LOCAL POLYNOMIAL CONVEXITY OF GRAPHS OF FUNCTIONS IN SEVERAL VARIABLES

Abstract. In that paper, we investigate the locally polynomial convexity of graphs of smooth functions in several variables. We also give a sufficient condition for real analytic function g defined near 0 in C which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on D for all disks D close enough to the origin in C

LOCAL POLYNOMIAL CONVEXITY OF GRAPHS OF FUNCTIONS

IN SEVERAL VARIABLES

KIEU PHUONG CHI

Abstract In that paper, we investigate the locally polynomial convexity of graphs

of smooth functions in several variables We also give a sufficient condition for real

analytic function g defined near 0 in C which behaves like zn near the origin so that

the algebra generated by z m and g is dense in the space of continuous functions on

D for all disks D close enough to the origin in C.

1 Introduction

We recall that for a given compact K in Cn, by ˆK we denote the polynomial convex hull of K i.e.,

ˆ

K = {z ∈ Cn : |p(z)| ≤ kpkK for every polynomial p in Cn}

We say that K is polynomially convex if ˆK = K A compact K is called locally polynomially convex at a ∈ K if there exists the closed ball B(a) centered at a such that B(a) ∩ K is polynomially convex The interest for studying polynomial convexity stems from the celebrated Oka-Weil approximation theorem (see [1], page 36) which states that holomorphic functions near a compact polynomially convex subset

of Cn can be uniformly approximated by polynomials in Cn A compact K ⊂ C is polynomially convex if is C \ K connected In higher dimensions, there is no such topological characterization of polynomially convex sets, and it is usual difficult to determine whether a given compact subset is polynomially convex By a well-known result of Wermer ([19]; see also [1], Theorem 17.1), every totally real manifold is locally polynomially convex Recall that a C1 smooth real manifold M is called totally real at

p ∈ M if the real tangent space TpM contains no complex line In this paper, we are concerned with local polynomial convexity at the origin of the graph Γf of a C2 smooth function f near 0 ∈ Cn such that f (0) = 0 By the theorem of Wermer just cited, we know that if ∂f

∂zi(0) 6= 0 for all i = 1, 2, , n then Γf is locally polynomially convex at the origin of Cn+1 Thus it remains to consider the case where ∂f

∂zi

(0) = 0 for some i Our study is motivated by a similar problem in one complex variable More precisely, let f be a C2 smooth function near 0 ∈ C such that f (0) = 0 Under certain condition

of f , one can show that Γf is locally polynomial convex at the origin of C2 The work

2010 Mathematics Subject Classification 46J10, 46J15, 47H10.

Key words and phrases polynomially convex, plurisubharmonic, totally real

1

associated with these direction of research is too numerous to list here; instead, the reader is referred to [2, 3, 20] and the references given therein

In the section 3, we will refine the technical from [6] to attack the problem in several variables For the readers convenience, we repeat a reasoning due to [6] First, we construct nonnegative smooth functions vanishing exactly on Γf These functions are, in general, plurisubharmonic only on open sets whose boundaries contain the origin Secondly,under some technical assumptions, we may add small strictly plurisubharmonic functions to obtain plurisubharmonic functions on certain open sets containing the (local) polynomially convex hull of Γf Finally, by invoking the nontrivial fact of about equivalence of plurisubharmonic hull and polynomial hulls, we can conclude that Γf is locally polynomially convex at the origin In this vein, we obtain some known results in one variable We also give some examples to show that our results are effective

In section 4, we shall present some results about locally uniform approximation of continuous function Let D be a small closed disk in the complex plane, centered at the origin and g be a C2 function on D which behaves like zn near the origin By [zm, g; D]

we denote the function algebra consisting of uniform limits on D of all polynomials in

zm and g Our goal finding conditions on g to ensure that [zm, g; D] = C(D), where C(D)is the set of continuous complex valued functions on D Of course a necessary condition is that the two functions zm and g must separate points of D However, this condition is far from sufficient Indeed, it can be shown that [zm, g; D] 6= C(D) for some choices of g (see [12, 16], ), while for other choices of g we have [zm, g; D] = C(D) ([9, 10], ) Our goal is giving generalized class of functions g defined near the origin in C such that [zm, g; D] = C(D) As in the previous work, we rely heavily on the theory of polynomial convexity It may be useful to recall the general scheme in proving [zm, g; D] = C(D) for appropriately chosen g Roughly speaking we consider the compact set ˜X which is inverse of X := {(zm, g) : z ∈ D} under the proper polynomial mapping (z, w) 7→ (zm, w) Then ˜X is a union of graphs (in C2) over D If

g behaves ”nicely” near the origin then we could show that each graph is polynomially convex Notice that we can not apply a well known result of Wermer as in [1] or [19], since each graph may fail to be totally real at the origin Thus in this case the compact ˜X is a union of polynomially convex compact sets If we could prove that

˜

X is polynomially convex, then by some known result about approximation on totally real compact sets (possibly with singularities) we could show that every continuous function on ˜X can be approximated uniformly by polynomials Since ˜X transforms nicely to X, by a well known lemma (see [11, 16]) we may deduce the same conclusion

on X Hence, it remains to decide the polynomial convexity of ˜X For this, we shall use an appropriate tool which is the version of Kallin lemma

Acknowledgment This work was done during a stay of the author at Vietnam Institute for Advance Study in Mathematics He wishes to express his gratitude to the institute for the support The author greatly thanks to Professor Nguyen Quang

2

Dieu for many stimulating conversations The author is supported by a grant from the NAFOSTED program

2 Some technical lemmas Consider the function g : [0, +∞) → R, is defined by

g(t) =

(

e−1t , t > 0

0 t = 0

For each x ≥ 0 set

χ(x) =

x

Z

0

g(t)dt

Let U be a open neighborhood of 0 in Cn, let the functions f = (f1, f2, , fm) ∈ C2(U ) and let X = { z, f1(z), , fm(z)
: z ∈ U } be the common graph of the functions fi

in Cn+m Consider the function F (z) =P

k=1|zn+k − fk(0z)|2 of class C2 in the region

U × Cm, where 0z = (z1, , zn) for all z = (z1, , zn, zn+1, , zm) ∈ Cn+m It is easy to see that the zero set of F coincides with X Let

LF(w) =

n+m

X

µ,ν=1

∂2F

∂zν∂zµwνwµ

be the Levi form of F and

Lf(0w) =

n

X

µ,ν=1

∂2f1

∂zν∂zνwµwν, ,

n

X

µ,ν=1

∂2fm

∂zν∂zνwµwν

 ,

where 0w = (w1, , wn) for all w = (w1, , wn+m) ∈ Cn+m For f = (f1, f2, , fm) ∈

C2(U ), we mean that

∂f

∂0z(

0

w) :=

n

X

ν=1

∂f1

∂zνwν,

n

X

ν=1

∂f2

∂zνwν, ,

n

X

ν=1

∂fm

∂zνwν



We have following the lemma

Lemma 2.1 The function

ϕ(z) = χ ◦ F (z)

is plurisubharmonic on

Ω = nz ∈ U × Cm :

∂f

∂0z(z)

0

w

2

+

 z0− f

|z0− f |,

∂(z0− f )

∂0z (z)

0

w

2

> 2Rez0− f (z), Lf(z)(0w)o, where z0 = (zn+1, , zn+m), for all z = (z1, , zn+m) ∈ Cn+m and w = (w1, , wn+m) ∈

Cn+m, w1w2 wn6= 0 Moreover, the following statements hold

i) If m = 1 then ϕ is plurisubharmonic on

3

Ω1 =nz ∈ U × C :

∂f

∂0z(z)

0

w

2

> Re(zn+1− f (z))Lf(z)(0w)o ii) If m = n = 1 then ϕ is plurisubharmonic on

Ω2 =n(z, w) ∈ U × C :

∂f

∂z(z)

2

> Re(w − f (z)) ∂

2f

∂z∂z(z)

o Proof It is easy to check χ(0) = χ0(0) = χ00(0) = 0,

χ0(x) = x2χ00(x), ∀x ≥ 0 and so that χ is a convex and increasing Since ϕ is convex, we reduce to show that F

is plurisubharmonic Consider the Levi form of F

LF(w) =

n+m

X

µ,ν=1

∂2F

∂zν∂zµwνwµ.

By the easy computations, we obtain

LF(w) =

m

X

k=1

n

X

ν=1

∂fk

∂zνwν

2

+

m

X

k=1

n+m

X

ν=1

∂(zn+k− fk)

∂zν wν

2

− 2Re

m

X

k=1

(zn+k− fk)

n

X

µ,ν=1

∂2fk

∂zν∂zµwνwµ.

So that in vector notation this takes the form

(1) LF(w) =

∂f

∂0z(z)

0

w

2

+

∂(z0− f )

∂z (z)w

2

− 2Rez0− f (z), Lf(0w) For the plurisubhamonicity of F , it is sufficient that LF(w) be nonnegative on complex tangent vectors w, these vectors are defined by the condition

n+m

X

k=1

∂F

∂zkwk= 0, which for the function F takes the form

∂0z

0

w=∂(z

0 − f )

∂z w, z

0− f

By applying Cauchy-Schwatz inequality, we obtain

∂(z0− f )

|z0 − f | ≥

∂(z0 − f )

∂z w, z

0− f =



z0 − f, ∂f

∂0z

0

w

 , and hence

∂(z0− f )

2

≥

z0 − f

|z0 − f |,

∂(z0− f )

∂0z (z)

0w

2

4

Since (1) and (3), we have

(4) LF(w) ≥

∂f

∂0z(z)

0

w

2

+

z0− f

|z0− f |,

∂(z0 − f )

∂0z (z)

0

w

2

− 2Rez0− f (z), Lf(z)(0w) Hence, the form LF(w) is nonnegative in the open set

n

z ∈ U × Cm :

∂f

∂0z

0

w

2

+

z0− f

|z0− f |,

∂(z0− f )

∂0z (z)

0

w

2

> 2Re z0− f (z), Lf(z)(0w)

o ,

i.e the function F is plurisubharmonic on Ω

If m = 1 then the right hand of (4) reduces to

2

∂f

∂0z(z)

0

w

2

− Re(zn+1− f (z))Lf(z)(0w)

so that F is plurisubharmonic on

Ω1 =nz ∈ U × C :

∂f

∂0z(z)

0

w

2

> Re(zn+1− f (z))Lf(z)(0w)o

If m = n = 1 then the fact is desired from Lf(z)(0w) = ∂

2f

∂z∂z(z)w for z ∈ U and

We need some the following fact, it is a consequence of the solution of the Levi problem in Cn (see [4], Theorem 4.3.3)

Lemma 2.2 ([6, 18]) Let K ⊂ Cn be compact

1) If K = ˆK then there exists a continuous plurisubharmonic function u on Cn such that u = 0 on K and u > 0 on Cn\ K

2) z ∈ ˆK if and only if u(z) 6 supKu for every plurisubharmonic function u on Cn The following lemma is due also from [6]

Lemma 2.3 Let K ⊂ Cn be compact and let U ⊂ Cn be an open neighborhood of ˆK Assume that there is a plurisubharmonic function u on U such that u = 0 on K Then

u = 0 on ˆK

Here an appropriate tool is the following version of Kallin lemma (see [16, 18]): Lemma 2.4 (Kallin’s Lemma) Suppose X1 and X2 are polynomially convex subsets

of Cn, suppose there is polynomial p mapping X1 and X2 into two polynomially convex subsets Y1 and Y2 of the complex plane such that 0 is a boundary point of both Y1 and

Y2 and with Y1∩ Y2 = {0} If p−1(0) ∩ (X1∪ X2) is polynomially convex, then X1∪ X2

is polynomially convex

Let X ⊂ Cn be a compact set and P (K) be the uniform closure in C(K) of all (restrictions to K) polynomials The following lemma is due from [17]

5

Lemma 2.5 Let X be a compact subset of C2, and let π : C2 → C2 be difined by π(z, w) = (zn, wm) Let π−1(X) = X11 ∪ X12 ∪ ∪ Xnm with X11 compact, and

Xkl = {exp(2π(k−1)m z, exp(2π(l−1)n w : (z, w) ∈ X11} for 1 ≤ k ≤ m, 1 ≤ l ≤ n If

P (π−1(X)) = C(π−1(X)) then P (X) = C(X)

3 Local polynomial convexity of graphs in Cn

Now we come to the main results of this work

Theorem 3.1 Let U be a open neighborhood of 0 in Cn, let the functions f ∈ C2(U ) Assume that there exists C2 smooth functions g defined on U satisfying the following conditions:

(i) |f |2 ≤ Re(f g) on U ;

(ii) |g(z)| ≤ |z|p, z ∈ U for some p ∈ N and ψ(z, w) = |w|2 − Re(wg(z)) is plurisubharmonic near the origin of Cn+1;

(iii)

∂f

∂z(z)w

2

+ Re f (z)Lf(z)w
> |g(z)Lf(z)(w)| for every w = (w1, , wn) ∈

Cn, w1w2 wn 6= 0 and z ∈ U \ {0}

Then Γf is locally polynomially convex at the origin in Cn+1 Furthermore, there exists an r > 0 small enough such that a continuous function on Xr := Γf ∩ B(0, r) can be approximated uniformly by polynomials

Remark 3.2 1) It follows from i) and ii) that |f | ≤ |g| and g(0) = 0 Combining this fact with iii) we arrive at

n

z ∈ U : ∂f

∂zk(0) = 0, k = 1, , n

o

consists at most the origin

2) If g is holomorphic on U then ψ(w, z) = |w|2− Re(wg(z)) is plurisubharmonic near the origin of Cn+1

Before taking up the proof of Theorem 3.1, we give some examples and corollaries, which mentions Theorem 3.1 is effective

Example 3.3 X = {(z1, z2, zm

1 +zn

2) : (z1, z2) ∈ C2}, m, n ∈ N and Y = {(z1, z2, z1z2) : (z1, z2) ∈ C2} are locally polynomial convex at 0 ∈ C3

Proof Consider f (z1, z2) = zm

1 + zn

2,for all (z1, z2) ∈ C2 Let g(z1, z2) = zm

1 + zn

2 for all (z1, z2) ∈ C2 It is easy to check that i) and ii) are satisfied for g It follows from

f (z1, z2) = zm

1 + zn

2 that Lf(z1, z2)(w1, w2) = 0 for all (z1, z2) ∈ C2 and (w1, w2) ∈ C2 Hence iii) is implied from the fact

∂f

∂z(z)w

2

=

∂f

∂z1(z1, z2)w1

2

+

∂f

∂z2(z1, z2)w2

2

= m2|z1|2(m−1)|w1|2+n2|z2|2(n−1)|w2|2 > 0 for all w = (w1, w2) ∈ C2 with w1w2 6= 0 Applying Theorem 3.1, we can deduce that

X is locally polynomial convex at 0 ∈ C3

For Y , we can conclude the same fact by considering f (z1, z2) = z1z2,for all (z1, z2) ∈ C2 and g(z1, z2) = z1z2 for all (z1, z2) ∈ C2 

6

Clearly, X, Y are not totally real at (0, 0, 0) ∈ C3 We can not apply Wermer’s result which mention in the introduction for locally polynomial convexity of X and Y Now,

we come to some consequence for the function in one variable

Corollary 3.4 Let U be a open neighborhood of 0 in C, let the functions f ∈ C2(U ) Assume that there exist C2 smooth function g defined on U satisfying the following conditions:

1) |f |2 ≤ Re(f g) on U ;

2) |g(z)| ≤ |z|p for some p ∈ N and ψ(z, w) = |w|2− Re wg(z)
is plurisubharmonic near the origin of C2;

3)

∂f

∂z

2

+ Ref ∂

2f

∂z∂z > |g

∂2f

∂z∂z| for all z ∈ U and z 6= 0

Then Γf is locally polynomially convex at the origin in C2 Furthermore, there exists

an r > 0 small enough such that a continuous function on Xr := Γf ∩ B(0, r) can be approximated uniformly by polynomials

The following corollary is main result of [6]

Corollary 3.5 [6] Let U be a open neighborhood of 0 in C, let the functions f ∈ C2(U ) Assume that there exist holomorphic function g on U and λ ∈ (0, 1) satisfying the following conditions:

(a) |f |2 ≤ Re(f g) on U ;

(b) g(0) = 0;

(c)λ

∂f

∂z

2

+ Ref ∂

2f

∂z∂z> |g

∂2f

∂z∂z| for all z ∈ U and z 6= 0

Then Γf is locally polynomially convex at the origin in C2 Furthermore, there exists

an r > 0 small enough such that a continuous function on Xr := Γf ∩ B(0, r) can be approximated uniformly by polynomials

Example 3.6 Let α be a complex number with |α| < 1 If n is a sufficiently enough integer then

S = {(z, zn+ αzzn−1) : z ∈ C}

is locally polynomially convex at the origin in C2

Proof We shall show that S satisfies Corollary 3.4 Consider the functions f (z) =

zn + αzzn−1 and g(z) = azn, with real number a > (1 + |α|)

2

1 − |α| Clearly, 2) holds In view of 1), we have

(5) |f (z)|2 = |zn+ αzzn−1|2

6 |z|2n(1 + |α|)2, ∀z ∈ C

and

(6) Re f (z)g(z)
= Reazn zn+ αzzn−1
≥ a1 − |α||z|2n

Since (5) and (6) we obtain

|f |2

6 Re(f g)

7

According to Corollary 3.4, it suffices checking the condition 3) Set ∆f = ∂

2f

∂z∂z We have

f (z) = zn+ αzzn−1 v ∆f (z) = α(n − 1)zn−2, ∀z ∈ C

Therefore

(7) |g(z)∆f (z)| = a|α|(n − 1)|z|2n−2, ∀z ∈ C

and



f (z)∆f (z)= Reα(n − 1)zn−2 zn+ αzzn−1


≥ |α|(n − 1)(|α| − 1)|z|2n−2

, ∀z ∈ C

On the other hand

(9)

∂f

∂z(z)

2

= |nzn−1+ αzn−1|2 ≥ (n − |α|)2|z|2n−2

≥ (n − 1)2|z|2n−2

, ∀z ∈ C

From (7), (8) and (9), we infer that the inequality

|g∆f | < |∂f

∂z|2+ Re(f ∆f ) for every z 6= 0 holds if

n > 1 + |α|(1 + a − |α|)

Therefore, the conditions of Corollary 3.4 are fulfilled with

n > 1 + |α|(1 + a − |α|)

In particular,

S = {(z, zn+ αzzn−1) : z ∈ C}

is locally polynomially convex at the origin in C2  Remark 3.7 1) We would to emphasize that if 1 > |α| >

√ 3

1 +√

3 then S is not satisfying Theorem 1.1 and Theorem 1.2 in [2]

2)If f (z) = zn+ o(|z|n) then f satisfies Corollary 3.4 in a small disk centered at 0 Let χ and F be mentioned as in the previous section The following lemma is key

of the proof of our main result

Lemma 3.8 Assume that the conditions of Theorem 3.1 are fulfilled Then, for every

ε > 0, there is δ(ε) > 0 such that the function

ϕε(z) = χ ◦ F (z) + ε|z|2

is plurisubharmonic on a ball

Bε =

n

z = (z1, , zn, zn+1, , zn+m) ∈ Cn+m: |z| < δ(ε)

o

8

Proof Let us ε > 0, we need show δ(ε) > 0 such that Lϕε is nonnegative form on

Bε =

n

z = (z1, , zn, zn+1, , zn+m) ∈ Cn+m: |z| < δ(ε)

o Since

Lϕε(w) = ε2|w|2+ Lϕ(w) = ε2|w|2+

n+m

X

µ,ν=1

∂2ϕ

∂zν∂zµwνwµ

we have that

(10) Lϕε(w) ≥ ε2|w|2− 2

n+m

X

µ,ν=1

∂2ϕ

∂zν∂zµ||w|2

On the other hand, we have

2ϕ

∂zν∂zµ =

∂2F

∂zν∂zµχ

0

F
+ ∂F

∂zν

∂F

∂zµχ

00

F

Since χ0(x) = x2χ00(x) for all x ∈ R+, we obtain

2ϕ

∂zν∂zµ = χ

00

F
F2 ∂

2F

∂zν∂zµ +

∂F

∂zν

∂F

∂zµ



It follows from i) and ii) that |f (z)| ≤ |g(z)| ≤ |z|p for all z ∈ U By this fact, we obtain

(13)

F (z) =X

k=1

|zn+k− fk(0z)|2

≤ 2(

n+m

X

k=n+1

|wk|2+

m

X

k=1

|fk(0z)|2) = 2(

n+m

X

k=n+1

|wk|2+ |f (0z)|2)

≤ 2(

n+m

X

k=n+1

|wk|2+ |g(0z)|2)

≤ 2(

n+m

X

k=n+1

|wk|2+ |0z|2p)

≤ 2(

n+m

X

k=n+1

|wk|2+

n

X

k=1

|0z|2) = 2|z|2,

where z = (z1, , zn, wn+1, , wn+m) ∈ Cn+m Since lim

x→0

χ00(x)

x = 0, we can choose

a > 0 such that χ00(x) < x for all |x| < a Combining this fact, (12) and (13), we can find a small neighborhood V of the origin of Cn such that

2ϕ

∂zν∂zµ| ≤ bνµ|z|2, z ∈ V

9

where bij is a positive constant independent on z Set b = max{bνµ} Since (10) and (14), we obtain

Lϕε(w) ≥ε2− 2

n+m

X

µ,ν=1

∂2ϕ

∂zν∂zµ||w|2 ≥ε2− 4(m + n)2b|z|2|w|2 Now, if we choose δ2(ε) = ε

2

4(m + n)2b then Lϕε(w) ≥ 0 on the ball

Bε =

n

z = (z1, , zn, zn+1, , zn+m) ∈ Cn+m: |z| < δ(ε)

o

Proof of Theorem 3.1 For r > 0, put Xr = Γf ∩ B(0, r) We claim that, for r > 0 small enough,

(15) Xˆr⊂ Kr =n(z1, , zn, zn+1) ∈ B(0, r) : |zn+1| ≤ |g(z)|o

Indeed, consider the function

ψ(z, w) = |w|2− Re wg(z)
for z ∈ Cn and w ∈ C Clearly

{(z, w) : z ∈ B(0, r), w ∈ C, ψ(z, w) ≤ 0} ⊂ Kr

By ii) we have that ψ is plurisubharmonic near the origin of Cn+1 Moreover, by part i) of the theorem we have ψ ≤ 0 on Xr for r > 0 small enough Thus by Lemma 2.3

we have that ψ ≤ 0 on ˆXr This proves our claim

Let χ be the function defined before in Lemma 2.1 Using Lemma 2.1, we can find

r > 0 small enough such that the function ϕ = χ ◦ F is plurisubharmonic on the open set

(16) Ωr =n(z1, , zn, zn+1) ∈ B(0, r) :

∂f

∂z(z)w

2

> Re(zn+1− f (z))Lf(z)(w))o Next, for (z, zn+1) ∈ Kr with z = (z1, , zn) 6= 0, by (iii) we have

Re(zn+1− f (z))Lf(z)w ≤ |zn+1Lf(z)w| − Re(f (z)Lf(z)w)

≤ |g(z)Lf(z)w| − Re(f (z)Lf(z)w)

<

∂f

∂z(z)w

2

for all w = (w1, , wn) ∈ Cn with w1w2 wn 6= 0 This implies that Kr\ {0} ⊂ Ωr

It follows from Lemma 3.8 and this fact, for every ε > 0, there exists a δ(ε) ∈ (0, r) (independent of r) such that the function

ϕε(z, zn+1) := χ(|zn+1− f (z)|2) + ε(|z|2+ |zn+1|2)

is plurisubharmonic on Ωr∪ B(0, δ(ε)), an open neighborhood of Xr Observe that

ϕε ≤ εr2 on Xr Therefore, applying Lemma 2.3 yields ϕε ≤ εr2 on ˆXr By letting

10

o Proof It is easy to check χ(0) = χ0(0) = χ00(0) = 0,

χ0(x) = x2χ00(x), ∀x ≥ and so that χ is a convex and increasing Since ϕ...

∂z w, z

0− f

By applying Cauchy-Schwatz inequality, we obtain

∂(z0− f )

|z0... Lf(0w) For the plurisubhamonicity of F , it is sufficient that LF(w) be nonnegative on complex tangent vectors w, these vectors are defined by the condition

n+m