On planar Beurling and Fourier transforms

0.15 MB·English

by  Haakan Hedenmalm

On planar Beurling and Fourier transforms
H˚akan Hedenmalm
7
0
0 Abstract. We study the Beurling and Fourier transforms on subspaces of L2(C) defined by an
2
invariance property with respect to the root-of-unity group. This leads to generalizations of
n these transformations acting unitarily on weighted L2-spaces over C.
a
J

4
2

1. Introduction
]
V
Beurling and Fourier transforms. In this note, we shall study certain extensions of the Beurling
C and Fourier transforms on the complex plane C. The Fourier transform of an appropriately area-
h. integrable function f is
t
a F[f](ξ)= e−2iRe[zξ¯]f(z)dA(z), ξ ∈C,
m ZC
[ while the Beurling transform is the singular integral operator

1 BC[f](z)=pv f(w) dA(w), z ∈C;
v ZC (w−z)2
6
here “pv” stands for “principal value”, and
9
6 dxdy
dA(z)= , z =x+iy,
1
π
0
is normalized area measure. The two transforms are connected via
7
/0 FBC[f](ξ)= ξξ¯F[f](ξ), ξ ∈C.
h
t By the Plancherel identity, F is a unitary transformation on L2(C), which is supplied with the
a
m standard norm
v: kfk2L2(C) =ZC|f(z)|2dA(z).
Xi Itis clearfromthis andthe aboverelationshipthatBC is unitaryonL2(C) aswell.We recallthat
anoperatorT acting ona complex Hilbert space H is unitary if T∗T =TT∗ =id, where T∗ is the
r
a adjoint and “id” is the identity operator. Expressed differently, that T is unitary means that T is
a surjective isometry.
Root of unity invariance. For N = 1,2,3,..., let A denote the N-th roots of unity, that is, the
N
collectionofall α∈C with αN =1. For n=1,...,N, we considerthe closedsubspace L2 (C) of
n,N
L2(C) consisting of functions f having the invariance property
(1.1) f(αz)=αnf(z), z ∈C, α∈A .
N
1991 Mathematics Subject Classification. Primary32A25,32A36;Secondary46E15, 47A15.
Key words and phrases. Beurlingtransform.
ResearchsupportedbytheGo¨ranGustafssonFoundation.
2 Hedenmalm
It is easy to see that f ∈L2 (C) if and only f ∈L2(C) is of the form
n,N
(1.2) f(z)=zng(zN), z ∈C,
where g some other complex-valued function.
We shall study the Beurling and Fourier transforms on the subspaces L2 (C). This will be
n,N
showntoleadtointerestinggeneralizationsofthesetransformstotheweightedspacesL2(C),with
θ
norm
kfk2 = |f(z)|2|z|2θdA(z).
L2θ(C) ZC
Here, θ is a real parameter. We apply the results obtained regarding the Beurling transform to
conformal mapping, and obtain Grunsky-type identities in the spirit of [3]. The Grunsky-type
identity obtained here implies the Prawitz inequality (see, e. g., [5], [6]) as a special case. The
Grunsky-type inequality that follows from the Grunsky-type identity is more or less equivalent to
the general Grunsky inequality that forms the backdrop to Louis de Branges’ work leading up to
the solution of the Bieberbach conjecture (see [1]).
2. The Beurling transform
The Cauchy transform. The Cauchy transform CC is the integral transform
f(w)
CC[f](z)= dA(w),
ZC w−z
defined for appropriately integrable functions. It is related to Beurling transform BC via
BC[f](z)=∂zCC[f](z),
where both sides are understood in the sense of distribution theory. Here, we use the notation
1 ∂ ∂ 1 ∂ ∂
∂ = −i , ∂¯ = +i .
z z
2(cid:18)∂x ∂y(cid:19) 2(cid:18)∂x ∂y(cid:19)
The Beurling transform and root-of-unity invariance. Fix an N =1,2,3,... and an n=1,...,N.
We suppose f ∈L2 (C). Then, by the change of variables formula,
n,N
f(w) αn
BC[f](z)=pv dA(w)=pv f(w)dA(w) =αn−2BC[f](α¯z), z ∈C,
ZC (w−z)2 ZC (αw−z)2
for α∈A . Taking the averageover A , we get the identity
N N
1 αn
BC[f](z)= pv f(w)dA(w), z ∈C.
N ZCαX∈AN (αw−z)2
A symmetric sum. Next, we study the sum
1 αn
F(z)= .
N 1−αz
αX∈AN
This sum has the symmetry property
F(βz)=β¯nF(z), β ∈A ,
N
which means that F has the form
F(z)=zN−nG(zN).
On planar Beurling and Fourier transforms 3
The function G then has a simple pole at 1, and is analytic everywhere else in the complex plane.
Moreover,F vanishes at infinity, so G vanishes there, too. This leaves us but one possibility, that
G has the form
C
G(z)= ,
1−z
where C is a constant. It is easily established that C =1. This leaves us with
1 αn zN−n
(2.1) F(z)= = , z ∈C.
N 1−αz 1−zN
αX∈AN
As a consequence, we get
1 αn N n−1
H(z):=F(z)+zF′(z)=[zF(z)]′ = =zN−n − .
N (1−αz)2 (cid:26)(1−zN)2 1−zN(cid:27)
αX∈AN
This allows us to compute the sum we need:
1 αn 1 w NzN n−1
= H =zn−2wN−n − .
N (αw−z)2 z2 (cid:18)z(cid:19) (cid:26)(zN −wN)2 zN −wN(cid:27)
αX∈AN
For f ∈L2 (C), we thus get the representation
n,N
NzN n−1
BC[f](z)=zn−2 pv − wN−nf(w)dA(w), z ∈C.
ZC(cid:26)(zN −wN)2 zN −wN(cid:27)
Let f and g be connected via (1.2), and implement this relationship into the above formula:
NzN n−1
(2.2) BC[f](z)=zn−2 pv − wN g(wN)dA(w), z ∈C.
ZC(cid:26)(zN −wN)2 zN −wN(cid:27)
A similar expression may be found for the Cauchy transform as well:
wN
(2.3) CC[f](z)=zn−N−1 g(wN)dA(w), z ∈C.
ZC wN −zN
The extended Beurling transform. Let TC denote the operator
1
TC[h](z)= CC[h](z),
z
and introduce, for 0≤θ ≤1, the modified Beurling transform
(2.4) BθC[h](z)=BC[h](z)+TC[h](z), z ∈C;
here,hisassumedtobe aniceenoughfunctionsothatthe aboveBeurlingandCauchytransforms
make sense. It is easy to check that with
zg(z)
h(z)= ,
|z|2−2/N
where g is connected to f via (1.2), we have
BC[f](z)=zN+n−2BC(n−1)/N[h](zN), z ∈C.
The fact that BC is an isometry becomes the norm identity
(2.5) |h(z)|2|z|2θdA(z)= Bθ[h](z) 2|z|2θdA(z),
C
ZC ZC(cid:12) (cid:12)
(cid:12) (cid:12)
4 Hedenmalm
where we suppose that θ = (n−1)/N. However, fractions of this type are dense in the interval
[0,1], so that (2.5) extends to all θ with 0 ≤ θ ≤ 1. In other words, for 0 ≤ θ ≤ 1, the operator
BθC is unitary on the space L2θ(C), which was defined earlier.It is known[7] that BC is a bounded
operator on L2(C) for −1 < θ < 1 (but not for θ = ±1). This means that for −1 < θ < 1, both
θ
terms in (2.4) are bounded operators on L (C). We suspect that the second term in (2.4), the
θ
operator TC, is compact with small spectrum.
Extension to real θ. We first note that M , multiplication by the independent variable, is an
z
isometric isomorphism L2 (C) → L2(C) for all real θ. Therefore, for integers k and 0 ≤ θ ≤ 1,
θ+1 θ
the operator
Bθ+k :=M−kBθMk
C z C z
is unitary on L2 (C). It supplies an extension of Bθ to all real θ. Note that since
θ+k
1 1 w
+ = ,
(w−z)2 w−z z(w−z)2
therenodisagreementarisingfromthe pointsθ =0andθ =1.For−1≤θ ≤0,Bθ takesthe form
C
BθC =BC+θT′C,
where
f
T′C[f](z)=CC (z).
(cid:20)z(cid:21)
3. Fourier transforms
The Fouriertransform and root-of-unity invariance. The Fourier transformof a function in L2(C)
is given by
F[f](ξ)= e−2iRe[zξ¯]f(z)dA(z), ξ ∈C.
ZC
By the Plancherel identity, we have
kF[f]kL2(C) =kfkL2(C), f ∈L2(C).
Now, suppose f ∈ L2 (C), so that f has the invariance property (1.1). Then, by the change of
n,N
variables formula,
F[f](ξ)= e−2iRe[zξ¯]f(z)dA(z)=αn e−2iRe[αzξ¯]f(z)dA(z)=αnF[f](α¯ξ), ξ ∈C,
ZC ZC
for α∈A . Taking the averageover A , we get the identity
N N
F[f](ξ)= E (zξ¯)f(z)dA(z), ξ ∈C,
n,N
ZC
where
1
E (z)= αne−2iRe[αz].
n,N
N
αX∈AN
This sum has the symmetry property
E (βz)=β¯nE (z),
n,N n,N
which means that E (z) has the form
n,N
E (z)=zN−nD (zN).
n,N n,N
On planar Beurling and Fourier transforms 5
The extended Fourier transform. We now introduce the generalized Fourier transform
|ξ|−2(n−1)/N
F [h](ξ)= D (zξ¯)h(z)dA(z).
n,N n,N
N ZC
The special case N = n = 1 gives the standard Fourier transform. We connect f and g via (1.2).
It is easy to check that with
zg(z)
h(z)= ,
|z|2−2/N
we get that
1
|f(z)|2dA(z)= |h(z)|2|z|2(n−1)/NdA(z).
ZC N ZC
Moreover,we have
F[f](ξ)=ξn−1ξ¯N−1F [h](ξN),
n,N
so that
|F[f](ξ)|2dA(ξ)= 1 F [h](ξ) 2|ξ|2(n−1)/N dA(ξ).
n,N
ZC N ZC(cid:12) (cid:12)
The Plancherel identity thus states that F (cid:12)is a unitar(cid:12)y transformation on L2 (C). The
n,N (n−1)/N
inverse transformation is quite similar:
F−1 [h](z)=(−1)N−nF [h] (−1)Nz .
n,N n,N
(cid:0) (cid:1)
We need to express the function D in a different manner. Since
n,N
+∞ (−i)j+k
e−2iRe[αz] =e−iαze−iα¯z¯= αj−kzjz¯k
j!k!
jX,k=0
and
+∞
1
αm = δ ,
m,Nl
N
αX∈AN l=X−∞
where delta stands for the Kronecker delta, we have
1 1 +∞ (−i)j+k
E (z)= αne−2iRe[αz] = zjz¯k αj−k+n
n,N
N N j!k!
αX∈AN jX,k=0 αX∈AN
+∞ +∞ (−i)j+k +∞ +∞(−1)k(−i)−n+Nl
= zjz¯kδ = zk−n+Nlz¯k
j−k+n,Nl
j!k! (k−n+Nl)!k!
l=X−∞jX,k=0 l=X−∞Xk=0
+∞ +∞ (−1)k(−i)−n+Nl
=z−n |z|2kzNl,
(k−n+Nl)!k!
l=X−∞kX=0
with the understanding that
1
=0, m=−1,−2,−3,....
m!
It now follows that
+∞ +∞ (−1)k(−i)−n+Nl
D (z)= |z|2k/N zl−1.
n,N
(k−n+Nl)!k!
l=X−∞Xk=0
6 Hedenmalm
We should mention that as E (z) is bounded by 1 in modulus, we have
n,N
D (z) ≤|z|n/N−1, z ∈C.
n,N
(cid:12) (cid:12)
The formula for D allows us(cid:12)to expres(cid:12)s the modified Fourier transform accordingly (provided
n,N
f(z)=O(|z|−m) as |z|→+∞ for every positive integer m):
|ξ|−2(n−1)/N +∞ +∞ (−1)k(−i)−n+Nl
F [f](ξ)= |ξ|2k/N ξl−1 |z|2k/Nzl−1f(z)dA(z).
n,N
N l=X−∞kX=0 (k−n+Nl)!k! ZC
Anapplication involvingthe confluenthypergeometricfunction. Next,we considerafunction f of
the form
f(z)=z¯m|z|2αe−β|z|2/N,
where m is an integer, α,β are real with β >0, and
n−1
m+2α+ >−1.
N
If we choose
n−1−r
α=− ,
N
where r is an integer, we obtain by calculation that
|ξ|−2(n−1)/Nξm +∞ (−1)k
F [f](ξ)=(−i)−n+N(m+1) |ξ|2k/N
n,N
N (k−n+N(m+1))!k!
Xk=0
× |z|2(k−n+r+1)/N+2me−β|z|2/N dA(z)
ZC
ξm|ξ|−2(n−1)/N +∞(k+r−n+N(m+1))! |ξ|2/N k
=(−i)−n+N(m+1) −
βr−n+1+N(m+1) (k−n+N(m+1))!k! (cid:18) β (cid:19)
Xk=0
ξm|ξ|−2(n−1)/N (N(m+1)+r−n)!
=(−i)−n+N(m+1)
βr−n+1+N(m+1) (N(m+1)−n)!
|ξ|2/N
× F N(m+1)−n+r+1;N(m+1)−n+1;− ,
1 1
(cid:18) β (cid:19)
where F stands for the standard confluent hypergeometric function. We have from one of the
1 1
classical identities that
F N(m+1)−n+r+1;N(m+1)−n+1;−|ξ|2/N =e−|ξ|2/N/β F −r;N(m+1)−n+1;|ξ|2/N ,
1 1 1 1
(cid:18) β (cid:19) (cid:18) β (cid:19)
where the right hand side is easy to compute for positive r, as the sum is then finite. As a result
of the unitarity of F , we find that (with β =1 and M =N(m+1)−n+1)
n,N
+∞ 2 (M +2r−1)![(M −1)!]2
F −r;M;t e−2ttM−1dt= ,
Z0 h1 1(cid:0) (cid:1)i 2M+2r[(M +r−1)!]2
which follows from formula 7.622 of [2] as a limit case.
On planar Beurling and Fourier transforms 7
4. Applications of Beurling transforms to conformal mapping
Transfertotheunitdisk.Weneedtointroducesomegeneralnotation.LetM denotetheoperator
F
of multiplication by the function F. We also need the Hilbert space L2(X) with the norm
θ
khk2 = |h(z)|2|z|2θdA(z),
L2θ(X) ZX
where X is some Borel measurable subset of C with positive area. In the sequel, we fix θ to the
interval 0 ≤ θ ≤ 1. Fix a bounded simply connected domain Ω in C, which contain the origin,
and let ϕ : D → Ω denote the conformal mapping with ϕ(0) = 0 and ϕ′(0) > 0. Let f ∈ L2(Ω),
and extend it to the whole complex plane so that it vanishes on C\Ω. Let B [f] denote the
Ω
restrictionto Ω of BC[f], anddo likewise to define the operatorsCΩ, TΩ,T′Ω, BθΩ, aswell asB−Ωθ.
We introduce transferredoperators on spaces over the unit disk in the following fashion. First, we
suppose f ∈L2(Ω). Then the associated function
θ
θ
ϕ(z)
(4.1) g(z)=ϕ¯′(z) f ◦ϕ(z), z ∈D,
(cid:20) z (cid:21)
belongs to L2(D), with equality of norms:
θ
kgkL2(D) =kfkL2(Ω).
θ θ
The transferred Cauchy transform is defined as follows:
ϕ(z) θ wϕ(z) θ ϕ′(w)
(4.2) Cθ[g](z)= C [f](z)= g(w)dA(w).
ϕ (cid:20) z (cid:21) Ω ZD(cid:20)zϕ(w)(cid:21) ϕ(w)−ϕ(z)
The transferred modified Beurling transform is defined analogously:
ϕ(z) θ
Bθ[g](z)=ϕ′(z) Bθ[f]◦ϕ(z)
ϕ (cid:20) z (cid:21) Ω
ϕ(z) θ θ ϕ′(z)
=ϕ′(z) B [f]◦ϕ(z)+ C [f]◦ϕ(z) =Bθ,0[g](z)+θ Cθ[g](z),
(cid:20) z (cid:21) (cid:26) Ω ϕ(z) Ω (cid:27) ϕ ϕ(z) ϕ
where
wϕ(z) θ ϕ′(z)ϕ′(w)
Bθ,0[g](z)=pv g(w)dA(w).
ϕ ZD(cid:20)zϕ(w)(cid:21) (ϕ(w)−ϕ(z))2
It is clear that Bθ is a norm contraction on L2(D). Let P be the integral operator
ϕ θ θ
1 θ
P [f](z)= + f(w)|w|2θdA(w);
θ ZD(cid:20)(1−zw¯)2 1−zw¯(cid:21)
it is the orthogonalprojectionto the subspaceof analyticfunctions in L2(D). As both Bθ andP
θ ϕ θ
are contractions on L2(D), so is their product P Bθ. It remains to represent the operator P Bθ
θ θ ϕ θ ϕ
in a reasonable fashion. The main observation is that
wϕ(z) θ ϕ′(z)ϕ′(w) 1 ϕ′(z) 1 1
= −θ − +O(1)
(cid:20)zϕ(w)(cid:21) (ϕ(w)−ϕ(z))2 (w−z)2 (cid:20)ϕ(z) z(cid:21)w−z
near the diagonal z =w, so that
wϕ(z) θ ϕ′(z)ϕ′(w) ϕ′(z) wϕ(z) θ ϕ′(w) 1 θ
(4.3) +θ = + +O(1),
(cid:20)zϕ(w)(cid:21) (ϕ(w)−ϕ(z))2 ϕ(z) (cid:20)zϕ(w)(cid:21) ϕ(w)−ϕ(z) (w−z)2 z(w−z)
8 Hedenmalm
again near the diagonal. We observe that in view of (4.3), we get the Grunsky-type identity
(4.4) PθBθϕ =Bθϕ−BD+PθBD+θPθTD−θTD.
To make the involved operators PθBD and PθTD appearing in the right hand side of (4.4) more
concrete, it is helpful to know that for λ∈D,
1 1 θ 1
P [f ](z)=λ¯|λ|2θ + tθdt, f (z)= ,
θ λ Z (cid:20)(1−tλ¯z)2 1−tλ¯z(cid:21) λ λ−z
0
while
1 1 θ 1
P [g ](z)=−θλ¯2|λ|2θ−2 + tθdt, g (z)= .
θ λ Z (cid:20)(1−tλ¯z)2 1−tλ¯z(cid:21) λ (λ−z)2
0
In view of these relations, we quickly verify that
PθBD+θPθTD =0.
The Grunsky-type identity (4.4) thus simplifies a bit:
(4.5) PθBθϕ =Bθϕ−BD−θTD.
The corresponding Grunsky-type inequality reads
(4.6) (cid:13)(cid:0)Bθϕ−BD−θTD(cid:1)[f](cid:13)L2θ(D) ≤kfkL2θ(D), f ∈L2θ(D).
To get a concrete exa(cid:13)mple of how the Grun(cid:13)sky-type inequality works, we pick
1 θ
f (z)=|z|−2θ − , z ∈D,
λ (cid:18)(1−z¯λ)2 1−z¯λ(cid:19)
and compute
λϕ(z) θ ϕ′(z)ϕ′(λ) 1
Bθϕ−BD−θTD [f](z)=(cid:20)zϕ(λ)(cid:21) (ϕ(λ)−ϕ(z))2 − (λ−z)2
(cid:0) (cid:1)
ϕ′(z) λϕ(z) θ ϕ′(λ) θ
+θ − .
ϕ(z) (cid:20)zϕ(λ)(cid:21) ϕ(λ)−ϕ(z) z(λ−z)
We see that (4.6) in this case assumes the form (0≤θ ≤1)
λϕ(z) θ ϕ′(z)ϕ′(λ) 1
(4.7) −
ZD(cid:12)(cid:12)(cid:20)zϕ(λ)(cid:21) (ϕ(λ)−ϕ(z))2 (λ−z)2
(cid:12)(cid:12) ϕ′(z) λϕ(z) θ ϕ′(λ) θ 2
+θ − |z|2θdA(z)
ϕ(z) (cid:20)zϕ(λ)(cid:21) ϕ(λ)−ϕ(z) z(λ−z)(cid:12)
(cid:12)
2 (cid:12)
1 θ (cid:12) 1 θ
≤ |f (z)|2|z|2θdA(z)= − |z|−2θdA(z)= − .
ZD λ ZD(cid:12)(cid:12)(1−z¯λ)2 1−z¯λ(cid:12)(cid:12) (1−|λ|2)2 1−|λ|2
(cid:12) (cid:12)
The special case λ=0 gives us the i(cid:12)nequality of Prawitz(cid:12)(see [5] and [6]; we assume ϕ′(0)=1):
θ−2 2
ϕ(z) 1
ϕ′(z) −1 |z|2θdA(z)≤ .
ZD(cid:12)(cid:12) (cid:20) z (cid:21) (cid:12)(cid:12) 1−θ
(cid:12) (cid:12)
(cid:12) (cid:12)
On planar Beurling and Fourier transforms 9
A dual version. We carry out the corresponding calculations on the basis of the fact that B−θ is
C
unitary on L2 (C) for 0≤θ ≤1. In analogy with the above treatment, we connect two functions
−θ
f,g via
−θ
ϕ(z)
(4.8) g(z)=ϕ¯′(z) f ◦ϕ(z), z ∈D.
(cid:20) z (cid:21)
Then f ∈L2 (Ω) if and only if g ∈L2 (D), with equality of norms:
−θ −θ
kgkL2(D) =kfkL2(Ω).
θ θ
The corresponding transferred Beurling transform assumes the form
−θ
ϕ(z)
B−θ[g](z)=ϕ′(z) B−θ[f]◦ϕ(z)
ϕ (cid:20) z (cid:21) Ω
−θ
ϕ(z) f g
=ϕ′(z) B [f]◦ϕ(z)−θC ◦ϕ(z) =B−θ,0[g](z)−θϕ′(z)C−θ (z),
(cid:20) z (cid:21) (cid:26) Ω Ω(cid:20)z(cid:21) (cid:27) ϕ ϕ (cid:20)ϕ(cid:21)
where B−θ,0 and C−θ are as before (just plug in −θ in place of θ in the corresponding formulæ).
ϕ ϕ
It is clear that B−θ is a contraction on L2 (D).
ϕ −θ
To cut a long story short, the Grunsky-type identity analogous to (4.5) reads
(4.9) P−θB−ϕθ =B−ϕθ−BD+θT′D.
Let P¯∗ be the operator
−θ
1 θ
P¯∗ [g](z)=|z|−2θ − g(w)dA(w);
−θ ZD(cid:18)(1−wz¯)2 1−wz¯(cid:19)
it is a contraction on L2(D), which can be written
θ
P¯∗−θ =M|z|−2θP¯−θM|z|2θ,
where P¯ denotes the orthogonal projection onto the antiholomorphic functions in L2 (D). By
−θ −θ
forming adjoints, we find that (4.9) states that
(4.10) BθϕP¯∗−θ =Bθϕ−BD−θTD.
We now combine (4.5) with (4.10):
(4.11) Bθϕ−BD−θTD =PθBθϕ =BθϕP¯∗−θ =PθBθϕP¯∗−θ.
This means that for full mappings ϕ, we have equality in the Grunsky-type inequality (4.6) if and
only if f(z) is of the form |z|−2θ times an antianalytic function. In particular, (4.7) is an equality
for full mappings.
References
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ternational Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 25–42, Amer. Math. Soc.,
Providence, RI,1987.
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Hedenmalm: Department of Mathematics, The Royal Institute of Technology, S – 100 44 Stock-
holm, SWEDEN
E-mail address: [email protected]