# On the vanishing rate of smooth CR functions

By Giuseppe Della Sala and Bernhard Lamel

## Abstract

Let be a lineally convex hypersurface of of finite type, . Then there exist non-trivial smooth CR functions on that are flat at , i.e. whose Taylor expansion about 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 be a smooth hypersurface containing . We recall that the space of germs of CR functions at , which we denote by , is the space of germs at of smooth funtions on which are annihilated by the CR vector fields. In a recent paper Reference1 (for the general case of integrable structures see Reference4) we showed that if a peak function at exists, then the “Borel map”

is onto (and possesses a continuous inverse). It is a natural question to determine the kernel of , 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 denote the Lewy hypersurface, given as

It is well known that every CR function defined near on extends holomorphically to a one-sided neighbourhood of ; i.e. is a holomorphic function for . If for some constants , and belonging to a neighborhood of in , then we say that decreases exponentially of order . The maximum principle implies that the extension of satisfies the same kind of estimate. A classical theorem in (-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 . Hence ; by induction one can show that for every derivative in the -directions, and thus we conclude that . On the other hand, the functions

give examples of functions decaying “of order ”, i.e. like for . Our goal in this paper is to generalize this observation to lineally convex hypersurfaces.

Let be a smooth hypersurface, given in coordinates near by a defining function

where are smooth functions such that , . Recall that is lineally convex, of finite order if for in a neighborhood of . Note that a lineally convex hypersurface is of finite commutator type.

We will need a bit more general notion: If is an open cone, we say that is lineally convex along of finite order if for all with and in a neighborhood of in .

Equivalently, for any and , the manifold is a lineally convex hypersurface of of order at most .

Let . If is lineally convex of finite order , then is a smooth, CR peaking function of finite type at for , in the sense specified in Reference1. If is lineally convex along the cone , the restriction of to for any is a CR peaking function for . Our aim in this paper is to understand the conditions on the order to which a smooth CR function defined on can vanish at without vanishing identically. These conditions will be expressed in terms of a comparison with the behavior of : we will show that (in a sense to be made precise below) the function 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 .

Let us start by giving a precise meaning to “decreasing like ”:

### Definition 1

We say that a function , defined in a neighborhood of in , is exponentially decreasing of order at if there exist and a neighborhood of in such that

Equivalently, we can require that for a certain we have

More generally, given an open cone we say that is exponentially decreasing (of order ) along if the restriction of to is exponentially decreasing of order for any .

Next, we introduce a class of functions for which we are able to show the existence of non-trivial CR functions that decrease at least as fast as . Essentially, what we ask is that be integrable in a neighborhood of ; for technical reasons, we also need some condition controlling the behavior of the first derivative.

### Definition 2

Consider a concave function of class having the following properties:

(1)

and thus, in particular, for ;

(2)

is monotone increasing.

We call admissible if for such a . For example, (corresponding to ) is an admissible function for all .

### Remark 1

The set of admissible functions thus defined is a convex cone. In the following, with no loss of generality, we will assume (up to, for example, replacing with ) that .

If is admissible, then for it is diverging at a much slower rate than , i.e. . We note, however, that the admissible set contains functions that diverge faster than for all . Indeed, if we define

then has such a property.

### Theorem 1

Let be lineally convex of finite order along an open cone , and let be a CR function of class , defined on a neighborhood of in , which is exponentially decreasing of order along . Then .

On the other hand, let be lineally convex of finite order (i.e. ) and let be an admissible function. Then there exists a non-trivial CR function of class , defined on a neighborhood of in , such that

for all .

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.

### Remark 2

Note that in particular, Theorem 1 tells us that the critical rate of decay as measured by the peaking function is , as observed in the case of the Lewy hypersurface above.

### Remark 3

One might wonder whether the lineal convexity assumption is actually needed in Theorem 1; our techniques do require this assumption at the moment. It is also easy to see that some kind of convexity assumption (or peaking) is needed for the existence of flat CR functions. For the validity of the first part of the theorem, we conjecture that minimality of is sufficient for the validity of a generalized Watson Lemma.

## 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 is a set of the form

We say that is a germ of a holomorphic function on if there exists a neighborhood of such that is holomorphic on . A germ of a holomorphic function on decreases exponentially of order if there exist ,

in a neighborhood of . 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 . This bound is in terms of an exponential rate of decrease whose order depends on the opening angle of the sector. For simplicity, though, we state the lemma for a half-plane.

#### Lemma 1 (Watson’s Lemma).

Let be a half-plane of and let be a germ of a holomorphic function on which is exponentially decreasing of order . Then .

For the proof of this version of Watson’s Lemma we refer to Reference5.

#### Remark 4

i)

There exists a plethora of functions which decrease exponentially of order on any proper subsector of the half-plane , e.g. the exponentials . In this sense, Lemma 1 is sharp.

ii)

More generally, if is a sector of opening angle , then any germ of a function decreasing exponentially of order near vanishes identically.

### 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.

#### Definition 3

Let be a vector-valued, uniformly continuous function defined on some domain . Its modulus of continuity is defined as

for all . The function is called Dini-continuous if . In particular, if is -Hölder for some positive , then it is Dini-continuous.

If and is of class , we say that it is Dini-smooth if its derivative is Dini-continuous. Accordingly, we call a (-dimensional) curve in Dini-smooth if it admits a Dini-smooth parametrization.

The Riemann map of a simply connected domain whose boundary is Dini-smooth is up to the boundary. This can be derived from the following result (see Reference2, Theorem 3.5):

#### Theorem 2

Let map the unit disc conformally onto the inner domain of a Dini-smooth Jordan curve. Then extends continuously to . Moreover, the extension of is non-vanishing on .

#### Corollary 1

Let be a (simply connected) domain whose boundary is Dini-smooth, , and be exponentially decreasing of order greater than at . Then .

#### Proof.

Let be the half-plane and let be a Riemann mapping such that . By Theorem 2, is up to the boundary and . In particular, for some constant we have for all in a neighborhood of in . Since is exponentially decreasing of order in , for some ,

when is close enough to . This implies that is exponentially decreasing of order in . Applying Lemma 1 we conclude that , and hence .

## 3. Proof of the main theorem

### 3.1. Vanishing of CR functions of exponential decay

We will need the following lemma:

#### Lemma 2

Let be lineally convex, and let be coordinates for in which can be expressed as in (Equation1). For a small neighborhood of in let , . Then for close enough to we have that and is compact.

#### Proof.

Choose a small and let ; moreover, let . If we choose for all such that , we get by (Equation1)

for all , which implies the first claim. For the second one, assume that is chosen in such a way that , and suppose that for a sequence of complex numbers we have . Up to a subsequence, by continuity we have , a contradiction. Thus, for small enough the compact subset is contained in , from which follows the second claim.

We turn now to the proof of the first claim of Theorem 1. We first observe that we can reduce to the case of . Indeed, let , and let . Then is a lineally convex hypersurface of finite type of , and is an exponentially decreasing CR function. It is enough to prove that for all , i.e. that vanishes on the set , which contains an open subset of . Since is minimal, this implies that vanishes identically.

Then let be the open subset of defined by . By well-known results, extends to a holomorphic function defined on and smooth up to the boundary. For any lying in a small neighborhood of we let ; then can be identified with the domain of with smooth boundary which is defined by .

We are going to show that the restriction of to each vanishes identically. This is sufficient to conclude that (hence ) by analytic continuation, because the union of the has non-empty interior in . So let be fixed; for any , we define . By Lemma 2, is a compact, non-empty set if is small enough. By the maximum principle, for any we have

Since the maximum distance of the points of to the origin approaches as , for a suitable we have

as , where we have used the fact that is exponentially decreasing of order . By Corollary 1, then, it follows that .

### 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 :

#### Lemma 3

Let be an admissible function as defined in Definition 2. Then the following hold:

(i)

is of class and ;

(ii)

and for ;

(iii)

for all ;

(iv)

for , the following holds:

#### Proof.

Since we are only interested in the behavior of for close to , in the following arguments we implicitly restrict ourselves to a neighborhood of in . Note that the facts that is of class , as and the last inequality in (iv) (due to the concavity of ) all follow directly from Definition 2. Computing the derivative of , we have

The concavity of implies that ; hence

This immediately implies assertions (i) and (ii). Now let . Since is increasing, we get

which is the claim at point (iii). To prove (iv), in view of (Equation2) it suffices to estimate . Let be defined in such a way that

By the concavity of , then, follows. Let . Then , and we have

The last inequality is again due to the concavity of (since , the affine function is decreasing). Since by assumption is monotone increasing (cf. Definition 2), this is the same as writing .

#### Lemma 4

Let be the half-plane , and let be as in Theorem 1. Then there exists a (simply connected) domain with the following properties:

the boundary of is Dini-smooth and ;

for all in a neighborhood of in .

#### Proof.

We consider real coordinates such that , and we define with . The boundary is Dini-smooth if and only if the derivative

is Dini-continuous. By point (iv) in Lemma 3, the modulus of continuity of can be estimated by a multiple of . The Dini-smoothness of thus follows from Definition 2. Also notice that, by definition, ; the first claim is then verified. As for the second one, let ; we have

Assume, first, that . Since , we obtain

as . If, on the other hand, , we estimate as follows:

(where we used the fact that ), which concludes the proof.

#### Lemma 5

Let , be as above. There is a non-vanishing holomorphic function , defined in a neighborhood of in and of class up to the boundary, such that

for all .

#### Proof.

Let be the domain constructed Lemma 4, and let be the mapping given by Theorem 2. Since the differential of does not vanish, for some , and in a neighborhood of one has . By construction, we have

for (here we used claim (iii) of Lemma 3). Hence, letting , we get

for , as desired. As for the smoothness of , first of all we note that it is clearly of class outside since this is the case for . Computing the first derivative gives

hence

as (see Remark 1). This shows that is of class .

#### Corollary 2

Let , be as in the previous lemma, and for define . For any fixed , there exists such that

for all .

#### Proof.

We note, first, that since is bounded the Cauchy estimates give, for any fixed ,

for a suitable constant . It follows that for .

Now, we have , where is a polynomial whose coefficients are determined by the Faa di Bruno formula. Thus, the polynomial estimates for each over imply that for a suitable and all . The conclusion then follows from Remark 1.

#### Lemma 6

Let be a smooth lineally convex hypersurface of finite type , and suppose that coordinates for are chosen as above. Then there exists a real function of class such that and, for a neighborhood of in ,

#### Proof.

We write a local defining equation for as in (Equation1). By the finite type hypothesis, then, it follows that (for a certain neighborhood of in )

for a suitable constant . Consider, for a fixed , the function as defined over . By looking at the derivative, it is easy to see that assumes a minimum in ; hence

where is a (negative) constant. This choice of , then, satisfies the conditions required by the lemma.

#### Remark 5

From the proof of the previous lemma it follows that we can choose of the form for a large enough .

Now we are in a position to prove the second claim of Theorem 1. Let be the domains defined as , , where is given by Lemma 6 and by the subsequent Remark 5.

#### Lemma 7

For any , denote by the distance between and . Then there exists such that for all .

#### Proof.

Consider the following local diffeomorphism of , defined in a neighborhood of :

where as in Remark 5. We note that is of class (at least) , in particular bi-Lipschitz. Therefore there exists a constant such that for all close enough to . Furthermore, we have and . Defining , we get for all .

Now, for any close enough to we get

for a large enough . In particular, choosing we have for around .

Since and , for we can write , which leads to the conclusion up to choosing a slightly larger exponent .

Since is of class , it is in particular Dini-smooth; let , be the inverse of the Riemann mapping . By Lemma 2 we deduce that is also of class up to the boundary and , so that for some constant we can write

for close enough to . We apply Lemma 5 with replaced by . Defining , it follows that is of class up to the boundary and that

Now, let ; we define a function by . By Lemma 6, yields by restriction a non-trivial CR function of class , defined over a neighborhood of in and clearly satisfying the estimate required by Theorem 1.

We claim that is in fact of class . Fix . Using the Faa di Bruno formula, we can compute the -th derivative of (note that of course only the ()-derivatives are relevant) as

for suitable polynomials . Now, if , , by Lemma 6 we have that , and thus by Lemma 7 it follows that for some . Moreover,

#### Lemma 8

For some we have , where is defined in Corollary 2.

#### Proof.

If , we have by definition for all . Using (Equation5), we deduce . Since , this is the same as writing for all ; hence we get for all . Choosing any we obtain the claim of the lemma (for close enough to ).

Following, now, the same lines as in the proof of Corollary 2, by the Cauchy estimates it follows that for ; thus because . Hence each term blows up at most polynomially in as , . On the other hand, by Corollary 2 we have (since ) that .

From the estimates above it follows that

for all as , . In conclusion as , , and since this holds for any we get that is of class .

#### Remark 6

Since the function so constructed extends holomorphically to a neighborhood of any except , it is in fact real-analytic on if is of class . Moreover, is only vanishing at .

## Acknowledgment

The authors would like to thank an anonymous referee who pointed us to the source Reference5. The Watson Lemma contained in that paper allowed us to strengthen the conclusion of the sufficiency part of Theorem 1.

## Mathematical Fragments

Equation (1)
Definition 2

Consider a concave function of class having the following properties:

(1)

and thus, in particular, for ;

(2)

is monotone increasing.

We call admissible if for such a . For example, (corresponding to ) is an admissible function for all .

Remark 1

The set of admissible functions thus defined is a convex cone. In the following, with no loss of generality, we will assume (up to, for example, replacing with ) that .

If is admissible, then for it is diverging at a much slower rate than , i.e. . We note, however, that the admissible set contains functions that diverge faster than for all . Indeed, if we define

then has such a property.

Theorem 1

Let be lineally convex of finite order along an open cone , and let be a CR function of class , defined on a neighborhood of in , which is exponentially decreasing of order along . Then .

On the other hand, let be lineally convex of finite order (i.e. ) and let be an admissible function. Then there exists a non-trivial CR function of class , defined on a neighborhood of in , such that

for all .

Lemma 1 (Watson’s Lemma).

Let be a half-plane of and let be a germ of a holomorphic function on which is exponentially decreasing of order . Then .

Theorem 2

Let map the unit disc conformally onto the inner domain of a Dini-smooth Jordan curve. Then extends continuously to . Moreover, the extension of is non-vanishing on .

Corollary 1

Let be a (simply connected) domain whose boundary is Dini-smooth, , and be exponentially decreasing of order greater than at . Then .

Lemma 2

Let be lineally convex, and let be coordinates for in which can be expressed as in (Equation1). For a small neighborhood of in let , . Then for close enough to we have that and is compact.

Lemma 3

Let be an admissible function as defined in Definition 2. Then the following hold:

(i)

is of class and ;

(ii)

and for ;

(iii)

for all ;

(iv)

for , the following holds:

Equation (2)
Lemma 4

Let be the half-plane , and let be as in Theorem 1. Then there exists a (simply connected) domain with the following properties:

the boundary of is Dini-smooth and ;

for all in a neighborhood of in .

Lemma 5

Let , be as above. There is a non-vanishing holomorphic function , defined in a neighborhood of in and of class up to the boundary, such that

for all .

Corollary 2

Let , be as in the previous lemma, and for define . For any fixed , there exists such that

for all .

Lemma 6

Let be a smooth lineally convex hypersurface of finite type , and suppose that coordinates for are chosen as above. Then there exists a real function of class such that and, for a neighborhood of in ,

Remark 5

From the proof of the previous lemma it follows that we can choose of the form for a large enough .

Lemma 7

For any , denote by the distance between and . Then there exists such that for all .

Equation (5)

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author={Lamel, B.},
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## Article Information

MSC 2010
Primary: 32V10 (CR functions), 32V20 (Analysis on CR manifolds), 32T40 (Peak functions)
Keywords
• CR function
• exponential decay
• Watson Lemma
Author Information
Giuseppe Della Sala
Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria
giuseppe.dellasala@univie.ac.at
MathSciNet
Bernhard Lamel
Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria
bernhard.lamel@univie.ac.at
MathSciNet

Both authors were supported by the START Prize Y377 of the Austrian Federal Ministry of Science and Research bmwf. The second author was also supported by the Austrian Science Fund FWF, Project P24878.

Communicated by
Frank Forstneric
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 1, Issue 3, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .