PDF remotecenter.postech.ac.kr

Contents lists available atScienceDirect

Journal of Mathematical Analysis and Applications

journal homepage:www.elsevier.com/locate/jmaa

Restriction of the Fourier transform to some complex curves

Jong-Guk Bak

^a,∗

, Seheon Ham

^b,∗

aDepartment of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea

bDepartment of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea

a r t i c l e i n f o

Article history:

Received 20 March 2013 Available online 11 August 2013 Submitted by R.H. Torres Keywords:

Fourier transforms of measures Complex curves

Fourier restriction problem Affine arclength measure

a b s t r a c t

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces inR^2d,d≥3. These surfaces are defined by a complex curveγ (^z)of simple type, which is given by a mapping of the form

z→γ (^z)=

z,^z2, . . . ,^zd−1, φ(^z)

whereφ(^z)is an analytic function on a domainΩ⊂C. This is regarded as a real mapping z=(^x,^y)→γ (^x,^y)^fromΩ⊂_R²toR^2d.

Our results cover the caseφ(^z)=_z^Nfor any nonnegative integerN, in all dimensions d≥3. The main result is a uniform estimate, valid whend=3, whereφ(^z)may be taken to be an arbitrary polynomial of degree at mostN. It is uniform in the sense that the operator norm is independent of the coefficients of the polynomial. These results are analogues of the uniform restricted strong type estimates in [5], valid for polynomial curves of simple type and some other classes of curves inR^d,d≥3.

©2013 Elsevier Inc. All rights reserved.

1. Introduction and statement of results

Let us consider a ‘complex curve’ of simple type inC^d,d

≥

2. By this we mean a mapping of the following form:

z

→ γ (

^z

) = 

z

,

^z2

, . . . ,

^zd−1

, φ(

^z

) 

,

^z

∈

_Ω (1.1)

where

φ(

^z

)

is an analytic function on a domainΩ

⊂

_C. We will regard this mapping as a 2-dimensional surface inR^2d, given by the real mapping

z

= (

^x

,

^y

) → γ (

^x

,

^y

) = 

x

,

^y

,

^x2

−

y²

,

^2xy

, . . . ,

^Re

(φ(

^z

)),

^Im

(φ(

^z

)) 

∈

_R⁶

.

In what follows we useCandR²interchangeably whenever there is no danger of confusion.

Let us consider a Fourier restriction estimate of the following form:



R²

| 

f

(γ (

^z

)) |

^q

w(

^z

)

^d

µ(

^z

)



1/q

≤

C_p

∥

f

∥

_Lp(R^2d) (1.2)

where



f

(ξ)

denotes the Fourier transform off

∈

_L^p

(

R^2d

)

, and the weight function

w(

^z

)

is given by

w(

^z

) = | τ(

^z

) |

^4/(d2+d)

,

^where

τ(

^z

) =

det

(γ

^′

(

^z

), . . . , γ

^(d)

(

^z

)).

^(1.3) Also,d

µ

denotes the surface measure given byd

µ(

^z

) =

d

µ(γ (

^z

)) =

dxdyforz

=

x

+

iy. Here,



f

(γ (

^z

))

^{stands for}



f

(γ (

^x

,

^y

))

^.

∗Corresponding author.

E-mail addresses:bak@postech.ac.kr(J.-G. Bak),hamsh@snu.ac.kr(S. Ham).

0022-247X/$ – see front matter©2013 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.jmaa.2013.07.073

For

γ

^{given by}(1.1), we have

τ(

^z

) =

c_d

φ

^(d)

(

^z

)

^withcd

=

2

! · · · (

^d

−

1

) !

. The expression

w(

^z

)

^d

µ(

^z

) = | τ(

^z

) |

^4/(d2+d)d

µ(

^z

)

is an analogue of the affine arclength measure for real curves (cf.[18,19,3]). See Section2for the optimality of this choice of measure.

Whend

=

2, Oberlin [23] proved the following.

Theorem 1.1 ([23]; Theorem 4 and Example 3). Let

γ (

^z

) = (

^z

, φ(

^z

))

^{, where}

φ(

^z

)

is an analytic function on an open set D

⊂

_C. Suppose that

φ

^′

(

^z

)

and the map

(

^z1

,

^z2

) → (

^z1

−

z₂

, φ(

^z1

) − φ(

^z2

))

both have generic multiplicities at most N on D and D², respectively.¹Then there is a constant C_p

(

^N

) < ∞

so that for all f

∈

L^p

(

R⁴

)

^,



D

| 

f

(γ (

^z

)) |

^q

| φ

^′′

(

^z

) |

^2/3d

µ(

^z

)



1/q

≤

C_p

(

^N

) ∥

f

∥

_Lp(R⁴) (1.4)

whenever1

/

^p

+

1

/(

^3q

) =

1,1

≤

p

<

⁴

/

^3.

See [10] for a related result for some 2-dimensional surfaces inR⁴which are not necessarily given by holomorphic functions, but which satisfy a certain nondegeneracy condition. (See also [17] for an analogous result for somek-dimensional surfaces inR^d, whered

=

2k.)

In this paper we obtain some positive results in higher dimensions. First let us assume that

γ (

^z

)

is in the form(1.1), where

φ(

^z

) =

z^N,z

∈

_C, for an integerN

≥

0.

Theorem 1.2. Given integers d

≥

_{3and N}

≥

_{0, let}

γ (

^z

)

^{be as in}(1.1), with

φ(

^z

) =

_z^N. Then there is a constant C

(

^N

) < ∞

_so that for all f

∈

L^pd,1

(

R^2d

)

^,



R²

| 

f

(γ (

^z

)) |

^pd

w(

^z

)

^d

µ(

^z

)



1/pd

≤

C

(

^N

) ∥

f

∥

_Lpd,1(R^2d) (1.5)

where

w(

^z

) = | φ

^(d)

(

^z

) |

^4/(d2+d)and p_d

= (

^d2

+

d

+

2

)/(

^d2

+

d

)

^. Moreover, there is a constant C_p

(

^N

) < ∞

such that



R²

| 

f

(γ (

^z

)) |

^q

w(

^z

)

^d

µ(

^z

)



1/q

≤

C_p

(

^N

) ∥

f

∥

_Lp(R^2d) (1.6)

whenever1

/

^p

+

2

/ [ (

^d2

+

d

)

^q

] =

1,1

≤

p

<

^pd.

These estimates (as well as those in the next theorem) are expected to be optimal on the Lorentz scale of exponents, in view of the analogous results in the real case (see [3] andTheorems 1.4and1.5). However, this seems to be quite difficult to show in the present context, where the (real) dimension of the surface isk

=

2. For instance, it is unknown if the estimate(1.15), which is dual to(1.6), fails forq

≤

q_d,d

≥

3, even whenf is a bump function and we are in the nondegenerate case (with

w =

1). This is related to the unsolved problem of determining the convergence exponent for the multi-dimensional Tarry’s problem. In this connection, compare the statements of Theorem 1.3 (fork

=

1) and Theorem 1.9 (fork

≥

2) in [1]. Notice that no information is available for the divergence of the integral in Theorem 1.9 (in [1]), while Theorem 1.3 (in [1]) gives the complete answer in the 1-dimensional case.

We show the sharpness of the condition 1

/

^p

+

2

/ [ (

^d2

+

d

)

^q

] =

1 at the end of this section (see under the heading ‘‘A homogeneity argument’’), and we also prove in Section2the optimality of the weight function

w(

^z

)

, given after(1.5).

Whend

=

3, we get a uniform estimate valid for anarbitrarypolynomial

φ(

^z

)

of degree at mostN.²This is an exact analogue ofTheorem 1.5for (real) curves, stated below.

Theorem 1.3. For d

=

3and N

≥

0, let

γ (

^z

) = (

^z

,

^z2

, φ(

^z

))

^{, where}

φ(

^z

)

is an arbitrary polynomial of degree at most N. Then there is a constant C

(

^N

) < ∞

, independent of the coefficients of

φ(

^z

)

, so that for all f

∈

L^7/6,1

(

R⁶

)

^,



R²

| 

f

(γ (

^z

)) |

^7/6

w(

^z

)

^d

µ(

^z

)



6/7

≤

C

(

^N

) ∥

f

∥

_L7/⁶,¹(R⁶) (1.7)

where

w(

^z

) = | φ

^′′′

(

^z

) |

^1/3.

Moreover, there is a constant C_p

(

^N

) < ∞

, independent of the coefficients of

φ(

^z

)

, such that



R²

| 

f

(γ (

^z

)) |

^q

w(

^z

)

^d

µ(

^z

)



¹/^q

≤

C_p

(

^N

) ∥

f

∥

_Lp(R⁶)

whenever1

/

^p

+

1

/(

^6q

) =

1,1

≤

p

<

^p3

=

7

/

^6.

1 Recall thatF:D⊂_R^k→_R^kis said to have generic multiplicityNif card[F⁻¹(^y)] ≤Nfor almost ally∈_R^k. Here, card[E]denotes the cardinality of the setE.

2 It will be interesting if one can show a version ofTheorem 1.3for higher dimensions (d≥4) as well as an analogue ofTheorem 1.4for complex curves.

One can show that the weight functions

w(

^z

)

^{in(1.5)and(1.7)}are sharp up to a multiplicative constant, as in the real case. SeeProposition 2.1.

To put things in perspective, let us now recall some analogous earlier results for curves inR^d.

Real curves.Lett

→ γ (

^t

)

be a curve inR^d, defined on an intervalIinR. Let us consider a Fourier restriction estimate of the following form:



I

| 

f

(γ (

^t

)) |

^q

w(

^t

)

^dt



1/q

≤

C

∥

f

∥

_Lp(R^d) (1.8)

where



f

(ξ)

denotes the Fourier transform off

∈

L^p

(

R^d

)

^and

w(

^t

) = | τ(

^t

) |

2

d2+d

,

^with

τ(

^t

) =

det

(γ

^′

(

^t

), . . . , γ

^(d)

(

^t

)).

^(1.9)

Here the measure

w(

^t

)

^dtis called the ‘affine arclength measure’ (cf.[18,19,3]). We are mostly interested inuniformestimates for(1.8), that is, we would like to take the constantCto be uniform over given classes of curves. Also, whenever appropriate we would like to proveglobalestimates, that is, forI

=

_Ror

(

⁰

, ∞ )

^.

For the history of this problem we refer the reader to [16,3,5] and the references therein. The endpoint versions of the Fourier restriction estimates(1.8)for some classes of curves were established in [5]. We shall now describe two such results.

The first concerns the case of ‘monomial’ curves of the form

t

→ γ

a

(

^t

) = (

^ta1

,

^ta2

, . . . ,

^tad

),

⁰

<

^t

< ∞

(1.10)

wherea

= (

^a1

, . . . ,

^ad

)

^{is a}d-tuple of arbitrary real numbers. Ford

≥

2, letp_d

= (

^d2

+

d

+

2

)/(

^d2

+

d

)

. The endpoint result is the following.

Theorem 1.4([5]).Let

w(

^t

)

^dt

= w

a

(

^t

)

dt denote the affine arclength measure for the curve (1.10), where

w(

^t

)

^{is given} by(1.9)with

γ = γ

a. Then, for d

≥

3, there is a constant C

(

^d

) < ∞

such that for all f

∈

L^pd,1

(

R^d

)

^,



∞ 0

| 

f

(γ

a

(

^t

)) |

^pd

w

a

(

^t

)

^dt



¹/^pd

≤

C

(

^d

) ∥

f

∥

_Lpd,1(R^d)

.

^(1.11)

The constant in(1.11)is uniform in the sense that it does not depend ona₁

,

^a2

, . . . ,

^ad. We would like to point out that the versions of(1.11)fail whend

=

2 (forp₂

=

4

/

3), even in the nondegenerate case and even when the target space is replaced byL¹

(

^I

; w

^dt

)

for a finite intervalI. (See [6]; see also Section 1 in [3].)

The

(

^Lp

,

^Lq

)

estimates, in the optimal range 1

≤

p

<

^pd,q

=

2p^′

/(

^d2

+

d

)

, follow by interpolating(1.11)and the

(

^L1

,

^L∞

)

estimate. These estimates were proved earlier in [3], following the work in [19]. (For a general result inR²see, for instance, [25] and the references therein.)

Similar results have been proved for some other classes of curves including the polynomial curves of ‘simple’ type given by

Γb

(

^t

) = 

t

,

^t2

, . . . ,

^td−1

,

^Pb

(

^t

) 

,

^t

∈

_R (1.12)

inR^d, whereP_bis an arbitrary polynomial of degreeN

≥

0, with the coefficients

(

^b0

, . . . ,

^bN

) =

b

∈

_R^N+1. Namely, P_b

(

^t

) = 

N

j=0b_jt^j. The affine arclength measure is given byW_b

(

^t

)

^{dt, whereW}b

(

^t

) = | τ(

^t

) |

^2/(d2+d)

= |

c_dP_b^(d)

(

^t

) |

^2/(d2+d) withc_d

=

2

! · · · (

^d

−

1

) !

. The endpoint estimate in this case is the following.

Theorem 1.5([5]).For d

≥

3, there is a constant C

(

^N

) < ∞

so that for all f

∈

L^pd,1

(

R^d

)

^{and b}

∈

_R^N+1,



∞

−∞

| 

f

(

Γb

(

^t

)) |

^pd_Wb

(

^t

)

^dt



1/pd

≤

_C

(

^N

) ∥

_f

∥

L^pd,1(R^d)

.

^(1.13)

BothTheorems 1.4and1.5are optimal with respect to the two Lorentz exponents occurring on both sides, if we consider them as weighted Lorentz norm estimates:L^pd,1

(

R^d

) →

L^pd,pd

(w

^dt

)

. In particular, the strong type

(

^Lpd

,

^Lpd

)

estimate fails.

This fact is an easy consequence of the corresponding result in [3] for the nondegenerate case, where it was shown that L^pd,1

(

R^d

)

was the smallest possible space andL^pd,pd

(w

^dt

)

the largest possible space on the scale of Lorentz spaces. Moreover, the weight functions

w

⁽

= w

aorW_b) are sharp up to a multiplicative constant. (See [5,24] and Section2.)

Remark 1.6. One can also consider general polynomial curves of the form

γ (

^t

) = (

^P1

(

^t

), . . . ,

^Pd

(

^t

))

, where eachP_jis a polynomial of degree at mostN. Dendrinos and Wright [14] established the uniform Jacobian estimate for the mapping

(

^t1

, . . . ,

^td

) → 

d

j=1

γ (

^tj

)

. This implies a uniform restriction estimate in the reduced range 1

≤

p

<

^pc

(

^d

) =

^d2+2d

d²+2d−2. (This range is commonly referred to as ‘Christ’s range’ of exponents.) This is the range where one does not need the ‘method of offspring curves’, hence the torsion bound is not needed here. In [5] (see Proposition 8.1 there) this range was extended a little by combining an argument of Drury [16] with a result of Stovall [27] on averaging operators.

The main obstacle for obtaining a uniform estimate in the full range, by means of the method of offspring curves, is that the second crucial estimate concerning thetorsionof the offspring curves (as described in the beginning of Section6) breaks

down for curves of non-simple type. At the moment the only known approach that gives the full range 1

≤

p

<

^pd(and also the restricted strong type forp

=

p_d) for curves of non-simple type is the method based on ‘exponential parametrization’, which originated in [19] and was used in [5] to proveTheorem 1.4. (See also [13] and the remark at the end of Section6of [5].) The dual estimate of (1.5). Letp^′denote the Hölder conjugate exponent, i.e. 1

/

^p

+

1

/

^p′

=

1. The dual estimate of(1.5)is the following weak type

(

^qd

,

^qd

)

estimate forq_d

=

p^′_d

= (

^d2

+

d

+

2

)/

^2:

∥

Tf

∥

_Lqd,∞(R^2d)

≤

C

(

^N

) ∥

f

∥

_Lqd(wdµ) (1.14)

whereTis given by Tf

(

^x

) =



R²

e^{ix·γ (z)}f

(

^z

)w(

^z

)

^d

µ(

^z

),

^x

∈

_R^2d

.

Recall that the mappingz

→ γ (

^z

)

is regarded as a 2-dimensional surface

(

^x

,

^y

) → γ (

^x

,

^y

)

ⁱⁿR^2d. In particular,x

· γ (

^z

)

denotes the dot product inR^2d.

By interpolating(1.14)with the

(

^L1

,

^L∞

)

estimate it follows that

∥

Tf

∥

_Lq(R^2d)

≤

C_q

(

^N

) ∥

f

∥

_Lp(w^dµ) (1.15)

for 1

/

^p

+ (

^d2

+

d

)/(

^2q

) =

1,q

>

^qd

=

p^′_d

= (

^d2

+

d

+

2

)/

^2.

A homogeneity argument.To see the necessity of the condition 1

/

^p

+ (

^d2

+

d

)/(

^2q

) =

1 for(1.15)or(1.14)to hold, we use the usual homogeneity argument. That is, we takef

= χ

BR, whereB_R

=

B

(

⁰

,

^R

)

is a ball inR². We see that

|

Tf

(

^x

) | ≥

cR

4(N−d) (d2+d)+2

χ

ER

(

^x

/

^a

)

for some small constanta

>

^{0, whereE}R

= [−

R⁻¹

,

^R−1

]

²

× [−

R⁻²

,

^R−2

]

²

× · · · × [−

R^−(d−1)

,

^R−(d−1)

]

²

× [−

R^−N

,

^R−N

]

². Hence, if(1.14)or(1.15)holds, then we must have

R

4(^N−d) d2+d +2

R^−2q(

d(^d−1)

2 +N)

≤

CR⁽

4(^N−d) d2+d +2)¹_p

, ∀

R

>

⁰

.

Thus, it follows that 1

/

^p

+ (

^d2

+

d

)/(

^2q

) =

1.

Organization of this paper.The optimality of the weight function

w(

^z

)

ⁱⁿTheorem 1.2orTheorem 1.3is proved in Section2.

Section3contains the proof of a lower bound for a Jacobian arising in the proof ofTheorem 1.2. A uniform lower bound for the Jacobian associated to curves of simple type with arbitrary polynomials

φ(

^z

)

is proved in Section4. There is also a short discussion about a sublevel set estimate for the complex Vandermonde determinant at the end of Section4. In Section5we state an interpolation theorem proved in [5].Theorem 1.3is proved in Section6. Finally, in Section7we indicate how to modify the latter argument to proveTheorem 1.2.

Notation.Adopting the usual convention, we letCorcrepresent strictly positive constants whose values may not be the same at each occurrence. These constants may usually depend onN,dandp, but they will always be independent off. (In addition, they are uniform over the class of

γ (

^z

)

^{given in}Theorem 1.3. In particular, they are independent of the coefficients of the polynomial

φ(

^z

)

throughout the proof of that result.) Their dependence on the parameters is sometimes indicated by a subscript or shown in parentheses. We writeA_.BorB_&Ato meanA

≤

CB, andA

≈

Bmeans bothA_.BandB_.A.

2. Optimality of the weight function

Letd

≥

2. Here we shall consider the more general mapping

γ (

^z

) = (φ

1

(

^z

), . . . , φ

d

(

^z

))

, where each

φ

jis an analytic function onΩ

⊂

_C. We continue to use the notation

τ(

^z

) =

det

(γ

^′

(

^z

), . . . , γ

^(d)

(

^z

))

. The following result is analogous to the one found in Section 2 of [5], which in turn is based on an argument in [24].

Proposition 2.1. Assume that for some p

∈ (

¹

,

^pd

]

and q

(

^p

) =

2p^′

/(

^d2

+

d

)

there is a constant B such that for all f

∈

L^p,1

(

R^2d

)

^,



Ω

| 

f

(γ (

^z

)) |

^q(p)

ω(

^z

)

^d

µ(

^z

)



1/q(p)

≤

B

∥

f

∥

_Lp,1(R^2d) (2.1)

where

ω(

^z

)

is a nonnegative, locally integrable weight function onΩ. Then there is a constant C_dsuch that

ω(

^z

) ≤

C_dB^q(p)

| τ(

^z

) |

4

d2+d a

.

^e

.

^z

∈

_Ω

.

^(2.2)

When

γ (

^z

)

^{is as in} (1.1), then we have

τ(

^z

) =

c_d

φ

^(d)

(

^z

)

, so that the last inequality becomes

ω(

^z

) ≤

C_dB^q(p)

| φ

^(d)

(

^z

) |

^4/(d2+d), as we wanted to show.

Proof. LetP

=

AQ

+

bbe a parallelepiped inR^2d, whereQ

= [−

¹₂

,

¹₂

]

^2d,b

∈

_R^2dandAis an invertible linear transformation onR^2d. Take



f

(ξ) =

exp

( − π |

A⁻¹

(ξ −

b

) |

²

)

^{. Then}

| 

f

(ξ) | ≥

c₀

>

^{0 for}

ξ ∈

P, andf

(

^x

) =

e^2πib·x

|

det

(

^A

) | ·

exp

( − π |

A^tx

|

²

)

^.

Since

|

P

| = |

det

(

^A

) |

, we have

∥

f

∥

_p,1

≈ |

P

|

^1/p′. Hence,(2.1)implies that



R²

χ

P

(γ (

^z

)) ω(

^z

)

^d

µ(

^z

) ≤

C

(

^d

)

^Bq(p)

|

P

|

^2/(d2+d)

.

^(2.3)

Since each

φ

j

(

^z

)

is analytic onΩ^{, so is}

τ(

^z

)

. Thus, we may assume

τ(

^z

)

has only isolated zeros. So, it is enough to show (2.2)at points where

τ(

^z

) ̸=

0. (Otherwise,

τ(

^z

)

is identically zero. We comment on this case at the end of this section.)

Fixa

∈

_Ω. We have

γ (

^a

+

z

) = γ (

^a

) +

d



j=1

z^j

j

! γ

^(j)

(

^a

) +

O

( |

z

|

^d+1

)

^(2.4)

forznear the origin. Now consider the linear mapping

(

^z1

, . . . ,

^zd

) →

_Φ

(

^z1

, . . . ,

^zd

) = γ (

^a

) +

d



j=1

z_j

j

! γ

^(j)

(

^a

).

^(2.5)

Writez_j

=

x_j

+

iy_j. For

ε >

^{0, letE}

= { (

^z1

, . . . ,

^zd

) : |

x_j

| ≤

2

ε

^j

, |

y_j

| ≤

2

ε

^j

,

¹

≤

j

≤

d

}

denote a rectangular box inR^2d. The imageP₁ofEunder this mapping is a parallelepiped inR^2d. Its volume

|

P₁

|

is given by

|

P₁

| =

2^2d

ε

^d2+d

·

J_RΦ

=

2^2d

ε

^d2+d

· |

detJ_CΦ

|

²

=

2^2d

ε

^d2+d

· | (

²

! · · ·

d

! )

^−1det

(γ

^′

(

^a

), . . . , γ

^(d)

(

^a

)) |

²

=

2^2d

(

²

! · · ·

d

! )

⁻²

ε

^d2+d

· | τ(

^a

) |

²

.

We used here the fact that the Jacobian of(2.5)as a real mapping is given byJ_RΦ

= |

detJ_CΦ

|

², whereJ_CΦis the holomorphic Jacobian matrix of the mapping(2.5). This is a consequence of Proposition 1.4.10 on p. 51 in [22].

If

τ(

^a

) ̸=

0, and if

ε = ε(

^a

) >

0 is sufficiently small, then we have

γ (

^a

+

z

) ∈

P₁ when

|

z

| ≤ ε

. In fact, since

γ

^′

(

^a

), . . . , γ

^(d)

(

^a

)

^spanC^d, it follows from(2.4)that

γ (

^a

+

z

) = γ (

^a

) +

d



j=1

z^j

+

z^dg_j

(

^z

,

^a

)

j

! γ

^(j)

(

^a

)

^(2.6)

for some functionsg_j

(

^z

,

^a

)

^{such thatg}j

(

^z

,

^a

) →

0 asz

→

0 forj

=

1

,

²

, . . . ,

^d.

Therefore, it follows from(2.3)that lim sup

ε→0

1

πε

²



|z|≤ε

ω(

^a

+

z

)

^d

µ(

^z

) ≤

C_dB^q(p)

| τ(

^a

) |

^4/(d2+d)

.

So the conclusion(2.2)follows by the Lebesgue differentiation theorem.

On the other hand, when

τ(

^a

) =

0, a slight modification of the above argument shows that



|z|≤ε

ω(

^a

+

z

)

^d

µ(

^z

) =

o

(ε

²

),

^as

ε →

0

.

Thus, when

τ(

^z

) ≡

0, we may conclude that

ω(

^z

)

is zero almost everywhere. (See Section 2 of [5] for more details.) 3. A lower bound for the Jacobian

Let us begin by making a definition.

Definition 3.1. LetNbe a nonnegative integer and letz₁

, . . . ,

^zdbe complex numbers. LetQ_Ndenote a homogeneous monic polynomial of degree N inz₁

, . . . ,

^zd, given by

Q_N

(

^z1

, . . . ,

^zd

) = 

α1+···+αd=N

z₁^α1

· · ·

z_d^αd

.

Here,

α

1

, . . . , α

dare nonnegative integers.

Thus,Q_Nis asymmetricpolynomial. We have the following properties ofQ_N. Lemma 3.2. Let d

≥

2and N

≥

1. Then

(

ⁱ

)

^Q0

(

^zd

, . . . ,

^z1

) =

1;

(

ⁱⁱ

)

^QN

(

^z3

,

^z1

) −

Q_N

(

^z2

,

^z1

) = (

^z3

−

z₂

)

^QN−1

(

^z3

,

^z2

,

^z1

)

^;

(

ⁱⁱⁱ

)

^QN

(

^zd

,

^zd−1

, . . . ,

^z1

) =

_QN

(

^zd

, . . . ,

^z2

) +

_QN−1

(

^zd

, . . . ,

^z2

)

^z1

+ · · · +

_Q1

(

^zd

, . . . ,

^z2

)

^z1^N−1

+

_z1^N_.

(

^iv

)

Moreover, we have

Q_N

(

^zd+1

,

^zd−1

, . . . ,

^z1

) −

Q_N

(

^zd

,

^zd−1

, . . . ,

^z1

) = (

^zd+1

−

z_d

)

^QN−1

(

^zd+1

, . . . ,

^z1

).

Proof. The properties (i)–(iii) are straightforward. To see that (iv) holds, we use induction ond. First, (ii) gives the case d

=

2. Now suppose that (iv) holds withdreplaced byd

−

1. That is, we assume

Q_N

(

^zd

,

^zd−2

, . . . ,

^z1

) −

Q_N

(

^zd−1

,

^zd−2

, . . . ,

^z1

) = (

^zd

−

z_d−1

)

^QN−1

(

^zd

, . . . ,

^z1

)

holds for somed

≥

3 and forN

≥

1. It follows from (iii) and this induction hypothesis that

Q_N

(

^zd+1

,

^zd−1

, . . . ,

^z1

) −

Q_N

(

^zd

,

^zd−1

, . . . ,

^z1

) =

Q_N

(

^zd+1

,

^zd−1

, . . . ,

^z2

) +

Q_N−1

(

^zd+1

,

^zd−1

, . . . ,

^z2

)

^z1

+ · · · +

z₁^N

− [

Q_N

(

^zd

,

^zd−1

, . . . ,

^z2

) +

Q_N−1

(

^zd

,

^zd−1

, . . . ,

^z2

)

^z1

+ · · · +

z₁^N

]

= (

^zd+1

−

z_d

) 

Q_N−1

(

^zd+1

,

^zd

, . . . ,

^z2

) +

Q_N−2

(

^zd+1

,

^zd

, . . . ,

^z2

)

^z1

+ · · · +

Q₁

(

^zd+1

,

^zd

, . . . ,

^z2

)

^zN1⁻²

+

z₁^N−1



= (

^zd+1

−

z_d

)

^QN−1

(

^zd+1

, . . . ,

^z1

)

which is the casedof (iv). Hence, (iv) holds for alld

≥

2 andN

≥

1.

We now turn to the proof of a lower bound for the Jacobian of a transformation that arises in the proof ofTheorem 1.2.

LetJ

(

^z1

, . . . ,

^zd

) =

J_C

(

^z1

, . . . ,

^zd

)

denote the determinant of the holomorphic Jacobian matrix of the mapping

(

^z1

, . . . ,

^zd

) →

₀

(

^z1

, . . . ,

^zd

) =

d



k=1

Γb

(

^zk

)

withz_k

=

z

+

h_k,h₁

=

0. Here,Γb

(

^z

) =

m⁻¹



m

j=1

γ (

^z

+

b_j

)

^{, wherem}

∈

_N, andb

= (

^b1

, . . . ,

^bm

) ∈

_C^m, withb₁

=

0.

Forh

= (

^h1

, . . . ,

^hd

) ∈

_C^d, letv

(

^h

) =

v

(

^h1

, . . . ,

^hd

) = 

1≤i<j≤d

|

h_i

−

h_j

|

denote the absolute value of the (complex) Vandermonde determinant.

Lemma 3.3. Let

γ (

^z

)

be given by(1.1)with

φ(

^z

) =

z^Nfor an integer N

≥

d with d

≥

2, and letΓb

(

^z

)

be defined as above. Set J

(

^z1

, . . . ,

^zd

) =

J_C

(

^z1

, . . . ,

^zd

) =

det

(

Γb^′

(

^z1

), . . . ,

Γb^′

(

^zd

))

where z_j

=

z

+

h_j

∈

_C,1

≤

j

≤

d, and h₁

=

0. ThenCmay be written as the union (ignoring a null-set) of C

(

^d

,

^N

)

^sectors∆ℓ with vertex at the origin such that for each1

≤ ℓ ≤

C

(

^d

,

^N

)

, and for each integer m

≥

1, we have

|

J

(

^z1

, . . . ,

^zd

) | ≥

c

(

^d

,

^N

)

^v

(

^h

)

^max



1 m

m



j=1

| φ

^(d)

(

^z

+

b_j

+

h_k

) | :

1

≤

k

≤

d



(3.1)

where z

+

b_j

+

h_k

∈

_∆ℓ. Here, C

(

^d

,

^N

)

^{and c}

(

^d

,

^N

)

are positive constants depending only on d and N.

Note thatv

(

^h

) =

v

(

^z1

, . . . ,

^zd

)

^{, sincez}i

−

z_j

=

h_i

−

h_j.

Proof. Let us writez_jk

=

z

+

b_j

+

h_k. Sinceh₁

=

0, we havez_j1

=

z

+

b_j. If we abbreviate



m j=1as



, we get J

(

^z1

, . . . ,

^zd

) =

det

(

Γ_b^′

(

^z

+

h₁

), . . . ,

Γ_b^′

(

^z

+

h_d

))

= (

^d

−

1

) !

N m^d−1

































1 1

· · ·

1

 (

^zj1

) 

(

^zj2

) · · ·  (

^zjd

)

 (

^zj1

)

²



(

^zj2

)

²

· · ·  (

^zjd

)

²

... ... ... ...

 (

^zj1

)

^d−2



(

^zj2

)

^d−2

· · ·  (

^zjd

)

^d−2

 (

^zj1

)

^N−1



(

^zj2

)

^N−1

· · ·  (

^zjd

)

^N−1

































= (

^d

−

1

) !

N m^d−1































1 0

· · ·

0

 (

^zj1

)

^mh2

· · ·

mh_d

 (

^zj1

)

^{2 h}2



Q₁

(

^zj2

,

^zj1

) · · ·

h_d



Q₁

(

^zjd

,

^zj1

)

... ... ... ...

 (

^zj1

)

^{d−2 h}2



Q_d−3

(

^zj2

,

^zj1

) · · ·

h_d



Q_d−3

(

^zjd

,

^zj1

)

 (

^zj1

)

^{N−1 h}2



Q_N−2

(

^zj2

,

^zj1

) · · ·

h_d



Q_N−2

(

^zjd

,

^zj1

)































.