On the vanishing rate of smooth CR functions

By Giuseppe Della Sala and Bernhard Lamel

Abstract

Let upper M be a lineally convex hypersurface of double-struck upper C Superscript n of finite type, 0 element-of upper M . Then there exist non-trivial smooth CR functions on upper 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 upper M subset-of double-struck upper C Superscript upper N be a smooth hypersurface containing 0 . We recall that the space of germs of CR functions at 0 , which we denote by upper C Subscript upper C upper R Superscript normal infinity Baseline left-parenthesis upper M comma 0 right-parenthesis , is the space of germs at 0 of smooth funtions on upper M 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 0 exists, then the “Borel map”

upper T 0 colon upper C Subscript upper C upper R Superscript normal infinity Baseline left-parenthesis upper M comma 0 right-parenthesis right-arrow double-struck upper C left-bracket left-bracket upper Z 1 comma ellipsis comma upper Z Subscript upper N Baseline right-bracket right-bracket

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

upper I m w equals double-vertical-bar z double-vertical-bar squared period

It is well known that every CR function alpha defined near 0 on upper M extends holomorphically to a one-sided neighbourhood of 0 element-of upper M ; i.e. alpha left-parenthesis z comma w right-parenthesis is a holomorphic function for upper I m w greater-than double-vertical-bar z double-vertical-bar squared . If StartAbsoluteValue alpha left-parenthesis z comma w right-parenthesis EndAbsoluteValue less-than upper A e Superscript minus StartFraction lamda Over StartAbsoluteValue w EndAbsoluteValue EndFraction for some constants upper A greater-than 0 , lamda greater-than 0 and left-parenthesis z comma w right-parenthesis belonging to a neighborhood of 0 in upper 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 left-parenthesis 0 comma w right-parenthesis equals 0 ; by induction one can show that upper D Subscript z Superscript beta Baseline alpha left-parenthesis 0 comma w right-parenthesis equals 0 for every derivative in the z -directions, and thus we conclude that alpha equals 0 . On the other hand, the functions

e Superscript minus left-parenthesis StartFraction i Over w EndFraction right-parenthesis Super Superscript gamma Superscript Baseline comma gamma less-than 1 comma

give examples of functions decaying “of order gamma ”, i.e. like e Superscript negative 1 slash StartAbsoluteValue w EndAbsoluteValue Super Superscript gamma for gamma less-than 1 . Our goal in this paper is to generalize this observation to lineally convex hypersurfaces.

Let upper M subset-of double-struck upper C Superscript n plus 1 be a smooth hypersurface, given in coordinates left-parenthesis z 1 comma period period period comma z Subscript n Baseline comma w right-parenthesis equals left-parenthesis z comma w right-parenthesis near 0 by a defining function

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel upper I m w equals h left-parenthesis z comma upper R e w right-parenthesis equals f left-parenthesis z right-parenthesis plus left-parenthesis upper R e w right-parenthesis g left-parenthesis z comma upper R e w right-parenthesis comma EndLayout

where f comma g are smooth functions such that f left-parenthesis 0 right-parenthesis equals 0 , g left-parenthesis 0 comma 0 right-parenthesis equals 0 . Recall that upper M is lineally convex, of finite order k greater-than-or-equal-to 2 if f left-parenthesis z right-parenthesis greater-than-or-equal-to StartAbsoluteValue z EndAbsoluteValue Superscript 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 script upper C subset-of upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis equals StartSet w equals 0 EndSet approximately-equals double-struck upper C Subscript z Superscript n is an open cone, we say that upper M is lineally convex along script upper C of finite order k if f left-parenthesis c v right-parenthesis greater-than-or-equal-to StartAbsoluteValue c EndAbsoluteValue Superscript k for all v element-of script upper C with StartAbsoluteValue v EndAbsoluteValue equals 1 and c in a neighborhood of 0 in double-struck upper C .

Equivalently, for any v element-of script upper C and double-struck upper C Subscript v Superscript 2 Baseline equals s p a n mathematical left-angle v comma partial-differential slash partial-differential w mathematical right-angle , the manifold upper M Subscript v Baseline equals double-struck upper C Subscript v Superscript 2 Baseline intersection upper M is a lineally convex hypersurface of double-struck upper C squared of order at most k .

Let psi equals minus i w Subscript vertical-bar Sub Subscript upper M Subscript . If upper M is lineally convex of finite order k , then psi is a smooth, CR peaking function of finite type at 0 for upper M , in the sense specified in Reference1. If upper M is lineally convex along the cone script upper C , the restriction of psi to upper M Subscript v for any v element-of script upper C is a CR peaking function for upper M Subscript v . Our aim in this paper is to understand the conditions on the order to which a smooth CR function defined on upper 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 Superscript negative 1 slash StartAbsoluteValue psi EndAbsoluteValue 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 Superscript negative 1 slash StartAbsoluteValue psi EndAbsoluteValue .

Let us start by giving a precise meaning to “decreasing like e Superscript negative 1 slash StartAbsoluteValue psi EndAbsoluteValue ”:

Definition 1

We say that a function phi , defined in a neighborhood of 0 in upper M , is exponentially decreasing of order 1 at 0 if there exist upper A comma lamda greater-than 0 and a neighborhood script upper V of 0 in upper M such that

StartAbsoluteValue phi left-parenthesis p right-parenthesis EndAbsoluteValue less-than-or-equal-to upper A e Superscript minus StartFraction lamda Over StartAbsoluteValue psi left-parenthesis p right-parenthesis EndAbsoluteValue EndFraction Baseline comma p element-of script upper V period

Equivalently, we can require that for a certain lamda greater-than 0 we have

StartAbsoluteValue phi left-parenthesis p right-parenthesis EndAbsoluteValue e Superscript StartFraction lamda Over StartAbsoluteValue psi left-parenthesis p right-parenthesis EndAbsoluteValue EndFraction Baseline right-arrow 0 normal a normal s p right-arrow 0 comma p element-of upper M period

More generally, given an open cone script upper C subset-of upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis we say that phi is exponentially decreasing (of order 1 ) along script upper C if the restriction of phi to upper M Subscript upper V is exponentially decreasing of order 1 for any v element-of script upper C .

Next, we introduce a class of functions beta colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus for which we are able to show the existence of non-trivial CR functions that decrease at least as fast as e Superscript minus beta left-parenthesis StartAbsoluteValue psi EndAbsoluteValue right-parenthesis . 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.

Definition 2

Consider a concave function gamma colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus of class upper C Superscript 1 having the following properties:

(1)

integral Subscript 0 Superscript 1 Baseline StartFraction gamma left-parenthesis t right-parenthesis Over t EndFraction d t less-than plus normal infinity comma and thus, in particular, gamma left-parenthesis t right-parenthesis right-arrow 0 for t right-arrow 0 ;

(2)

t gamma prime left-parenthesis t right-parenthesis is monotone increasing.

We call beta colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus admissible if beta left-parenthesis t right-parenthesis equals gamma left-parenthesis t right-parenthesis slash t for such a gamma . For example, beta left-parenthesis t right-parenthesis equals 1 slash t Superscript a (corresponding to gamma left-parenthesis t right-parenthesis equals t Superscript 1 minus a ) is an admissible function for all 0 less-than a less-than 1 .

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 beta left-parenthesis t right-parenthesis with beta left-parenthesis t right-parenthesis plus 1 slash StartRoot t EndRoot ) that beta left-parenthesis t right-parenthesis greater-than-or-equal-to 1 slash StartRoot t EndRoot .

If beta left-parenthesis t right-parenthesis is admissible, then for t right-arrow 0 it is diverging at a much slower rate than 1 slash t , i.e. beta left-parenthesis t right-parenthesis slash left-parenthesis 1 slash t right-parenthesis right-arrow 0 . We note, however, that the admissible set contains functions that diverge faster than 1 slash t Superscript a for all 0 less-than a less-than 1 . Indeed, if we define

gamma left-parenthesis t right-parenthesis equals sigma-summation Underscript script l equals 1 Overscript normal infinity Endscripts StartFraction 1 Over 2 Superscript 2 script l Baseline EndFraction t Superscript 1 slash 2 Super Superscript script l Superscript Baseline comma

then beta left-parenthesis t right-parenthesis equals gamma left-parenthesis t right-parenthesis slash t has such a property.

Theorem 1

Let upper M be lineally convex of finite order k along an open cone script upper C subset-of upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis , and let phi be a CR function of class upper C Superscript 1 , defined on a neighborhood of 0 in upper M , which is exponentially decreasing of order 1 along script upper C . Then phi identical-to 0 .

On the other hand, let upper M be lineally convex of finite order (i.e. script upper C equals upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis ) and let beta colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus be an admissible function. Then there exists a non-trivial CR function eta of class upper C Superscript normal infinity , defined on a neighborhood script upper V of 0 in upper M , such that

StartAbsoluteValue eta left-parenthesis p right-parenthesis EndAbsoluteValue less-than-or-equal-to e Superscript minus beta left-parenthesis StartAbsoluteValue psi left-parenthesis p right-parenthesis EndAbsoluteValue right-parenthesis

for all p element-of script upper V .

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 psi is 1 , 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 upper M 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 upper S subset-of double-struck upper C is a set of the form

upper S equals StartSet z element-of double-struck upper C colon alpha less-than arg z less-than beta EndSet period

We say that f is a germ of a holomorphic function on upper S if there exists a neighborhood upper U of 0 such that f is holomorphic on upper S intersection upper U . A germ f of a holomorphic function on upper S decreases exponentially of order k if there exist upper C comma lamda greater-than 0 ,

StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue less-than upper C e Superscript minus StartFraction lamda Over StartAbsoluteValue z EndAbsoluteValue Super Superscript k Superscript EndFraction Baseline comma

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 upper 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.

Lemma 1 (Watson’s Lemma).

Let upper S be a half-plane of double-struck upper C and let f be a germ of a holomorphic function on upper S which is exponentially decreasing of order 1 . Then f identical-to 0 .

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 1 on any proper subsector of the half-plane upper S , e.g. the exponentials e Superscript negative mu slash z . In this sense, Lemma 1 is sharp.

ii)

More generally, if upper S Subscript alpha comma beta Baseline equals StartSet z colon alpha less-than arg z less-than beta EndSet is a sector of opening angle beta minus alpha , then any germ of a function decreasing exponentially of order left-parenthesis beta minus alpha right-parenthesis slash pi near 0 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 phi be a vector-valued, uniformly continuous function defined on some domain upper A subset-of double-struck upper R Superscript n . Its modulus of continuity omega colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus is defined as

omega left-parenthesis delta right-parenthesis equals sup left-brace StartAbsoluteValue phi left-parenthesis x right-parenthesis minus phi left-parenthesis y right-parenthesis EndAbsoluteValue colon x comma y element-of upper A comma StartAbsoluteValue x minus y EndAbsoluteValue less-than-or-equal-to delta right-brace

for all delta element-of double-struck upper R Superscript plus . The function phi is called Dini-continuous if integral Subscript 0 Superscript 1 Baseline omega left-parenthesis t right-parenthesis slash t less-than plus normal infinity . In particular, if phi is alpha -Hölder for some positive alpha , then it is Dini-continuous.

If upper A subset-of double-struck upper R and phi is of class upper C Superscript 1 , we say that it is Dini-smooth if its derivative phi prime is Dini-continuous. Accordingly, we call a ( 1 -dimensional) curve in double-struck upper C Dini-smooth if it admits a Dini-smooth parametrization.

The Riemann map of a simply connected domain whose boundary is Dini-smooth is upper C Superscript 1 up to the boundary. This can be derived from the following result (see Reference2, Theorem 3.5):

Theorem 2

Let script upper R map the unit disc upper D conformally onto the inner domain normal upper Omega of a Dini-smooth Jordan curve. Then script upper R prime extends continuously to upper D overbar . Moreover, the extension of script upper R prime is non-vanishing on b upper D .

Corollary 1

Let normal upper Omega be a (simply connected) domain whose boundary is Dini-smooth, 0 element-of b normal upper Omega , and f element-of script upper O left-parenthesis normal upper Omega right-parenthesis be exponentially decreasing of order greater than 1 at 0 . Then f identical-to 0 .

Proof.

Let upper H be the half-plane and let script upper R be a Riemann mapping upper H right-arrow normal upper Omega such that script upper R left-parenthesis 0 right-parenthesis equals 0 . By Theorem 2, script upper R is upper C Superscript 1 up to the boundary and script upper R prime left-parenthesis 0 right-parenthesis not-equals 0 . In particular, for some constant upper C greater-than 0 we have StartAbsoluteValue script upper R left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C StartAbsoluteValue z EndAbsoluteValue for all z in a neighborhood of 0 in upper H . Since f is exponentially decreasing of order 1 in normal upper Omega , for some upper A comma lamda greater-than 0 ,

StartAbsoluteValue f left-parenthesis script upper R left-parenthesis z right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to upper A e Superscript minus StartFraction lamda Over StartAbsoluteValue script upper R left-parenthesis z right-parenthesis EndAbsoluteValue EndFraction Baseline less-than-or-equal-to e Superscript minus StartFraction lamda slash upper C Over StartAbsoluteValue z EndAbsoluteValue EndFraction

when z is close enough to 0 . This implies that f ring script upper R is exponentially decreasing of order 1 in upper H . Applying Lemma 1 we conclude that f ring script upper R identical-to 0 , and hence f identical-to 0 .

3. Proof of the main theorem

3.1. Vanishing of CR functions of exponential decay

We will need the following lemma:

Lemma 2

Let upper M be lineally convex, and let left-parenthesis z comma w right-parenthesis be coordinates for double-struck upper C Superscript n plus 1 in which upper M can be expressed as in (Equation1). For a small neighborhood upper U of 0 in double-struck upper C Superscript n plus 1 let upper U 1 equals upper U intersection StartSet upper I m w greater-than h left-parenthesis z comma upper R e w right-parenthesis EndSet , upper U 2 equals upper U minus upper U overbar Subscript 1 . Then for c element-of double-struck upper C close enough to 0 we have that StartSet w equals c EndSet intersection upper U 2 not-equals normal empty-set and upper U intersection StartSet w equals c EndSet intersection upper M is compact.

Proof.

Choose a small epsilon greater-than 0 and let upper S Subscript epsilon Baseline equals StartSet z element-of double-struck upper C Superscript n Baseline colon StartAbsoluteValue z EndAbsoluteValue equals epsilon EndSet ; moreover, let upper M Subscript epsilon Baseline equals max left-brace StartAbsoluteValue g left-parenthesis z comma upper R e w right-parenthesis EndAbsoluteValue colon z element-of upper S Subscript epsilon Baseline comma StartAbsoluteValue upper R e w EndAbsoluteValue less-than-or-equal-to epsilon right-brace . If we choose delta less-than epsilon Superscript k Baseline slash 2 upper M Subscript epsilon for all w such that StartAbsoluteValue upper R e w EndAbsoluteValue less-than delta , upper I m w less-than epsilon Superscript k Baseline slash 2 we get by (Equation1)

upper I m w minus h left-parenthesis z comma upper R e w right-parenthesis equals upper I m w minus f left-parenthesis z right-parenthesis minus upper R e w dot g left-parenthesis z comma upper R e w right-parenthesis less-than epsilon Superscript k Baseline slash 2 minus epsilon Superscript k Baseline plus delta upper M Subscript epsilon Baseline less-than 0

for all z element-of upper S Subscript epsilon , which implies the first claim. For the second one, assume that upper U is chosen in such a way that upper U overbar intersection upper M intersection StartSet w equals 0 EndSet equals StartSet 0 EndSet , and suppose that for a sequence c Subscript j Baseline right-arrow 0 of complex numbers we have p Subscript j Baseline element-of b upper U intersection upper M intersection StartSet w equals c Subscript j Baseline EndSet not-equals normal empty-set . Up to a subsequence, by continuity we have p Subscript j Baseline right-arrow p element-of b upper U intersection upper M intersection StartSet w equals 0 EndSet , a contradiction. Thus, for small enough c the compact subset upper U overbar intersection StartSet w equals c EndSet intersection upper M is contained in upper U , 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 n equals 1 . Indeed, let v element-of script upper C subset-of upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis , and let double-struck upper C Subscript v Superscript 2 Baseline equals s p a n left-brace partial-differential slash partial-differential w comma v right-brace . Then upper M Subscript v Baseline equals upper M intersection double-struck upper C Subscript v Superscript 2 is a lineally convex hypersurface of finite type of double-struck upper C Subscript v Superscript 2 , and phi Subscript v Baseline equals phi Subscript vertical-bar Sub Subscript upper M Sub Sub Subscript v Subscript is an exponentially decreasing CR function. It is enough to prove that phi Subscript v Baseline identical-to 0 for all v element-of script upper C , i.e. that phi vanishes on the set union Underscript v element-of script upper C Endscripts upper M Subscript v , which contains an open subset of upper M . Since upper M is minimal, this implies that phi vanishes identically.

Then let upper U be the open subset of double-struck upper C squared defined by StartSet upper I m w greater-than h left-parenthesis z comma upper R e w right-parenthesis EndSet . By well-known results, phi extends to a holomorphic function phi overTilde defined on upper U and smooth up to the boundary. For any a element-of double-struck upper C lying in a small neighborhood of 0 we let script upper U Subscript a Baseline equals upper U intersection StartSet z equals a w EndSet ; then script upper U Subscript a can be identified with the domain of double-struck upper C Subscript w with smooth boundary which is defined by StartSet upper I m w greater-than h left-parenthesis a w comma upper R e w right-parenthesis EndSet .

We are going to show that the restriction of phi overTilde to each script upper U Subscript a vanishes identically. This is sufficient to conclude that phi overTilde identical-to 0 (hence phi identical-to 0 ) by analytic continuation, because the union of the script upper U Subscript a has non-empty interior in double-struck upper C squared . So let a be fixed; for any c element-of script upper U Subscript a subset-of double-struck upper C Subscript w , we define gamma Subscript c Baseline equals upper M intersection StartSet w equals c EndSet . By Lemma 2, gamma Subscript c is a compact, non-empty set if c is small enough. By the maximum principle, for any w 0 element-of script upper U we have

StartAbsoluteValue ModifyingAbove phi With tilde left-parenthesis a w 0 comma w 0 right-parenthesis EndAbsoluteValue less-than-or-equal-to max Underscript z element-of gamma Subscript w 0 Baseline Endscripts StartAbsoluteValue phi left-parenthesis z comma w 0 right-parenthesis EndAbsoluteValue period

Since the maximum distance of the points of gamma Subscript w 0 to the origin approaches 0 as w 0 right-arrow 0 , for a suitable lamda greater-than 0 we have

StartAbsoluteValue ModifyingAbove phi With tilde left-parenthesis a w 0 comma w 0 right-parenthesis EndAbsoluteValue e Superscript StartFraction lamda Over StartAbsoluteValue w 0 EndAbsoluteValue EndFraction Baseline less-than-or-equal-to max Underscript z element-of gamma Subscript w 0 Baseline Endscripts StartAbsoluteValue phi left-parenthesis z comma w 0 right-parenthesis EndAbsoluteValue e Superscript StartFraction lamda Over StartAbsoluteValue w 0 EndAbsoluteValue EndFraction Baseline right-arrow 0

as w 0 right-arrow 0 , where we have used the fact that phi is exponentially decreasing of order 1 . By Corollary 1, then, it follows that ModifyingAbove phi With tilde left-parenthesis a w 0 comma w 0 right-parenthesis identical-to 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 :

Lemma 3

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

(i)

beta is of class upper C Superscript 1 and beta prime less-than-or-equal-to 0 ;

(ii)

t beta left-parenthesis t right-parenthesis right-arrow 0 and t squared beta prime left-parenthesis t right-parenthesis right-arrow 0 for t right-arrow 0 ;

(iii)

beta left-parenthesis t right-parenthesis less-than-or-equal-to upper C beta left-parenthesis upper C t right-parenthesis for all upper C greater-than 1 ;

(iv)

for 0 less-than t 1 less-than t 2 , the following holds: StartAbsoluteValue t 2 squared beta prime left-parenthesis t 2 right-parenthesis minus t 1 squared beta prime left-parenthesis t 1 right-parenthesis EndAbsoluteValue less-than-or-equal-to 2 left-parenthesis t 2 beta left-parenthesis t 2 right-parenthesis minus t 1 beta left-parenthesis t 1 right-parenthesis right-parenthesis less-than-or-equal-to 2 left-parenthesis t 2 minus t 1 right-parenthesis beta left-parenthesis t 2 minus t 1 right-parenthesis period

Proof.

Since we are only interested in the behavior of beta left-parenthesis t right-parenthesis for t close to 0 , in the following arguments we implicitly restrict ourselves to a neighborhood of 0 in double-struck upper R Superscript plus . Note that the facts that beta is of class upper C Superscript 1 , t beta left-parenthesis t right-parenthesis equals gamma left-parenthesis t right-parenthesis right-arrow 0 as t right-arrow 0 and the last inequality in (iv) (due to the concavity of gamma ) all follow directly from Definition 2. Computing the derivative of beta , we have

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel beta prime left-parenthesis t right-parenthesis equals StartFraction gamma prime left-parenthesis t right-parenthesis Over t EndFraction minus StartFraction gamma left-parenthesis t right-parenthesis Over t squared EndFraction period EndLayout

The concavity of gamma implies that 0 less-than gamma prime left-parenthesis t right-parenthesis less-than-or-equal-to gamma left-parenthesis t right-parenthesis slash t ; hence

StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel 0 less-than-or-equal-to minus beta prime left-parenthesis t right-parenthesis less-than-or-equal-to StartFraction gamma left-parenthesis t right-parenthesis Over t squared EndFraction equals StartFraction beta left-parenthesis t right-parenthesis Over t EndFraction period EndLayout

This immediately implies assertions (i) and (ii). Now let upper C greater-than 1 . Since gamma is increasing, we get

beta left-parenthesis t right-parenthesis equals StartFraction gamma left-parenthesis t right-parenthesis Over t EndFraction equals upper C StartFraction gamma left-parenthesis t right-parenthesis Over upper C t EndFraction less-than-or-equal-to upper C StartFraction gamma left-parenthesis upper C t right-parenthesis Over upper C t EndFraction equals upper C beta left-parenthesis upper C t right-parenthesis comma

which is the claim at point (iii). To prove (iv), in view of (Equation2) it suffices to estimate StartAbsoluteValue t 2 gamma prime left-parenthesis t 2 right-parenthesis minus t 1 gamma prime left-parenthesis t 1 right-parenthesis EndAbsoluteValue . Let t 3 greater-than 0 be defined in such a way that

gamma left-parenthesis t 1 right-parenthesis minus gamma prime left-parenthesis t 1 right-parenthesis left-parenthesis t 1 minus t 3 right-parenthesis equals gamma left-parenthesis t 2 right-parenthesis minus gamma prime left-parenthesis t 2 right-parenthesis left-parenthesis t 2 minus t 3 right-parenthesis period

By the concavity of gamma , then, t 1 less-than t 3 less-than t 2 follows. Let tau equals t 1 minus t 3 . Then tau less-than 0 , and we have

gamma left-parenthesis t 2 right-parenthesis minus gamma left-parenthesis t 1 right-parenthesis equals gamma prime left-parenthesis t 2 right-parenthesis left-parenthesis tau plus t 2 minus t 1 right-parenthesis minus gamma prime left-parenthesis t 1 right-parenthesis tau greater-than gamma prime left-parenthesis t 2 right-parenthesis t 2 minus gamma prime left-parenthesis t 1 right-parenthesis t 1 period

The last inequality is again due to the concavity of gamma (since gamma prime left-parenthesis t 2 right-parenthesis less-than gamma prime left-parenthesis t 1 right-parenthesis , the affine function x right-arrow gamma prime left-parenthesis t 2 right-parenthesis left-parenthesis x plus t 2 minus t 1 right-parenthesis minus gamma prime left-parenthesis t 1 right-parenthesis x is decreasing). Since by assumption t gamma prime left-parenthesis t right-parenthesis is monotone increasing (cf. Definition 2), this is the same as writing StartAbsoluteValue gamma prime left-parenthesis t 2 right-parenthesis t 2 minus gamma prime left-parenthesis t 1 right-parenthesis t 1 EndAbsoluteValue less-than gamma left-parenthesis t 2 right-parenthesis minus gamma left-parenthesis t 1 right-parenthesis .

Lemma 4

Let upper H be the half-plane StartSet z element-of double-struck upper C colon upper R e z greater-than 0 EndSet , and let beta be as in Theorem 1. Then there exists a (simply connected) domain normal upper Omega Subscript beta Baseline subset-of upper H with the following properties:

the boundary of normal upper Omega Subscript beta is Dini-smooth and 0 element-of b normal upper Omega Subscript beta ;

upper R e StartFraction 1 Over z EndFraction greater-than-or-equal-to beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis for all z in a neighborhood of 0 in normal upper Omega Subscript beta .

Proof.

We consider real coordinates left-parenthesis x comma y right-parenthesis such that z equals x plus i y , and we define normal upper Omega Subscript beta Baseline equals StartSet x greater-than phi left-parenthesis y right-parenthesis EndSet with phi left-parenthesis y right-parenthesis equals 2 y squared beta left-parenthesis StartAbsoluteValue y EndAbsoluteValue right-parenthesis . The boundary b normal upper Omega Subscript beta is Dini-smooth if and only if the derivative

StartFraction d phi Over d y EndFraction left-parenthesis y right-parenthesis equals 4 y beta left-parenthesis StartAbsoluteValue y EndAbsoluteValue right-parenthesis plus 2 normal s normal i normal g normal n left-parenthesis y right-parenthesis y squared StartFraction d beta Over d y EndFraction left-parenthesis StartAbsoluteValue y EndAbsoluteValue right-parenthesis

is Dini-continuous. By point (iv) in Lemma 3, the modulus of continuity of d phi slash d y can be estimated by a multiple of t beta left-parenthesis t right-parenthesis equals gamma left-parenthesis t right-parenthesis . The Dini-smoothness of phi thus follows from Definition 2. Also notice that, by definition, phi left-parenthesis 0 right-parenthesis equals 0 ; the first claim is then verified. As for the second one, let z element-of normal upper Omega Subscript beta ; we have

StartStartFraction left-parenthesis upper R e StartFraction 1 Over z EndFraction right-parenthesis OverOver beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis EndEndFraction equals StartFraction x Over beta left-parenthesis StartRoot x squared plus y squared EndRoot right-parenthesis left-parenthesis x squared plus y squared right-parenthesis EndFraction period

Assume, first, that x greater-than-or-equal-to StartAbsoluteValue y EndAbsoluteValue . Since beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis less-than-or-equal-to beta left-parenthesis x right-parenthesis , we obtain

StartStartFraction left-parenthesis upper R e StartFraction 1 Over z EndFraction right-parenthesis OverOver beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis EndEndFraction greater-than-or-equal-to StartFraction 1 Over 2 x beta left-parenthesis x right-parenthesis EndFraction right-arrow plus normal infinity

as x right-arrow 0 . If, on the other hand, phi left-parenthesis y right-parenthesis less-than x less-than-or-equal-to StartAbsoluteValue y EndAbsoluteValue , we estimate as follows:

StartStartFraction left-parenthesis upper R e StartFraction 1 Over z EndFraction right-parenthesis OverOver beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis EndEndFraction greater-than-or-equal-to StartFraction phi left-parenthesis y right-parenthesis Over 2 y squared beta left-parenthesis StartAbsoluteValue y EndAbsoluteValue right-parenthesis EndFraction equals 1

(where we used the fact that beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis less-than-or-equal-to beta left-parenthesis StartAbsoluteValue y EndAbsoluteValue right-parenthesis ), which concludes the proof.

Lemma 5

Let beta , upper H be as above. There is a non-vanishing holomorphic function alpha , defined in a neighborhood upper U of 0 in upper H and of class upper C Superscript 1 up to the boundary, such that

StartAbsoluteValue alpha left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to e Superscript minus beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis

for all z element-of upper U .

Proof.

Let normal upper Omega Subscript beta be the domain constructed Lemma 4, and let script upper R be the mapping given by Theorem 2. Since the differential of script upper R does not vanish, for some upper C greater-than 1 , upper D greater-than 0 and z in a neighborhood upper U of 0 one has upper D StartAbsoluteValue z EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue script upper R left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C StartAbsoluteValue z EndAbsoluteValue . By construction, we have

upper R e StartFraction 1 Over script upper R left-parenthesis z right-parenthesis EndFraction greater-than-or-equal-to beta left-parenthesis StartAbsoluteValue script upper R left-parenthesis z right-parenthesis EndAbsoluteValue right-parenthesis greater-than-or-equal-to beta left-parenthesis upper C StartAbsoluteValue z EndAbsoluteValue right-parenthesis greater-than-or-equal-to StartFraction 1 Over upper C EndFraction beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis

for z element-of upper U (here we used claim (iii) of Lemma 3). Hence, letting alpha left-parenthesis z right-parenthesis equals e Superscript negative upper C slash script upper R left-parenthesis z right-parenthesis , we get

StartAbsoluteValue alpha left-parenthesis z right-parenthesis EndAbsoluteValue equals e Superscript minus upper C upper R e StartFraction 1 Over script upper R left-parenthesis z right-parenthesis EndFraction Baseline less-than-or-equal-to e Superscript minus beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis

for z element-of upper U , as desired. As for the smoothness of alpha , first of all we note that it is clearly of class upper C Superscript 1 outside 0 since this is the case for 1 slash script upper R left-parenthesis z right-parenthesis . Computing the first derivative gives

alpha prime left-parenthesis z right-parenthesis equals upper C e Superscript minus StartFraction upper C Over script upper R left-parenthesis z right-parenthesis EndFraction Baseline StartFraction script upper R prime left-parenthesis z right-parenthesis Over script upper R squared left-parenthesis z right-parenthesis EndFraction semicolon

hence

StartAbsoluteValue alpha prime left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C prime e Superscript minus beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis Baseline StartFraction 1 Over StartAbsoluteValue z EndAbsoluteValue squared EndFraction right-arrow 0

as z right-arrow 0 (see Remark 1). This shows that alpha is of class upper C Superscript 1 .

Corollary 2

Let alpha , upper H be as in the previous lemma, and for kappa greater-than 0 define upper H Subscript kappa Baseline equals StartSet z element-of double-struck upper C colon upper R e z greater-than StartAbsoluteValue z EndAbsoluteValue Superscript kappa Baseline EndSet . For any fixed j element-of double-struck upper N , there exists upper C Subscript j comma kappa Baseline greater-than 0 such that

StartAbsoluteValue alpha Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C Subscript j comma kappa Baseline e Superscript minus StartFraction 1 Over StartRoot StartAbsoluteValue z EndAbsoluteValue EndRoot EndFraction

for all z element-of upper H Subscript kappa .

Proof.

We note, first, that since script upper R is bounded the Cauchy estimates give, for any fixed i element-of double-struck upper N ,

StartAbsoluteValue script upper R Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C prime Subscript i Baseline StartFraction 1 Over left-parenthesis upper R e z right-parenthesis Superscript i Baseline EndFraction comma z element-of upper H comma

for a suitable constant upper C prime Subscript i . It follows that StartAbsoluteValue script upper R Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C prime Subscript i Baseline slash StartAbsoluteValue z EndAbsoluteValue Superscript i kappa for z element-of upper H Subscript kappa .

Now, we have alpha Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis z right-parenthesis equals alpha left-parenthesis z right-parenthesis upper P Subscript j Baseline left-parenthesis 1 slash script upper R left-parenthesis z right-parenthesis comma script upper R prime left-parenthesis z right-parenthesis comma period period period comma script upper R Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis z right-parenthesis right-parenthesis , where upper P Subscript j Baseline element-of double-struck upper R left-bracket x 0 comma x 1 comma period period period comma x Subscript j Baseline right-bracket is a polynomial whose coefficients are determined by the Faa di Bruno formula. Thus, the polynomial estimates for each StartAbsoluteValue script upper R Superscript left-parenthesis i right-parenthesis Baseline EndAbsoluteValue over upper H Subscript kappa imply that StartAbsoluteValue alpha Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C double-prime Subscript j comma kappa Baseline e Superscript minus beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis Baseline slash StartAbsoluteValue z EndAbsoluteValue Superscript script l for a suitable script l equals script l left-parenthesis j comma kappa right-parenthesis greater-than 0 and all z element-of upper H Subscript kappa . The conclusion then follows from Remark 1.

Lemma 6

Let upper M be a smooth lineally convex hypersurface of finite type k , and suppose that coordinates left-parenthesis z comma w right-parenthesis for double-struck upper C Superscript n plus 1 are chosen as above. Then there exists a real function r of class upper C Superscript 1 comma StartFraction 1 Over k minus 1 EndFraction such that r left-parenthesis 0 right-parenthesis equals 0 and, for a neighborhood upper U of 0 in double-struck upper C Superscript n plus 1 ,

left-parenthesis upper M minus StartSet 0 EndSet right-parenthesis intersection upper U subset-of StartSet left-parenthesis z comma w right-parenthesis element-of upper U colon upper I m w greater-than r left-parenthesis upper R e w right-parenthesis EndSet period

Proof.

We write a local defining equation for upper M as in (Equation1). By the finite type hypothesis, then, it follows that (for a certain neighborhood upper U prime of 0 in double-struck upper C Superscript n plus 1 )

StartLayout 1st Row with Label left-parenthesis 4 right-parenthesis EndLabel left-parenthesis upper M minus StartSet 0 EndSet right-parenthesis intersection upper U prime subset-of StartSet left-parenthesis z comma w right-parenthesis element-of upper U Superscript prime Baseline colon upper I m w greater-than StartAbsoluteValue z EndAbsoluteValue Superscript k Baseline minus upper A left-parenthesis StartAbsoluteValue upper R e w EndAbsoluteValue StartAbsoluteValue z EndAbsoluteValue plus StartAbsoluteValue upper R e w EndAbsoluteValue squared right-parenthesis EndSet EndLayout

for a suitable constant upper A greater-than 0 . Consider, for a fixed u greater-than 0 , the function p Subscript u Baseline left-parenthesis x right-parenthesis equals x Superscript k Baseline minus upper A u x minus upper A u squared as defined over StartSet x greater-than 0 EndSet . By looking at the derivative, it is easy to see that p Subscript u assumes a minimum in x 0 equals left-parenthesis upper A u slash k right-parenthesis Superscript 1 slash left-parenthesis k minus 1 right-parenthesis ; hence

p Subscript u Baseline left-parenthesis x right-parenthesis greater-than-or-equal-to r left-parenthesis u right-parenthesis equals upper A prime u Superscript 1 plus StartFraction 1 Over k minus 1 EndFraction Baseline minus upper A u squared normal f normal o normal r x greater-than 0 comma

where upper A prime equals minus left-parenthesis k minus 1 right-parenthesis left-parenthesis upper A slash k right-parenthesis Superscript k slash left-parenthesis k minus 1 right-parenthesis is a (negative) constant. This choice of r , then, satisfies the conditions required by the lemma.

Remark 5

From the proof of the previous lemma it follows that we can choose r of the form r left-parenthesis u right-parenthesis equals minus upper D u Superscript 1 plus StartFraction 1 Over k minus 1 EndFraction for a large enough upper D greater-than 0 .

Now we are in a position to prove the second claim of Theorem 1. Let normal upper Omega 2 subset-of normal upper Omega 1 subset-of double-struck upper C Subscript w be the domains defined as normal upper Omega 1 equals StartSet upper I m w greater-than 2 r left-parenthesis upper R e w right-parenthesis EndSet , normal upper Omega 2 equals StartSet upper I m w greater-than r left-parenthesis upper R e w right-parenthesis EndSet , where r is given by Lemma 6 and by the subsequent Remark 5.

Lemma 7

For any w element-of double-struck upper C , denote by delta 1 left-parenthesis w right-parenthesis the distance between w and b normal upper Omega 1 . Then there exists kappa greater-than 0 such that delta 1 left-parenthesis w right-parenthesis greater-than-or-equal-to StartAbsoluteValue w EndAbsoluteValue Superscript kappa for all w element-of normal upper Omega 2 .

Proof.

Consider the following local diffeomorphism of double-struck upper C approximately-equals double-struck upper R squared , defined in a neighborhood of 0 :

psi colon double-struck upper R squared right-arrow double-struck upper R squared comma psi left-parenthesis upper R e w comma upper I m w right-parenthesis equals left-parenthesis upper R e w comma upper I m w minus 2 r left-parenthesis upper R e w right-parenthesis right-parenthesis

where r left-parenthesis upper R e w right-parenthesis equals minus upper D left-parenthesis upper R e w right-parenthesis Superscript 1 plus StartFraction 1 Over k minus 1 EndFraction as in Remark 5. We note that psi is of class (at least) upper C Superscript 1 , in particular bi-Lipschitz. Therefore there exists a constant upper E greater-than 0 such that upper E Superscript negative 1 Baseline d left-parenthesis w 1 comma w 2 right-parenthesis less-than-or-equal-to d left-parenthesis psi left-parenthesis w 1 right-parenthesis comma psi left-parenthesis w 2 right-parenthesis right-parenthesis less-than-or-equal-to upper E d left-parenthesis w 1 comma w 2 right-parenthesis for all w 1 comma w 2 element-of double-struck upper C close enough to 0 . Furthermore, we have psi left-parenthesis normal upper Omega 1 right-parenthesis equals normal upper Omega prime 1 equals StartSet upper I m w greater-than 0 EndSet and psi left-parenthesis normal upper Omega 2 right-parenthesis equals normal upper Omega prime 2 equals StartSet upper I m w greater-than minus r left-parenthesis upper R e w right-parenthesis EndSet . Defining delta prime 1 left-parenthesis w right-parenthesis equals normal d normal i normal s normal t left-parenthesis w comma b normal upper Omega prime 1 right-parenthesis , we get delta prime 1 left-parenthesis w right-parenthesis equals upper I m w for all w element-of normal upper Omega prime 2 .

Now, for any w element-of normal upper Omega prime 2 close enough to 0 we get

StartLayout 1st Row 1st Column StartAbsoluteValue w EndAbsoluteValue squared 2nd Column equals left-parenthesis upper R e w right-parenthesis squared plus left-parenthesis upper I m w right-parenthesis squared less-than left-parenthesis left-parenthesis negative r right-parenthesis Superscript negative 1 Baseline left-parenthesis upper I m w right-parenthesis right-parenthesis squared plus left-parenthesis upper I m w right-parenthesis squared 2nd Row 1st Column Blank 2nd Column equals upper D Superscript negative 2 Baseline left-parenthesis upper I m w right-parenthesis Superscript StartFraction 2 k minus 2 Over k EndFraction Baseline plus left-parenthesis upper I m w right-parenthesis squared less-than upper D prime left-parenthesis upper I m w right-parenthesis Superscript StartFraction 2 k minus 2 Over k EndFraction Baseline equals upper D prime delta prime 1 left-parenthesis w right-parenthesis Superscript StartFraction 2 k minus 2 Over k EndFraction EndLayout

for a large enough upper D prime greater-than 0 . In particular, choosing kappa greater-than k slash left-parenthesis k minus 1 right-parenthesis we have StartAbsoluteValue w EndAbsoluteValue Superscript kappa Baseline less-than delta prime 1 left-parenthesis w right-parenthesis for w element-of normal upper Omega prime 2 around 0 .

Since StartAbsoluteValue w EndAbsoluteValue less-than-or-equal-to upper E StartAbsoluteValue psi left-parenthesis w right-parenthesis EndAbsoluteValue and delta 1 left-parenthesis w right-parenthesis greater-than-or-equal-to upper E Superscript negative 1 Baseline delta prime 1 left-parenthesis psi left-parenthesis w right-parenthesis right-parenthesis , for w element-of normal upper Omega 2 we can write StartAbsoluteValue w EndAbsoluteValue Superscript kappa Baseline less-than-or-equal-to upper E Superscript kappa Baseline StartAbsoluteValue psi left-parenthesis w right-parenthesis EndAbsoluteValue Superscript kappa Baseline less-than upper E Superscript kappa Baseline delta prime 1 left-parenthesis psi left-parenthesis w right-parenthesis right-parenthesis less-than-or-equal-to upper E Superscript kappa plus 1 Baseline delta 1 left-parenthesis w right-parenthesis , which leads to the conclusion up to choosing a slightly larger exponent kappa .

Since b normal upper Omega 1 is of class upper C Superscript 1 comma StartFraction 1 Over k minus 1 EndFraction , it is in particular Dini-smooth; let script upper Q colon normal upper Omega 1 right-arrow upper H , script upper Q left-parenthesis 0 right-parenthesis equals 0 be the inverse of the Riemann mapping script upper R colon upper H right-arrow normal upper Omega 1 . By Lemma 2 we deduce that script upper Q is also of class upper C Superscript 1 up to the boundary and script upper Q prime left-parenthesis 0 right-parenthesis not-equals 0 , so that for some constant upper C greater-than 1 we can write

StartLayout 1st Row with Label left-parenthesis 5 right-parenthesis EndLabel upper C Superscript negative 1 Baseline StartAbsoluteValue w 1 minus w 2 EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue script upper Q left-parenthesis w 1 right-parenthesis minus script upper Q left-parenthesis w 2 right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C StartAbsoluteValue w 1 minus w 2 EndAbsoluteValue EndLayout

for w 1 comma w 2 close enough to 0 . We apply Lemma 5 with beta replaced by beta 1 equals upper C beta . Defining eta 1 left-parenthesis w right-parenthesis equals alpha left-parenthesis script upper Q left-parenthesis w right-parenthesis right-parenthesis , it follows that eta 1 is of class upper C Superscript 1 up to the boundary and that

StartAbsoluteValue eta 1 left-parenthesis w right-parenthesis EndAbsoluteValue equals StartAbsoluteValue alpha left-parenthesis script upper Q left-parenthesis w right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to e Superscript minus beta 1 left-parenthesis StartAbsoluteValue script upper Q left-parenthesis w right-parenthesis EndAbsoluteValue right-parenthesis Baseline less-than-or-equal-to e Superscript minus upper C beta left-parenthesis upper C StartAbsoluteValue w EndAbsoluteValue right-parenthesis Baseline less-than-or-equal-to e Superscript minus beta left-parenthesis StartAbsoluteValue w EndAbsoluteValue right-parenthesis Baseline period

Now, let normal upper Omega equals double-struck upper C Superscript n Baseline times normal upper Omega 1 equals StartSet left-parenthesis z comma w right-parenthesis element-of double-struck upper C Superscript n plus 1 Baseline colon upper I m w greater-than 2 r left-parenthesis upper R e w right-parenthesis EndSet ; we define a function eta element-of script upper O left-parenthesis normal upper Omega right-parenthesis intersection upper C Superscript 1 Baseline left-parenthesis normal upper Omega overbar right-parenthesis by eta left-parenthesis z comma w right-parenthesis equals eta 1 left-parenthesis w right-parenthesis . By Lemma 6, eta yields by restriction a non-trivial CR function of class upper C Superscript 1 , defined over a neighborhood of 0 in upper M and clearly satisfying the estimate required by Theorem 1.

We claim that eta Subscript vertical-bar upper M is in fact of class upper C Superscript normal infinity . Fix j element-of double-struck upper N . Using the Faa di Bruno formula, we can compute the j -th derivative of eta (note that of course only the ( partial-differential slash partial-differential w )-derivatives are relevant) as

eta Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis z comma w right-parenthesis equals sigma-summation Underscript i equals 1 Overscript j Endscripts alpha Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis script upper Q left-parenthesis w right-parenthesis right-parenthesis upper P Subscript i Baseline left-parenthesis script upper Q prime left-parenthesis w right-parenthesis comma period period period comma script upper Q Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis w right-parenthesis right-parenthesis

for suitable polynomials upper P Subscript i Baseline element-of double-struck upper Z left-bracket x 1 comma period period period comma x Subscript j Baseline right-bracket . Now, if p element-of upper M , p equals left-parenthesis z Subscript p Baseline comma w Subscript p Baseline right-parenthesis , by Lemma 6 we have that w Subscript p Baseline element-of normal upper Omega 2 , and thus by Lemma 7 it follows that w Subscript p Baseline element-of normal upper Omega Subscript kappa colon equals StartSet w element-of normal upper Omega 1 colon delta 1 left-parenthesis w right-parenthesis greater-than-or-equal-to StartAbsoluteValue w EndAbsoluteValue Superscript kappa Baseline EndSet for some kappa greater-than 0 . Moreover,

Lemma 8

For some kappa prime greater-than kappa we have script upper Q left-parenthesis normal upper Omega Subscript kappa Baseline right-parenthesis subset-of upper H Subscript kappa prime , where upper H Subscript kappa prime is defined in Corollary 2.

Proof.

If w element-of normal upper Omega Subscript kappa , we have by definition StartAbsoluteValue w minus w prime EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue w EndAbsoluteValue Superscript kappa for all w prime element-of b normal upper Omega 1 . Using (Equation5), we deduce StartAbsoluteValue script upper Q left-parenthesis w right-parenthesis minus script upper Q left-parenthesis w prime right-parenthesis EndAbsoluteValue greater-than-or-equal-to upper C Superscript negative 1 Baseline StartAbsoluteValue w minus w prime EndAbsoluteValue greater-than-or-equal-to upper C Superscript negative 1 Baseline StartAbsoluteValue w EndAbsoluteValue Superscript kappa Baseline greater-than-or-equal-to upper C Superscript negative kappa minus 1 Baseline StartAbsoluteValue script upper Q left-parenthesis w right-parenthesis EndAbsoluteValue Superscript kappa . Since script upper Q left-parenthesis b normal upper Omega 1 right-parenthesis equals b upper H , this is the same as writing StartAbsoluteValue script upper Q left-parenthesis w right-parenthesis minus w double-prime EndAbsoluteValue greater-than-or-equal-to upper C Superscript negative kappa minus 1 Baseline StartAbsoluteValue script upper Q left-parenthesis w right-parenthesis EndAbsoluteValue Superscript kappa for all w double-prime element-of b upper H ; hence we get upper R e script upper Q left-parenthesis w right-parenthesis greater-than-or-equal-to upper C Superscript negative kappa minus 1 Baseline StartAbsoluteValue script upper Q left-parenthesis w right-parenthesis EndAbsoluteValue Superscript kappa for all w element-of normal upper Omega Subscript kappa . Choosing any kappa prime greater-than kappa we obtain the claim of the lemma (for w close enough to 0 ).

Following, now, the same lines as in the proof of Corollary 2, by the Cauchy estimates it follows that StartAbsoluteValue script upper Q Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis w right-parenthesis EndAbsoluteValue less-than-or-equal-to upper F Subscript i Baseline slash delta 1 left-parenthesis w right-parenthesis Superscript i for w element-of normal upper Omega 1 ; thus StartAbsoluteValue script upper Q Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis w Subscript p Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to upper F Subscript i Baseline slash StartAbsoluteValue w Subscript p Baseline EndAbsoluteValue Superscript i kappa because w Subscript p Baseline element-of normal upper Omega Subscript kappa . Hence each term StartAbsoluteValue upper P Subscript i Baseline left-parenthesis script upper Q prime left-parenthesis w Subscript p Baseline right-parenthesis comma period period period comma script upper Q Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis w Subscript p Baseline right-parenthesis right-parenthesis EndAbsoluteValue blows up at most polynomially in 1 slash StartAbsoluteValue w Subscript p Baseline EndAbsoluteValue as p right-arrow 0 , p element-of upper M . On the other hand, by Corollary 2 we have (since script upper Q left-parenthesis w Subscript p Baseline right-parenthesis element-of upper H Subscript kappa prime ) that StartAbsoluteValue alpha Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis script upper Q left-parenthesis w Subscript p Baseline right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C Subscript i Baseline e Superscript minus StartFraction 1 Over StartRoot StartAbsoluteValue script upper Q left-parenthesis w Super Subscript p Superscript right-parenthesis EndAbsoluteValue EndRoot EndFraction Baseline less-than-or-equal-to upper C Subscript i Baseline e Superscript minus StartFraction 1 Over upper C Super Superscript 1 slash 2 Superscript StartRoot StartAbsoluteValue w Super Subscript p Superscript EndAbsoluteValue EndRoot EndFraction .

From the estimates above it follows that

StartAbsoluteValue alpha Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis script upper Q left-parenthesis w Subscript p Baseline right-parenthesis right-parenthesis EndAbsoluteValue dot StartAbsoluteValue upper P Subscript i Baseline left-parenthesis script upper Q prime left-parenthesis w Subscript p Baseline right-parenthesis comma period period period comma script upper Q Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis w Subscript p Baseline right-parenthesis right-parenthesis EndAbsoluteValue right-arrow 0

for all 1 less-than-or-equal-to i less-than-or-equal-to j as p right-arrow 0 , p element-of upper M . In conclusion eta Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis p right-parenthesis right-arrow 0 as p right-arrow 0 , p element-of upper M , and since this holds for any j element-of double-struck upper N we get that eta Subscript vertical-bar upper M is of class upper C Superscript normal infinity .

Remark 6

Since the function eta so constructed extends holomorphically to a neighborhood of any p element-of upper M except 0 , it is in fact real-analytic on upper M minus StartSet 0 EndSet if upper M is of class upper C Superscript omega . Moreover, eta left-parenthesis p right-parenthesis is only vanishing at p equals 0 .

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)
StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel upper I m w equals h left-parenthesis z comma upper R e w right-parenthesis equals f left-parenthesis z right-parenthesis plus left-parenthesis upper R e w right-parenthesis g left-parenthesis z comma upper R e w right-parenthesis comma EndLayout
Definition 2

Consider a concave function gamma colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus of class upper C Superscript 1 having the following properties:

(1)

integral Subscript 0 Superscript 1 Baseline StartFraction gamma left-parenthesis t right-parenthesis Over t EndFraction d t less-than plus normal infinity comma and thus, in particular, gamma left-parenthesis t right-parenthesis right-arrow 0 for t right-arrow 0 ;

(2)

t gamma prime left-parenthesis t right-parenthesis is monotone increasing.

We call beta colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus admissible if beta left-parenthesis t right-parenthesis equals gamma left-parenthesis t right-parenthesis slash t for such a gamma . For example, beta left-parenthesis t right-parenthesis equals 1 slash t Superscript a (corresponding to gamma left-parenthesis t right-parenthesis equals t Superscript 1 minus a ) is an admissible function for all 0 less-than a less-than 1 .

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 beta left-parenthesis t right-parenthesis with beta left-parenthesis t right-parenthesis plus 1 slash StartRoot t EndRoot ) that beta left-parenthesis t right-parenthesis greater-than-or-equal-to 1 slash StartRoot t EndRoot .

If beta left-parenthesis t right-parenthesis is admissible, then for t right-arrow 0 it is diverging at a much slower rate than 1 slash t , i.e. beta left-parenthesis t right-parenthesis slash left-parenthesis 1 slash t right-parenthesis right-arrow 0 . We note, however, that the admissible set contains functions that diverge faster than 1 slash t Superscript a for all 0 less-than a less-than 1 . Indeed, if we define

gamma left-parenthesis t right-parenthesis equals sigma-summation Underscript script l equals 1 Overscript normal infinity Endscripts StartFraction 1 Over 2 Superscript 2 script l Baseline EndFraction t Superscript 1 slash 2 Super Superscript script l Superscript Baseline comma

then beta left-parenthesis t right-parenthesis equals gamma left-parenthesis t right-parenthesis slash t has such a property.

Theorem 1

Let upper M be lineally convex of finite order k along an open cone script upper C subset-of upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis , and let phi be a CR function of class upper C Superscript 1 , defined on a neighborhood of 0 in upper M , which is exponentially decreasing of order 1 along script upper C . Then phi identical-to 0 .

On the other hand, let upper M be lineally convex of finite order (i.e. script upper C equals upper T 0 Superscript c Baseline left-parenthesis upper M right-parenthesis ) and let beta colon double-struck upper R Superscript plus Baseline right-arrow double-struck upper R Superscript plus be an admissible function. Then there exists a non-trivial CR function eta of class upper C Superscript normal infinity , defined on a neighborhood script upper V of 0 in upper M , such that

StartAbsoluteValue eta left-parenthesis p right-parenthesis EndAbsoluteValue less-than-or-equal-to e Superscript minus beta left-parenthesis StartAbsoluteValue psi left-parenthesis p right-parenthesis EndAbsoluteValue right-parenthesis

for all p element-of script upper V .

Lemma 1 (Watson’s Lemma).

Let upper S be a half-plane of double-struck upper C and let f be a germ of a holomorphic function on upper S which is exponentially decreasing of order 1 . Then f identical-to 0 .

Theorem 2

Let script upper R map the unit disc upper D conformally onto the inner domain normal upper Omega of a Dini-smooth Jordan curve. Then script upper R prime extends continuously to upper D overbar . Moreover, the extension of script upper R prime is non-vanishing on b upper D .

Corollary 1

Let normal upper Omega be a (simply connected) domain whose boundary is Dini-smooth, 0 element-of b normal upper Omega , and f element-of script upper O left-parenthesis normal upper Omega right-parenthesis be exponentially decreasing of order greater than 1 at 0 . Then f identical-to 0 .

Lemma 2

Let upper M be lineally convex, and let left-parenthesis z comma w right-parenthesis be coordinates for double-struck upper C Superscript n plus 1 in which upper M can be expressed as in (Equation1). For a small neighborhood upper U of 0 in double-struck upper C Superscript n plus 1 let upper U 1 equals upper U intersection StartSet upper I m w greater-than h left-parenthesis z comma upper R e w right-parenthesis EndSet , upper U 2 equals upper U minus upper U overbar Subscript 1 . Then for c element-of double-struck upper C close enough to 0 we have that StartSet w equals c EndSet intersection upper U 2 not-equals normal empty-set and upper U intersection StartSet w equals c EndSet intersection upper M is compact.

Lemma 3

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

(i)

beta is of class upper C Superscript 1 and beta prime less-than-or-equal-to 0 ;

(ii)

t beta left-parenthesis t right-parenthesis right-arrow 0 and t squared beta prime left-parenthesis t right-parenthesis right-arrow 0 for t right-arrow 0 ;

(iii)

beta left-parenthesis t right-parenthesis less-than-or-equal-to upper C beta left-parenthesis upper C t right-parenthesis for all upper C greater-than 1 ;

(iv)

for 0 less-than t 1 less-than t 2 , the following holds: StartAbsoluteValue t 2 squared beta prime left-parenthesis t 2 right-parenthesis minus t 1 squared beta prime left-parenthesis t 1 right-parenthesis EndAbsoluteValue less-than-or-equal-to 2 left-parenthesis t 2 beta left-parenthesis t 2 right-parenthesis minus t 1 beta left-parenthesis t 1 right-parenthesis right-parenthesis less-than-or-equal-to 2 left-parenthesis t 2 minus t 1 right-parenthesis beta left-parenthesis t 2 minus t 1 right-parenthesis period

Equation (2)
StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel beta prime left-parenthesis t right-parenthesis equals StartFraction gamma prime left-parenthesis t right-parenthesis Over t EndFraction minus StartFraction gamma left-parenthesis t right-parenthesis Over t squared EndFraction period EndLayout
Lemma 4

Let upper H be the half-plane StartSet z element-of double-struck upper C colon upper R e z greater-than 0 EndSet , and let beta be as in Theorem 1. Then there exists a (simply connected) domain normal upper Omega Subscript beta Baseline subset-of upper H with the following properties:

the boundary of normal upper Omega Subscript beta is Dini-smooth and 0 element-of b normal upper Omega Subscript beta ;

upper R e StartFraction 1 Over z EndFraction greater-than-or-equal-to beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis for all z in a neighborhood of 0 in normal upper Omega Subscript beta .

Lemma 5

Let beta , upper H be as above. There is a non-vanishing holomorphic function alpha , defined in a neighborhood upper U of 0 in upper H and of class upper C Superscript 1 up to the boundary, such that

StartAbsoluteValue alpha left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to e Superscript minus beta left-parenthesis StartAbsoluteValue z EndAbsoluteValue right-parenthesis

for all z element-of upper U .

Corollary 2

Let alpha , upper H be as in the previous lemma, and for kappa greater-than 0 define upper H Subscript kappa Baseline equals StartSet z element-of double-struck upper C colon upper R e z greater-than StartAbsoluteValue z EndAbsoluteValue Superscript kappa Baseline EndSet . For any fixed j element-of double-struck upper N , there exists upper C Subscript j comma kappa Baseline greater-than 0 such that

StartAbsoluteValue alpha Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C Subscript j comma kappa Baseline e Superscript minus StartFraction 1 Over StartRoot StartAbsoluteValue z EndAbsoluteValue EndRoot EndFraction

for all z element-of upper H Subscript kappa .

Lemma 6

Let upper M be a smooth lineally convex hypersurface of finite type k , and suppose that coordinates left-parenthesis z comma w right-parenthesis for double-struck upper C Superscript n plus 1 are chosen as above. Then there exists a real function r of class upper C Superscript 1 comma StartFraction 1 Over k minus 1 EndFraction such that r left-parenthesis 0 right-parenthesis equals 0 and, for a neighborhood upper U of 0 in double-struck upper C Superscript n plus 1 ,

left-parenthesis upper M minus StartSet 0 EndSet right-parenthesis intersection upper U subset-of StartSet left-parenthesis z comma w right-parenthesis element-of upper U colon upper I m w greater-than r left-parenthesis upper R e w right-parenthesis EndSet period
Remark 5

From the proof of the previous lemma it follows that we can choose r of the form r left-parenthesis u right-parenthesis equals minus upper D u Superscript 1 plus StartFraction 1 Over k minus 1 EndFraction for a large enough upper D greater-than 0 .

Lemma 7

For any w element-of double-struck upper C , denote by delta 1 left-parenthesis w right-parenthesis the distance between w and b normal upper Omega 1 . Then there exists kappa greater-than 0 such that delta 1 left-parenthesis w right-parenthesis greater-than-or-equal-to StartAbsoluteValue w EndAbsoluteValue Superscript kappa for all w element-of normal upper Omega 2 .

Equation (5)
StartLayout 1st Row with Label left-parenthesis 5 right-parenthesis EndLabel upper C Superscript negative 1 Baseline StartAbsoluteValue w 1 minus w 2 EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue script upper Q left-parenthesis w 1 right-parenthesis minus script upper Q left-parenthesis w 2 right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C StartAbsoluteValue w 1 minus w 2 EndAbsoluteValue EndLayout

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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
Additional Notes

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 .
Copyright Information
Copyright 2014 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
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