 1 , interpret them as maps of class W 1,p , and then use elliptic bootstrapping (with respect to Sobolev norms) and Sobolev embedding to show that the curves are in fact smooth. In order to establish that M is a manifold of the expected dimension, one needs to verify that ∂ J is a Fredholm operator, so that one can apply the theorem of Sard-Smale. By a perturbation argument it may suffice to do this for the linear Cauchy-Riemann operator ∂. This Fredholm property, as we shall see presently, holds for Sobolev spaces, but it is violated in the C k -realm.
Consider a bounded linear operator T : E → F between Banach spaces with dim ker T < ∞. Let E 1 be a closed complement of ker T in E. It then follows from the open mapping theorem, applied to T | E1 : E 1 → T (E), that T has a closed image if and only if we have an estimate x E ≤ c T x F for all x ∈ E 1 . The image T (E) being closed is a necessary condition for coker T to be finite, i.e. the Fredholm property of T .
Thus, whether or not ∂ has a closed image (when regarded as an operator between certain Banach spaces of functions) is equivalent to the existence or the failure of the Calderón-Zygmund estimate in the corresponding norms, provided ker ∂ is finitedimensional in the given setting.
Here is an example how to use Proposition 2 to show the failure of the Fredholm property in the C k -theory of J-holomorphic discs. Write D ⊂ C for the closed unit disc, and C 1 R (D, C) for the space of C 1 -maps D → C with real boundary values. We may regard C 1 c (B 1/2 , C) as a subspace of C 1 R (D, C), and C 0 c (B 1/2 , C) as a subspace of C 0 (D, C). Notice that the ambient spaces are Banach spaces, but the subspaces are not closed.

4.2.
Failure of the Fredholm property in C k . We now want to use Sikorav's example to demonstrate by a specific example that the image of ∂ : C 1 R (D, C) → C 0 (D, C) is not closed, and thus give a more concrete proof of Corollary 3. Proposition 4. The weak derivative ∂(ψf ) ∈ C 0 (D, C) -with f as in (1), and ψ the cut-off function from Section 3.2 -is in the closure of im ∂, but not itself an element of that image.
We first present a 'classical' argument using mollification, and then an alternative approach using the sequence (f ν ) introduced in Section 3.2.

4.3.
Proof by mollification. We begin by analysing Sikorav's example a little more carefully.
Lemma 5. The function f defined in (1) is an element of W 1,p (B 1/2 , C) for any p ∈ [1, ∞), and the weak derivatives ∂f and ∂f may be assumed to coincide with the actual derivatives on B 1/2 \ {0}.
Proof. (i) First we are going to show that the weak ∂-and ∂-derivatives of f are as claimed. We consider ∂f ; for ∂f the argument is completely analogous. For ε > 0, let χ ε ∈ C ∞ (C) be a cut-off function with χ ε ≡ 0 on B ε , and χ ε ≡ 1 outside B 2ε .
Set f ε = χ ε f ∈ C ∞ (B 1/2 ). For any test function ϕ ∈ C ∞ c (B 1/2 ), integration by parts gives We may assume that |∂χ ε | ≤ c 2ε for some constant c. For |z| ≤ ε and |z| ≥ 2ε, the derivative ∂χ ε vanishes identically. It follows that Now, ∂f ε converges pointwise on B 1/2 \ {0} to ∂f . Hence, provided the functions f /z and ∂f are integrable, we can take the limit ε ց 0 in (4) and conclude with the Lebesgue dominated convergence theorem that so ∂f constitutes the weak ∂-derivative of f . (ii) It remains to show that the functions f /z, ∂f and ∂f are in L p (B 1/2 , C) for any p ∈ [1, ∞). Both ∂f and the function z → 1/ log |z| −2 extend continuously to B 1/2 , so we need only show that For r ∈ (0, 1/2) we have 0 < log log r −2 < log r −2 = 2 log r −1 , so it suffices to show that r → log r −1 is an L p -function on the interval (0, 1/2). In fact, this function is even L p -integrable on (0, 1), as can be seen by the substitution t = log r −1 , t ∈ (0, ∞), which yields a transformation to the Γ-function. For with r = e −t and dr = −e −t dt we have We conclude that the function z → log log |z| −2 is in L p (B 1/2 , C).
For the basic theory of mollifiers we use presently in the first proof of Proposition 4, see [3,Sections C.5 and 5.3].
The function f can be defined on B 0.6 , so for ε < 0.1 we can define the mollifi- where dλ 2 w denotes the 2-dimensional Lebesgue measure with respect to the variable w. As ε ց 0, the function f ε converges to f in W 1,p (B 1/2 , C), and for p > 2 this convergence is uniform in C 0 (B 1/2 , C) by the Sobolev embedding theorem.
Since ∂f is continuous on B 1/2 , we have uniformly on B 1/2 as ε ց 0. Now let ψ be the cut-off function defined in Section 3.2, and set This function is compactly supported in B 1/2 , and we may regard it as an element uniformly on D, and this limit equals the weak ∂-derivative of ψf . But ∂(ψf ) -regarded as an element of C 0 (D, C) -does not equal ∂h for any h ∈ C 1 R (D, C), for otherwise we would have ∂(ψf − h) = 0 (in the distributional sense), which by the regularity of the ∂-operator would entail that ψf − h is of class C ∞ , contradicting the fact that ψf is not of class C 1 . Indeed, any weak solution u ∈ L 1 loc of the equation ∂u = 0 is also a weak solution of the Laplace equation ∆u = 0, and hence harmonic (and, in particular, smooth) by Weyl's lemma.

4.4.
Proof using the sequence (f ν ). We now give a more explicit construction of a sequence in im ∂ ⊂ C 0 (D, C) with limit not contained in that image, based directly on the sequence (f ν ) presented in Section 3.2.