# On the vanishing rate of smooth CR functions

## 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 ( complex analysis known as Watson’s Lemma (see Lemma -dimensional)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 and thus we conclude that -directions, 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 for some positive -Hölder 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 ( curve in -dimensional) 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 . derivative of -th (note that of course only the ( are relevant) as )-derivatives

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.