A Counterexample to $C^{k}$ -regularity for the Newlander-Nirenberg Theorem

Yao, Liding
University of Wisconsin-Madison

Abstract

We give an example of $C^{k}$ -integrable almost complex structure that does not admit a corresponding $C^{k+1}$ -complex coordinate system.

The celebrated Newlander-Nirenberg theorem [5] states that given an integrable almost complex structure, it is locally induced by some complex coordinate system.

Malgrange [4] proved the existence of such complex coordinate chart when the almost complex structure is not smooth, and he obtained the following sharp Hölder regularity for this chart:

Theorem 1 (Sharp Newlander-Nirenberg).

Let $k\in\mathbb{Z}_{+}$ and $0<\alpha<1$ , let $M^{2n}$ be a $C^{k+1,\alpha}$ -manifold endowed with a $C^{k,\alpha}$ -almost complex structure $J:TM\to TM$ . If $J$ is integrable, then for any $p\in M$ there is a $C^{k+1,\alpha}$ -complex coordinate chart $(w^{1},\dots,w^{n}):U\to\mathbb{C}^{n}$ near $p$ such that $J\frac{\partial}{\partial w^{j}}=i\frac{\partial}{\partial w^{j}}$ for $j=1,\dots,n$ .

There are several equivalent characterizations for integrability. One of which is the vanishing of the Nijenhuis tensor $N_{J}(X,Y):=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]$ .

Our main theorem is to show that this is not true when $\alpha\in\{0,1\}$ :

Theorem 2.

Let $k,n\in\mathbb{Z}_{+}$ . There is a $C^{k}$ -integrable almost complex structure $J$ on $\mathbb{R}^{2n}$ , such that there is no $C^{k,1}$ -complex coordinate chart $w:U\subset\mathbb{R}^{2n}\to\mathbb{C}^{n}$ near $0$ satisfying $J\frac{\partial}{\partial w^{j}}=i\frac{\partial}{\partial w^{j}}$ for $j=1,\dots,n$ .

For convenience we use the viewpoint of eigenbundle of $J$ : Set $\mathcal{V}_{n}=\coprod_{p}\{v\in\mathbb{C}T_{p}\mathbb{R}^{2n}:J_{p}v=iv\}$ . So $J\frac{\partial}{\partial w^{j}}=i\frac{\partial}{\partial w^{j}}$ for all $j$ iff $d\bar{w}^{1},\dots,d\bar{w}^{n}$ spans $\mathcal{V}_{n}^{\bot}|_{U}\leq\mathbb{C}T^{*}\mathbb{R}^{2n}|_{U}$ . And $J$ is integrable if and only if $X,Y\in\Gamma(\mathcal{V}_{n})\Rightarrow[X,Y]\in\Gamma(\mathcal{V}_{n})$ for all complex vector fields $X,Y$ . See [1] Chapter 1 for details.

First we can restrict our focus to the 1-dimensional case:

Proof of 1-dim $\Rightarrow$ n-dim.

Suppose $J_{1}$ is a $C^{k}$ -almost complex structure on $\mathbb{C}^{1}_{z^{1}}$ (not compatible with the standard complex structure), such that near $0$ , there is no $C^{k,1}$ -complex coordinate $\varphi$ satisfying $\mathcal{V}_{1}^{\bot}=\operatorname{Span}d\bar{\varphi}$ in the domain. Here $\mathcal{V}_{1}^{\bot}$ is the dual eigenbundle of $J_{1}$ .

Denote $\theta=\theta(z^{1})$ as a $C^{k}$ 1-form on $\mathbb{C}^{1}_{z^{1}}$ that spans $\mathcal{V}_{1}^{\bot}$ .

Consider $\mathbb{R}^{2n}\simeq\mathbb{C}^{n}_{(z^{1},\dots,z^{n})}$ . We identify $\theta$ as the $C^{k}$ 1-form on $\mathbb{R}^{2n}$ . Take an $n$ -dim almost complex structure on $\mathbb{R}^{2n}$ such that the dual of eigenbundle $\mathcal{V}_{n}^{\bot}$ is spanned by $\theta,d\bar{z}^{2},\dots,d\bar{z}^{n}$ .

In other words, $\mathcal{V}_{n}$ is the “tensor” of $\mathcal{V}_{1}$ with the standard complex structure of $\mathbb{C}^{n-1}_{(z^{2},\dots,z^{n})}$ .

If $w=(w^{1},\dots,w^{n})$ is a corresponding $C^{k,1}$ -complex chart for $\mathcal{V}_{n}$ near $0$ , then there is a $1\leq j_{0}\leq n$ such that $d\bar{w}^{j_{0}}\not\equiv 0\pmod{d\bar{z}^{2},\dots,d\bar{z}^{n}}$ near $0$ . In other words, we have linear combinations $d\bar{w}^{j_{0}}=\lambda_{1}\theta+\lambda_{2}d\bar{z}^{2}+\dots+\lambda_{n}d\bar{z}^{n}$ for some non-vanishing function $\lambda_{1}(z^{1},\dots,z^{n})$ near $z=0$ .

Therefore $w^{j_{0}}(\cdot,0^{n-1})$ is a complex coordinate chart defined near $z^{1}=0\in\mathbb{R}^{2}$ whose differential spans $\mathcal{V}_{n}^{\bot}|_{\mathbb{R}^{2}_{z^{1}}}\cong\mathcal{V}_{1}^{\bot}$ near $0$ . By our assumption on $\mathcal{V}_{1}$ , we have $w^{j_{0}}(\cdot,0)\notin C^{k,1}$ . So $w^{j_{0}}\notin C^{k,1}$ , which means $w\notin C^{k,1}$ . ∎

Now we focus on the one-dimensional case. Note that a 1-dim structure is automatically integrable.

Fix $k\geq 1$ . Define an almost complex structure by setting its eigenbundle $\mathcal{V}_{1}\leq\mathbb{C}T\mathbb{R}^{2}$ equals to the span of $\frac{\partial}{\partial z}+a(z)\frac{\partial}{\partial\bar{z}}$ , where $a\in C^{k}(\mathbb{R}^{2};\mathbb{C})$ has compact support that satisfies the following:

(i)

$a\in C^{\infty}_{\mathrm{loc}}(\mathbb{R}^{2}\backslash\{0\};\mathbb{C})$ ;
(ii)

$\partial_{z}^{-1}\partial_{\bar{z}}a\notin C^{k-1,1}$ near 0;
(iii)

$za\in C^{k+1}(\mathbb{R}^{2};\mathbb{C})$ and $z^{-1}a\in C^{k-1}(\mathbb{R}^{2};\mathbb{C})$ (which implies $a(z)=o(|z|)$ as $z\to 0$ );
(iv)

$\operatorname{supp}a\subset\mathbb{B}^{2}$ ;
(v)

$\|a\|_{C^{0}}<\delta_{0}$ for some small enough $\delta_{0}>0$ (take $\delta_{0}=10^{-1}$ will be ok).

Here we take $\partial_{z}^{-1}$ to be the conjugated Cauchy-Green operator on the unit disk¹¹1Throughout the paper, $\mathbb{B}^{2}=B^{2}(0,1)$ refers to the unit disk in $\mathbb{C}^{1}$ .:

\displaystyle\partial_{z}^{-1}\phi(z)=\partial_{z,\mathbb{B}^{2}}^{-1}\phi(z):=\frac{1}{\pi}\int_{\mathbb{B}^{2}}\frac{\phi(\xi+i\eta)d\xi d\eta}{\bar{z}-\xi+i\eta}.

We use notation $\partial_{z}^{-1}$ because it is an right inverse of $\partial_{z}$ . And $\partial_{z}^{-1}:C^{m,\beta}(\mathbb{B}^{2};\mathbb{C})\to C^{m+1,\beta}(\mathbb{B}^{2};\mathbb{C})$ is bounded linear for all $m\in\mathbb{Z}_{\geq 0}$ , $0<\beta<1$ . See [8] theorem 1.32 in section 8.1 (page 56), or [2] lemma 2.3.4 for example.

We can take $\operatorname{supp}a\subset\mathbb{B}^{2}$ such that when $|z|<\frac{1}{2}$ ,

\textstyle a(z):=\frac{1}{100}\bar{z}^{k+1}\partial_{z}\big{(}(-\log|z|)^{\frac{1}{2}}\big{)}=\frac{1}{100}\partial_{z}\big{(}\bar{z}^{k+1}(-\log|z|)^{\frac{1}{2}}\big{)}.

(1)

Note that for this $a$ we have $a(z)=O\big{(}|z|^{k}(-\log|z|)^{-\frac{1}{2}}\big{)}=o(|z|^{k})$ .

Remark 3.

Roughly speaking, Property (i) $\operatorname{Singsupp}a=\{0\}$ says that the regularity of $a(z)$ corresponds to the vanishing order of $a$ at $0$ . To some degree, by multiplying with $a(z)$ , a function gains some regularity at the origin.

We check Property (ii) that $\partial_{z}^{-1}\partial_{\bar{z}}a\notin C^{k-1,1}$ here.

Lemma 4.

Let $a(z)$ be given by (1), and let $\chi\in C_{c}^{\infty}(\frac{1}{2}\mathbb{B}^{2})$ satisfies $\chi\equiv 1$ in a neighborhood of 0. Then $\partial_{z}^{-1}(\chi a_{\bar{z}})\notin C^{k-1,1}$ near $z=0$ .

Proof.

Denote $b(z):=\bar{z}^{k+1}(-\log|z|)^{\frac{1}{2}}$ , so $b\in C^{\infty}(\frac{1}{2}\mathbb{B}^{2}\backslash\{0\};\mathbb{C})$ , $\chi a=\frac{1}{100}\chi b_{z}$ and $\chi a_{\bar{z}}=\frac{1}{100}\chi b_{z\bar{z}}$ .

First we show that $\partial_{z}^{-1}(\chi b_{z\bar{z}})-b_{\bar{z}}\in C^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})$ . We write

\partial_{z}^{-1}(\chi b_{z\bar{z}})-b_{\bar{z}}=\partial_{z}^{-1}\partial_{z}(\chi b_{\bar{z}})-\chi b_{\bar{z}}-(1-\chi)b_{\bar{z}}-\partial_{z}^{-1}(\chi_{z}b_{\bar{z}}).

Since $\partial_{z}\partial_{z}^{-1}\partial_{z}(\chi b_{\bar{z}})=\partial_{z}(\chi b_{\bar{z}})$ , we know $\partial_{z}^{-1}\partial_{z}(\chi b_{\bar{z}})-\chi b_{\bar{z}}$ is anti-holomorphic. By Cauchy integral formula we get $\partial_{z}^{-1}\partial_{z}(\chi b_{\bar{z}})-\chi b_{\bar{z}}\in C^{\infty}$ .

By assumption $0\notin\operatorname{supp}\chi_{z}$ , $0\notin\operatorname{supp}(1-\chi)$ and $b\in C^{\infty}(\frac{1}{2}\mathbb{B}^{2}\backslash\{0\};\mathbb{C})$ , we know $\chi_{z}b_{\bar{z}}\in C_{c}^{\infty}$ and $(1-\chi)b_{\bar{z}}\in C^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})$ .

Note that when acting on functions supported in the unit disk, $\partial_{z}^{-1}=\frac{1}{\pi\bar{z}}\ast(\cdot)$ is a convolution operator with kernel $\frac{1}{\pi\bar{z}}\in L^{1}$ , so $\partial_{z}^{-1}(\chi_{z}b_{\bar{z}})=\frac{1}{\pi\bar{z}}\ast(\chi_{z}b_{\bar{z}})\in C^{\infty}$ .

It remains to show $b_{\bar{z}}\notin C^{k-1,1}$ near $0$ . Indeed one has

\partial_{\bar{z}}^{k+1}(\bar{z}^{k+1}(-\log|z|)^{\frac{1}{2}})=(k+1)!(-\log|z|)^{\frac{1}{2}}+O(1)\quad,\text{ as }z\to 0,

because by Leibniz rule

		$\displaystyle\textstyle\partial_{\bar{z}}^{k}\partial_{\bar{z}}\big{(}\bar{z}^{k+1}(-\log\|z\|)^{\frac{1}{2}}\big{)}=\sum_{j=0}^{k+1}{k+1\choose j}\partial_{\bar{z}}^{k+1-j}(z^{k+1})\cdot\partial_{\bar{z}}^{j}(-\log\|z\|)^{\frac{1}{2}}$
	$\displaystyle=$	$\displaystyle\textstyle(k+1)!(-\log\|z\|)^{\frac{1}{2}}+\sum_{j=1}^{k+1}O(z^{j})O\big{(}z^{-j}(-\log\|z\|)^{-\frac{1}{2}}\big{)}=(k+1)!(-\log\|z\|)^{\frac{1}{2}}+O\big{(}(-\log\|z\|)^{-\frac{1}{2}}\big{)}.$

∎

Now assume $w:\tilde{U}\subset\mathbb{R}^{2}\to\mathbb{C}$ is a 1-dim $C^{1}$ -complex coordinate chart defined near $0$ that represents $\mathcal{V}_{1}$ , then $\operatorname{Span}d\bar{w}=\operatorname{Span}(d\bar{z}-adz)|_{\tilde{U}}=\mathcal{V}_{1}^{\bot}|_{\tilde{U}}$ . So $d\bar{w}=\bar{w}_{\bar{z}}d\bar{z}+\bar{w}_{z}dz=\bar{w}_{\bar{z}}(d\bar{z}-adz)$ , that is,

\displaystyle\frac{\partial w}{\partial\bar{z}}(z)+\bar{a}(z)\frac{\partial w}{\partial z}(z)=0,\qquad z\in{\tilde{U}}.

Remark 5.

It is worth noticing that $\partial_{w}\neq\partial_{z}+a\partial_{\bar{z}}$ . Indeed $\partial_{w}$ is only a scalar multiple of $\partial_{z}+a\partial_{\bar{z}}$ .

Note that $w_{z}(0)\neq 0$ because $(d\bar{z}-adz)|_{0}=d\bar{z}|_{0}\in\operatorname{Span}d\bar{w}|_{0}$ . So by multiplying $w_{z}(0)^{-1}$ , we can assume $w_{z}(0)=1$ without loss of generality. Then $f:=\log\partial_{z}w$ is a well-defined function in a smaller neighborhood $U\subset\tilde{U}$ of $0$ , which solves

\frac{\partial f}{\partial\bar{z}}(z)+\overline{a(z)}\frac{\partial f}{\partial z}(z)=-\frac{\partial\bar{a}}{\partial z}(z)\quad\Big{(}=-\overline{\frac{\partial a}{\partial\bar{z}}(z)}\Big{)},\qquad z\in U.

(2)

Property (v) indicates that the operator $\partial_{\bar{z}}+\bar{a}\partial_{z}$ is a first order elliptic operator. Therefore we can consider a second order divergence form elliptic operator

L:=\partial_{z}(\partial_{\bar{z}}+\bar{a}\partial_{z})

whose coefficients are $C^{k}$ globally and are $C^{\infty}$ outside the origin.

By the classical Schauder’s estimate (see [7] Theorem 4.2, or [3] Chapter 6 & 8), we have the following:

Lemma 6 (Schauder’s interior estimate).

Assume $u,\psi\in C^{0,1}(\mathbb{B}^{2};\mathbb{C})$ satisfy $Lu=\psi_{z}$ . Let $U\subset\mathbb{R}^{2}$ be a neighborhood of $0$ . The following hold:

(a)

If $\psi\in C^{\infty}_{\mathrm{loc}}(U\backslash\{0\};\mathbb{C})$ , then $u\in C^{\infty}_{\mathrm{loc}}(U\backslash\{0\};\mathbb{C})$ .
(b)

If $\psi\in C^{k-1,1}(\mathbb{R}^{2};\mathbb{C})$ , then $u\in C^{k,1-\varepsilon}_{\mathrm{loc}}(\mathbb{R}^{2};\mathbb{C})$ for all $0<\varepsilon<1$ .

Our Theorem 2 for 1-dim case is done by the following proposition:

Proposition 7.

For any neighborhood $U\subset\mathbb{R}^{2}_{z}$ of $0$ , there is no $f\in C^{k-1,1}(U;\mathbb{C})$ solving (2).

Proof.

Suppose there is a neighborhood $U\subset\frac{1}{2}\mathbb{B}^{2}$ of the origin, and a solution $f\in C^{k-1,1}(U;\mathbb{C})$ to (2).

Applying Lemma 6 (a) on (2) with $u=f$ and $\psi=-\bar{a}_{z}$ , we know that $f\in C^{\infty}(U\backslash\{0\};\mathbb{C})$ .

Take $\chi\in C_{c}^{\infty}(U,[0,1])$ such that $\chi\equiv 1$ in a smaller neighborhood of $0$ . Denote

g(z):=\chi(z)f(z),\qquad h(z):=\bar{z}g(z).

So $g,h$ are $C^{k-1,1}$ -functions defined in $\mathbb{R}^{2}$ that are also smooth away from 0, and satisfy the following:

g_{\bar{z}}+\bar{a}g_{z}=\chi_{\bar{z}}f+\chi_{z}\bar{a}f-\chi\bar{a}_{z},

(3)

\qquad h_{\bar{z}}+\bar{a}h_{z}=g+\bar{z}(\chi_{\bar{z}}f+\chi_{z}\bar{a}f)-\chi\bar{z}\bar{a}_{z}.

(4)

By construction $\nabla\chi\equiv 0$ holds in a neighborhood of $0$ , so $\chi_{\bar{z}}f+\chi_{z}\bar{a}f\in C_{c}^{\infty}(U;\mathbb{C})$ .

Under our assumption that $f\in C^{k-1,1}(U;\mathbb{C})$ , then the key is to show that

(I)

$h\in C^{k,1-\varepsilon}_{c}(\mathbb{R}^{2};\mathbb{C})$ , $\forall\varepsilon\in(0,1)$ . This implies:
(II)

$\bar{a}g_{z}\in C^{k-1,1-\varepsilon}_{c}(\mathbb{R}^{2};\mathbb{C})$ , $\forall\varepsilon\in(0,1)$ .

(I) By assumption $\bar{z}(\chi_{\bar{z}}f+\chi_{z}\bar{a}f)\in C^{\infty}_{c}$ , $g\in C^{k-1,1}$ , and by Property (iii), $\chi\bar{z}\bar{a}_{z}\in C^{k}(\mathbb{R}^{2};\mathbb{C})$ .

Applying Lemma 6 (b) to (4), with $u=h$ and $\psi=g+\bar{z}(\chi_{\bar{z}}f+\chi_{z}\bar{a}f)-\chi\bar{z}\bar{a}_{z}\in C^{k}_{c}(\mathbb{R}^{2};\mathbb{C})$ , we get $h\in C^{k,1-\varepsilon}(\mathbb{B}^{2};\mathbb{C})$ , for all $0<\varepsilon<1$ .

(II) When $k\geq 2$ , we know $z^{-1}a\in C^{k-1}$ for Property (iii). So for any $\varepsilon\in(0,1)$ , one has $\bar{a}g_{z}\in C^{k-1,1-\varepsilon}$ because

\nabla_{z,\bar{z}}(\bar{a}g_{z})=\bar{a}\cdot(\partial_{z}g_{z},\partial_{\bar{z}}g_{z})+g_{z}\nabla_{z,\bar{z}}\bar{a}=\bar{z}^{-1}\bar{a}\cdot(h_{zz},h_{z\bar{z}}-g_{z})+O(C^{k-2,1})\in C^{k-2,1-\varepsilon}.

When $k=1$ , for any $z_{1},z_{2}\in\mathbb{R}^{2}\backslash\{0\}$ , note that $g_{\bar{z}}$ is smooth outside the origin, so $\bar{a}g_{z}\in C^{0,1-\varepsilon}$ :

		$\displaystyle\|\bar{a}g_{z}(z_{1})-\bar{a}g_{z}(z_{2})\|\leq\|\bar{a}(z_{1})\|\|g_{z}(z_{1})-g_{z}(z_{2})\|+\|\bar{a}(z_{1})-\bar{a}(z_{2})\|\|g_{z}(z_{2})\|$
	$\displaystyle\leq$	$\displaystyle\|\bar{z}_{1}^{-1}\bar{a}(z_{1})\|\|\bar{z}_{1}g_{z}(z_{1})-\bar{z}_{2}g_{z}(z_{2})\|+\|\bar{z}_{1}^{-1}\bar{a}(z_{1})\|\|\bar{z}_{1}-\bar{z}_{2}\|\|g_{z}(z_{2})\|+\|\bar{a}(z_{1})-\bar{a}(z_{2})\|\|g_{z}(z_{2})\|$
	$\displaystyle\leq$	$\displaystyle\\|z^{-1}a\\|_{C^{0}}\\|\nabla h\\|_{C^{1-\varepsilon}}\|z_{1}-z_{2}\|^{1-\varepsilon}+\\|z^{-1}a\\|_{C^{0}}\\|g\\|_{C^{0,1}}\|z_{1}-z_{2}\|+\\|a\\|_{C^{1}}\\|g\\|_{C^{0,1}}\|z_{1}-z_{2}\|.$

Here as a remark, $|\bar{a}g_{z}(z_{1})-\bar{a}g_{z}(z_{2})|\lesssim_{a,g,h}|z_{1}-z_{2}|^{1-\varepsilon}$ still makes sense when $z_{1}$ or $z_{2}=0$ , though $\bar{g}_{z}(0)$ may not be defined. Indeed $\lim\limits_{z\to 0}\bar{a}g_{z}(z)$ exists because $\bar{a}g_{z}$ itself has bounded $C^{0,1-\varepsilon}$ -oscillation on $\mathbb{B}^{2}\backslash\{0\}$ , and then the limit defines the value of $\bar{a}g_{z}$ at $z=0$ .

So for either case of $k$ , we have $\bar{a}g_{z}\in C^{k-1,1-\varepsilon}(\mathbb{R}^{2};\mathbb{C})$ for all $\varepsilon\in(0,1)$ .

Based on consequence (II), for (3) we have

g=(g-\partial_{\bar{z}}^{-1}g_{\bar{z}})+\partial_{\bar{z}}^{-1}(\chi_{\bar{z}}f+\chi_{z}\bar{a}f)-\partial_{\bar{z}}^{-1}(\bar{a}g_{z})-\partial_{\bar{z}}^{-1}(\chi\bar{a}_{z}),\qquad\text{in }\mathbb{B}^{2}.

(5)

The right hand side of (5) consists of four terms, the first to the third are all $C^{k}$ , while the last one is not $C^{k-1,1}$ . We explain these as follows:

•

Since $\partial_{z}(g-\partial_{\bar{z}}^{-1}g_{\bar{z}})=0$ in $\mathbb{B}^{2}$ , we know $g-\partial_{\bar{z}}^{-1}g_{\bar{z}}$ is anti-holomorphic, which is smooth in $\mathbb{B}^{2}$ .
•

By assumption $\chi_{\bar{z}}f+\chi_{z}\bar{a}f\in C^{\infty}_{c}(\mathbb{R}^{2};\mathbb{C})$ , so $\partial_{\bar{z}}^{-1}(\chi_{\bar{z}}f+\chi_{z}\bar{a}f)\in C^{\infty}(\mathbb{B}^{2};\mathbb{C})$ as well.
•

By consequence (II) $\bar{a}g_{z}\in C^{k-1,1-\varepsilon}(\mathbb{R}^{2};\mathbb{C})$ , for all $\varepsilon\in(0,1)$ , so $\partial_{\bar{z}}^{-1}(\bar{a}g_{z})\in C^{k,1-\varepsilon}\subset C^{k}(\mathbb{B}^{2};\mathbb{C})$ .
•

However by Lemma 4, $\partial_{\bar{z}}^{-1}(\chi\bar{a}_{z})\notin C^{k-1,1}$ near 0.

Combining each term to the right hand side of (5), we know $g\notin C^{k-1,1}$ near 0. Contradiction! ∎

Remark 8.

The key to the proof is the non-surjectivity of $\partial_{z}:C^{k}\to C^{k-1}$ , which we use to construct a function $a(z)$ such that $a(0)=0$ , $\operatorname{Singsupp}a=\{0\}$ , and $\partial_{z}^{-1}\partial_{\bar{z}}a\notin C^{k}$ .

Remark 9.

For positive integer $k$ , Malgrange’s sharp estimate of Theorem 1 still holds for Zygmund spaces $\mathscr{C}^{k}=B^{k}_{\infty\infty}$ , that is, given $J\in\mathscr{C}^{k}$ , there exists a $\mathscr{C}^{k+1}$ -coordinate chart $(w^{1},\dots,w^{n})$ , such that $J\frac{\partial}{\partial w^{j}}=i\frac{\partial}{\partial w^{j}}$ . One can also see [6] for details. A reason why our proof does not give a counterexample for Zygmund spaces, is that there does exist a $f\in\mathscr{C}^{k}$ defined in a neighborhood of 0 that solves (2).

Acknowledgement

The author would like to express his appreciation to his advisor Prof. Brian T. Street for his help.

References

[1] Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie. An introduction to involutive structures, volume 6 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.
[2] So-Chin Chen and Mei-Chi Shaw. Partial differential equations in several complex variables, volume 19 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001.
[3] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
[4] B. Malgrange. Sur l’intégrabilité des structures presque-complexes. In Symposia Mathematica, Vol. II (INDAM, Rome, 1968), pages 289–296. Academic Press, London, 1969.
[5] A. Newlander and L. Nirenberg. Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2), 65:391–404, 1957.
[6] Brian Street. Sharp regularity for the integrability of elliptic structures. J. Funct. Anal., 278(1):108290, 2020.
[7] Michael E. Taylor. Partial differential equations III. Nonlinear equations, volume 117 of Applied Mathematical Sciences. Springer, New York, second edition, 2011.
[8] I. N. Vekua. Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962.

University of Wisconsin-Madison, Department of Mathematics, 480 Lincoln Dr.,
Madison, WI, 53706
[email protected]

MSC 2010: 35J46 (Primary), 32Q60 and 35F05 (Secondary)

		$\displaystyle\|\bar{a}g_{z}(z_{1})-\bar{a}g_{z}(z_{2})\|\leq\|\bar{a}(z_{1})\|\|g_{z}(z_{1})-g_{z}(z_{2})\|+\|\bar{a}(z_{1})-\bar{a}(z_{2})\|\|g_{z}(z_{2})\|$
	$\displaystyle\leq$	$\displaystyle\|\bar{z}_{1}^{-1}\bar{a}(z_{1})\|\|\bar{z}_{1}g_{z}(z_{1})-\bar{z}_{2}g_{z}(z_{2})\|+\|\bar{z}_{1}^{-1}\bar{a}(z_{1})\|\|\bar{z}_{1}-\bar{z}_{2}\|\|g_{z}(z_{2})\|+\|\bar{a}(z_{1})-\bar{a}(z_{2})\|\|g_{z}(z_{2})\|$
	$\displaystyle\leq$	$\displaystyle\\|z^{-1}a\\|_{C^{0}}\\|\nabla h\\|_{C^{1-\varepsilon}}\|z_{1}-z_{2}\|^{1-\varepsilon}+\\|z^{-1}a\\|_{C^{0}}\\|g\\|_{C^{0,1}}\|z_{1}-z_{2}\|+\\|a\\|_{C^{1}}\\|g\\|_{C^{0,1}}\|z_{1}-z_{2}\|.$

A Counterexample to CkC^{k}-regularity for the Newlander-Nirenberg Theorem

Abstract

Theorem 1 (Sharp Newlander-Nirenberg).

Theorem 2.

Proof of 1-dim ⇒\Rightarrow n-dim.

Remark 3.

Lemma 4.

Proof.

Remark 5.

Lemma 6 (Schauder’s interior estimate).

Proposition 7.

Proof.

Remark 8.

Remark 9.

Acknowledgement

References

A Counterexample to $C^{k}$ -regularity for the Newlander-Nirenberg Theorem

Proof of 1-dim $\Rightarrow$ n-dim.