Let $M$ be a lineally convex hypersurface of $\mathbb{C}^n$ of finite type, $0\in M$. Then there exist non-trivial smooth CR functions on $M$ that are flat at $0$, i.e. whose Taylor expansion about $0$ vanishes identically. Our aim is to characterize the rate at which flat CR functions can decrease without vanishing identically. As it turns out, non-trivial CR functions cannot decay arbitrarily fast, and a possible way of expressing the critical rate is by comparison with a suitable exponential of the modulus of a local peak function.
1. Introduction and statement of results
Let $M\subset \mathbb{C}^N$ be a smooth hypersurface containing $0$. We recall that the space of germs of CR functions at $0$, which we denote by $C^\infty _{CR} (M,0)$, is the space of germs at $0$ of smooth funtions on $M$ which are annihilated by the CR vector fields. In a recent paper Reference 1 (for the general case of integrable structures see Reference 4) we showed that if a peak function at $0$ exists, then the “Borel map”
is onto (and possesses a continuous inverse). It is a natural question to determine the kernel of $T_0$, i.e. describe (germs of) flat CR functions. In this paper, we shall find a critical rate of decay for such flat functions for the case of a lineally convex hypersurface.
In order to introduce our main result, we first discuss a particular example. Let $M$ denote the Lewy hypersurface, given as
It is well known that every CR function $\alpha$ defined near $0$ on $M$ extends holomorphically to a one-sided neighbourhood of $0\in M$; i.e. $\alpha (z,w)$ is a holomorphic function for $\operatorname {Im}w > \| z\|^2$. If $|\alpha (z,w)| < Ae^{- \frac{\lambda }{|w|}}$ for some constants $A>0$,$\lambda >0$ and $(z,w)$ belonging to a neighborhood of $0$ in $M$, then we say that $\alpha$ decreases exponentially of order $1$. The maximum principle implies that the extension of $\alpha$ satisfies the same kind of estimate. A classical theorem in ($1$-dimensional) complex analysis known as Watson’s Lemma (see Lemma 1) tells us that a function decreasing in that way in a half-plane is necessarily $0$. Hence $\alpha (0,w) = 0$; by induction one can show that $D_z^\beta \alpha (0,w)=0$ for every derivative in the $z$-directions, and thus we conclude that $\alpha =0$. On the other hand, the functions
give examples of functions decaying “of order $\gamma$”, i.e. like $e^{-1/|w|^\gamma }$ for $\gamma < 1$. Our goal in this paper is to generalize this observation to lineally convex hypersurfaces.
Let $M\subset \mathbb{C}^{n+1}$ be a smooth hypersurface, given in coordinates $(z_1,\ldots ,z_n,w) = (z,w)$ near $0$ by a defining function
where $f,g$ are smooth functions such that $f(0)=0$,$g(0,0)=0$. Recall that $M$ is lineally convex, of finite order $k\geq 2$ if $f(z)\geq |z|^k$ for $z$ in a neighborhood of $0$. Note that a lineally convex hypersurface is of finite commutator type.
We will need a bit more general notion: If $\mathcal{C} \subset T_0^c(M) = \{ w = 0\} \cong \mathbb{C}^n_z$ is an open cone, we say that $M$ is lineally convex along $\mathcal{C}$ of finite order $k$ if $f(cv)\geq |c|^k$ for all $v\in \mathcal{C}$ with $|v|=1$ and $c$ in a neighborhood of $0$ in $\mathbb{C}$.
Equivalently, for any $v\in \mathcal{C}$ and $\mathbb{C}^2_v = \operatorname {span}\langle v, \partial /\partial w\rangle$, the manifold $M_v=\mathbb{C}^2_v\cap M$ is a lineally convex hypersurface of $\mathbb{C}^2$ of order at most $k$.
Let $\psi =- i w_{|_M}$. If $M$ is lineally convex of finite order $k$, then $\psi$ is a smooth, CR peaking function of finite type at $0$ for $M$, in the sense specified in Reference 1. If $M$ is lineally convex along the cone $\mathcal{C}$, the restriction of $\psi$ to $M_v$ for any $v\in \mathcal{C}$ is a CR peaking function for $M_v$. Our aim in this paper is to understand the conditions on the order to which a smooth CR function defined on $M$ can vanish at $0$ without vanishing identically. These conditions will be expressed in terms of a comparison with the behavior of $\psi$: we will show that (in a sense to be made precise below) the function $e^{-1/|\psi |}$ represents the critical rate of decrease for CR functions; that is, we will prove that any CR function that decreases at that speed must vanish, while there exist many non-trivial ones which decrease to a rate “closely” approaching $e^{-1/|\psi |}$.
Let us start by giving a precise meaning to “decreasing like $e^{-1/|\psi |}$”:
Next, we introduce a class of functions $\beta :\mathbb{R}^+\to \mathbb{R}^+$ for which we are able to show the existence of non-trivial CR functions that decrease at least as fast as $e^{-\beta (|\psi |)}$. Essentially, what we ask is that $\beta$ be integrable in a neighborhood of $0$; for technical reasons, we also need some condition controlling the behavior of the first derivative.
For the proof of Theorem 1, we follow the line of argument already used in the introductory example. In order to overcome the additional difficulties from the more general geometry considered here, we will adapt Watson’s Lemma to suitable domains (see Corollary 1). In order to prove the second part of the theorem, we will introduce a certain form of enveloping product domain of sufficient smoothness in Lemma 6. Before we start with the preparations, let us give some additional remarks.
2. Preparations
We will show that the statement of Theorem 1 is a consequence of certain results in one complex variable, which we review here.
2.1. Watson’s Lemma
A sector $S\subset \mathbb{C}$ is a set of the form
$$\begin{equation*} S = \{ z\in \mathbb{C}\colon \alpha < \arg z < \beta \}. \end{equation*}$$
We say that $f$ is a germ of a holomorphic function on $S$ if there exists a neighborhood $U$ of $0$ such that $f$ is holomorphic on $S\cap U$. A germ $f$ of a holomorphic function on $S$ decreases exponentially of order $k$ if there exist $C,\lambda >0$,
$$\begin{equation*} |f (z) | < C e^{-\frac{\lambda }{|z|^k}}, \end{equation*}$$
in a neighborhood of $0$. In comparing growth rates on sectors, one has a choice of fixing the growth rate and comparing on closed subsectors or fixing the sector and comparing with a strictly greater rate. We shall choose to follow the second path here. Watson’s Lemma gives an exact bound of the maximum order of decrease of a non-zero germ on a sector $S$. This bound is in terms of an exponential rate of decrease whose order $k$ depends on the opening angle of the sector. For simplicity, though, we state the lemma for a half-plane.
For the proof of this version of Watson’s Lemma we refer to Reference 5.
2.2. Smooth extension of Riemann maps
We will also need some results about the behavior of holomorphic maps at the boundary, in particular, regarding the extension of the first derivative.
The Riemann map of a simply connected domain whose boundary is Dini-smooth is $C^1$ up to the boundary. This can be derived from the following result (see Reference 2, Theorem 3.5):
3. Proof of the main theorem
3.1. Vanishing of CR functions of exponential decay
We will need the following lemma:
We turn now to the proof of the first claim of Theorem 1. We first observe that we can reduce to the case of $n=1$. Indeed, let $v\in \mathcal{C}\subset T_0^c(M)$, and let $\mathbb{C}^2_v= \operatorname {span}\left\{\partial /\partial w, v\right\}$. Then $M_v= M\cap \mathbb{C}^2_v$ is a lineally convex hypersurface of finite type of $\mathbb{C}^2_v$, and $\varphi _v=\varphi _{|_{M_v}}$ is an exponentially decreasing CR function. It is enough to prove that $\varphi _v\equiv 0$ for all $v\in \mathcal{C}$, i.e. that $\varphi$ vanishes on the set $\bigcup _{v\in \mathcal{C}} M_v$, which contains an open subset of $M$. Since $M$ is minimal, this implies that $\varphi$ vanishes identically.
Then let $U$ be the open subset of $\mathbb{C}^2$ defined by $\{\operatorname {Im}w > h(z,\operatorname {Re}w)\}$. By well-known results, $\varphi$ extends to a holomorphic function $\widetilde{\varphi }$ defined on $U$ and smooth up to the boundary. For any $a\in \mathbb{C}$ lying in a small neighborhood of $0$ we let $\mathcal{U}_a = U\cap \{z=aw\}$; then $\mathcal{U}_a$ can be identified with the domain of $\mathbb{C}_w$ with smooth boundary which is defined by $\{\operatorname {Im}w > h(aw,\operatorname {Re}w)\}$.
We are going to show that the restriction of $\widetilde{\varphi }$ to each $\mathcal{U}_a$ vanishes identically. This is sufficient to conclude that $\widetilde{\varphi }\equiv 0$ (hence $\varphi \equiv 0$) by analytic continuation, because the union of the $\mathcal{U}_a$ has non-empty interior in $\mathbb{C}^2$. So let $a$ be fixed; for any $c\in \mathcal{U}_a\subset \mathbb{C}_w$, we define $\gamma _c = M\cap \{w=c\}$. By Lemma 2, $\gamma _c$ is a compact, non-empty set if $c$ is small enough. By the maximum principle, for any $w_0\in \mathcal{U}$ we have
as $w_0\to 0$, where we have used the fact that $\varphi$ is exponentially decreasing of order $1$. By Corollary 1, then, it follows that $\widetilde{\varphi }(aw_0,w_0)\equiv 0$.
3.2. Existence of non-trivial CR functions with admissible decay
Now, we focus on the second claim of Theorem 1. As before, we are going to derive it from a result in one complex variable, but first we need to establish some properties of $\beta$:
Now we are in a position to prove the second claim of Theorem 1. Let $\Omega _2\subset \Omega _1\subset \mathbb{C}_w$ be the domains defined as $\Omega _1=\{\operatorname {Im}w>2r(\operatorname {Re}w)\}$,$\Omega _2=\{\operatorname {Im}w>r(\operatorname {Re}w)\}$, where $r$ is given by Lemma 6 and by the subsequent Remark 5.
Since $b\Omega _1$ is of class $C^{1,\frac{1}{k-1}}$, it is in particular Dini-smooth; let $\mathcal{Q}:\Omega _1\to H$,$\mathcal{Q}(0)=0$ be the inverse of the Riemann mapping $\mathcal{R}:H\to \Omega _1$. By Lemma 2 we deduce that $\mathcal{Q}$ is also of class $C^1$ up to the boundary and $\mathcal{Q}'(0)\neq 0$, so that for some constant $C>1$ we can write
for $w_1,w_2$ close enough to $0$. We apply Lemma 5 with $\beta$ replaced by $\beta _1=C\beta$. Defining $\eta _1(w)=\alpha (\mathcal{Q}(w))$, it follows that $\eta _1$ is of class $C^1$ up to the boundary and that
Now, let $\Omega = \mathbb{C}^n\times \Omega _1 = \{(z,w)\in \mathbb{C}^{n+1}: \operatorname {Im}w>2r(\operatorname {Re}w) \}$; we define a function $\eta \in \mathcal{O}(\Omega )\cap C^1(\overline{\Omega })$ by $\eta (z,w) = \eta _1(w)$. By Lemma 6, $\eta$ yields by restriction a non-trivial CR function of class $C^1$, defined over a neighborhood of $0$ in $M$ and clearly satisfying the estimate required by Theorem 1.
We claim that $\eta _{|M}$ is in fact of class $C^\infty$. Fix $j\in \mathbb{N}$. Using the Faa di Bruno formula, we can compute the $j$-th derivative of $\eta$ (note that of course only the ($\partial /\partial w$)-derivatives are relevant) as
for suitable polynomials $P_i\in \mathbb{Z}[x_1,\ldots ,x_j]$. Now, if $p\in M$,$p=(z_p,w_p)$, by Lemma 6 we have that $w_p\in \Omega _2$, and thus by Lemma 7 it follows that $w_p\in \Omega _\kappa := \{w\in \Omega _1: \delta _1(w)\geq |w|^\kappa \}$ for some $\kappa >0$. Moreover,
Following, now, the same lines as in the proof of Corollary 2, by the Cauchy estimates it follows that $|\mathcal{Q}^{(i)}(w)|\leq F_i / \delta _1(w)^i$ for $w\in \Omega _1$; thus $|\mathcal{Q}^{(i)}(w_p)|\leq F_i / |w_p|^{i\kappa }$ because $w_p\in \Omega _\kappa$. Hence each term $|P_i \left(\mathcal{Q}'(w_p), \ldots , \mathcal{Q}^{(j)}(w_p)\right)|$ blows up at most polynomially in $1/|w_p|$ as $p\to 0$,$p\in M$. On the other hand, by Corollary 2 we have (since $\mathcal{Q}(w_p)\in H_{\kappa '}$) that $|\alpha ^{(i)}(\mathcal{Q}(w_p))| \leq C_i e^{-\frac{1}{\sqrt {|\mathcal{Q}(w_p)|}}}\leq C_i e^{-\frac{1}{C^{1/2}\sqrt {|w_p|}}}$.
for all $1\leq i\leq j$ as $p\to 0$,$p\in M$. In conclusion $\eta ^{(j)}(p)\to 0$ as $p\to 0$,$p\in M$, and since this holds for any $j\in \mathbb{N}$ we get that $\eta _{|M}$ is of class $C^\infty$.
Acknowledgment
The authors would like to thank an anonymous referee who pointed us to the source Reference 5. The Watson Lemma contained in that paper allowed us to strengthen the conclusion of the sufficiency part of Theorem 1.
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