Extremal quasiconformality vs bounded rational approximation
We show that, on most of the hyperbolic simply connected domains, the weighty bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to the integral approximation). The obstructions are caused by the features of extremal quas...
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| citation_txt | Extremal quasiconformality vs bounded rational approximation / S.L. Krushkal // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 181-199. — Бібліогр.: 26 назв. — англ. |
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| description | We show that, on most of the hyperbolic simply connected domains, the weighty bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to the integral approximation). The obstructions are caused by the features of extremal quasiconformality.
The paper is devoted to the 100th anniversary of Georgii Dmitrievich Suvorov, my first university adviser and teacher. He was an outstanding mathematician and a widely talented, extremely great human being.
|
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Український математичний вiсник
Том 16 (2019), № 2, 181 – 199
Extremal quasiconformality
vs rational approximation
Samuel L. Krushkal
(Presented by V. Gutlyanskĭı)
The paper is devoted to the 100th anniversary of Georgii Dmitrievich
Suvorov, my first university adviser and teacher. He was an outstanding
mathematician and a widely talented, extremely great human being.
Abstract. We show that on most of the hyperbolic simply connected
domains the weighty bounded rational approximation in a natural sup
norm is possible only for a very sparse set of holomorphic functions (in
contrast to integral approximation). The obstructions are caused by the
features of extremal quasiconformality.
2010 MSC. Primary: 30C62, 30C75, 30E10; Secondary: 30F45, 30F60,
32G15.
Key words and phrases. Rational approximation, holomorphic func-
tion, quasiconformal maps, quasicircles, universal Teichmüller space,
Schwarzian derivative, Strebel point, Grunsky coefficients.
1. Results
This paper gives a link of geometric function theory to weighty boun-
ded rational approximation of holomorphic functions in sup norms and
shows how the intrinsic features of extremal quasiconformal maps and
universal Teichmüller space provide strong obstructions to such an ap-
proximation. The situation is completely different from the integral ap-
proximation.
1.1. Introductory remarks
The classical directions in rational interpolation of holomorphic func-
tions investigated by many authors concern mainly the uniform inter-
polation of functions holomorphic in the inner points of the closed set
Received 02.06.2019
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
182 Extremal quasiconformality vs bounded rational...
X on the Riemann sphere Ĉ = C ∪ {∞} and continuous on X by ra-
tional functions with poles off X and its combination with interpola-
tion (see, e.g., [26]). The second approach was originated by Walsh
(see [2,25]) and has recently been extended in [11] to the functional space
A−∞ =
∪
q>0A
−q over the Dini domains (with topology of the inductive
limit), where A−q(D) is the Banach space of holomorphic functions in a
domain D with norm ∥f∥ = supD δD(z)|f(z)|; here δD(z) = dist(z, ∂D)
denotes the Euclidean distance from the point z ∈ D to the boundary.
More generally, one considers a space F of holomorphic functions f
in a domain D ⊂ Ĉ. For any given collections of points
An = {anj}nj=0 ⊂ D, Bn = {bnj}nj=1 ⊂ Ĉ \D (n = 1, 2, . . . ),
there exists a unique rational function rn,f of degree n, with poles at
Bn, interpolating to f at An, counting multiplicities. The problem is to
select these collections An and Bn so that for all f the interpolants rn,f
converge to f in the topology of F .
It is established in [11] (by sweeping out appropriate measures and
application of the potential methods) that for f ∈ A−q(D), q > 0, the
interpolants rn,f are convergent (under appropriate conditions) to f in
A−q′ norm, where q′(q) ≫ q.
1.2. Weakened rational approximation in sup norm
Let L be an oriented quasiconformal Jordan curve (quasicircle) on the
Riemann sphere Ĉ = C ∪ {∞} with the interior and exterior domains D
and D∗, and let p ≥ 2 be an integer. Denote by λD(z)|dz| the hyperbolic
metric of D of Gaussian curvature −4 and consider the Banach spaces
Ap(D), Bp(D) of holomorphic functions (quadratic differentials) φ with
norms
∥φ∥Bp = sup
D
λD(z)
−p|φ(z)|, ∥φ∥Ap =
∫∫
D
λD(z)
2−p|φ(z)|dxdy,
respectively; due to [5], the space Bp is dual to Ap.
Note that for simply connected domains (with more than one bound-
ary points) not containing inside the infinite point,
1
4
≤ λD(z)δD(z) ≤ 1, (1)
where the right hand inequality follows from the Schwarz lemma and the
left from Koebe’s 1
4 theorem (so the spaces A−p mentioned above are
obtained by renormalization of Bp).
We also shall use the notations
S. L. Krushkal 183
D = {z : |z| < 1}, D∗ = {z ∈ Ĉ : |z| > 1};
H = {z : Im z > 0}, H∗ = {z : Im z < 0}.
We start with the following general
Theorem 1. Let D ⊂ Ĉ be a domain with quasiconformal boundary
L. Then for any function f ∈ Bp(D) there exists a sequence of rational
functions with poles of order two on L of the form
rn(z) =
n∑
1
cj
(z − aj)2
,
n∑
1
|cj | > 0, (2)
such that lim
n→∞
∥rn − φ∥Bp+1(D) = 0.
In the case of the half-plane (or disk), this theorem is strengthened
as follows.
Theorem 2. For any φ ∈ Bp(H) there exists a sequence of rational
functions (2) with real poles aj and real coefficients cj, convergent to φ
in Bp+1(H).
One can see from the proof of Theorem 1 that the weight exponent
p + 1 is not sharp; though it is not clear whether this exponent can be
replaces by p+ ϵ(p) with 0 < ϵ(p) < 1.
The limit case ϵ = 0 has intrinsic interest. Then the assertion on
convergence fails for ϵ = 0, because the space B2 of bounded holomorphic
quadratic differentials φdz2 is not separable for any Riemann surface of
infinite genus, and similar for all Bp.
This was established by functional-analytic methods but can be es-
tablished also from geometric features generated by Thurston’s theorem
on existence of uncountable many conformally rigid domains (see [3,24]).
Such domains correspond to the isolated points of the set U\T in B2(H),
where U denotes the set of the Schwarzian derivatives
Sw(z) =
(w′′(z)
w′(z)
)′
− 1
2
(w′′(z)
w′(z)
)2
of all univalent functions on H and T is the universal Teichmüller space
modeled by a bounded domain in B2(H) (formed by f having quasicon-
formal extension to Ĉ).
Note also that any φ ∈ B2(D) can be regarded as the Schwarzian of
a locally univalent function in D and determines this function up to a
Moebius transformation of Ĉ.
184 Extremal quasiconformality vs bounded rational...
1.3. Main theorems
Our aim is to show that on most of the hyperbolic simply connected
domains the rational approximation in B2 is possible only for a very
sparse subset of functions.
Theorem 3. For any simply connected domain D ⊂ Ĉ with quasiconfor-
mal boundary L, whose conformal mapping function gD : H → D satisfies
∥SgD∥B2 < 1/2, the set of functions φ ∈ B2(D) approximated in B2 norm
by general rational functions with poles of order two on L,
rn(z) =
n∑
1
cj
(z − aj)2
+
n∑
1
c′j
z − aj
,
n∑
1
|cj | > 0, (3)
is nonwhere dense in the space B2(D).
This theorem is a consequence of some deep results concerning the
extremal maps and the Grunsky operator given by
Theorem 4. There exists a constant c0 > 0 such that for any simply
connected domain D ⊂ Ĉ with quasiconformal boundary L and such that
∥SgD∥B2 < 1/2, and for any rational function rn with poles of order two
on L of the form (3) and norm ∥rn∥B2(D) < c0, we have the equalities
κD(w) = k(w) = ∥rn∥B2(D), (4)
where κD(w) and k(w) denote the Grunsky and Teichmüller norms of
(appropriately normalized) univalent solution w : D → Ĉ of the Schwar-
zian equation Sf = rn.
Note that the indicated constant c0 does not depend on D. In the
case of a disk (half-plane), one can take c0 = 1/2.
Theorem 4 shows that all Schwarzians φ = rn with ∥rn∥B2 < c0 are
not Strebel points in the universal Teichmüller space T (in other words,
the conformal maps f with Sf = rn do not have Teichmüller extremal
extensions onto the complementary domain D∗ = Ĉ \ D). Hence, such
points cannot be dense in B2(D).
Note also that, in view of the first equality in (4), the dilatation k(f)
is attained on the squares of holomorphic abelian differentials ωdz on D.
Theorem 3 is an immediate consequence of the equalities (4) in view
of either of two basic results on openness and density in the universal
S. L. Krushkal 185
Teichmüller space T: the result of [14] for points with non-equal Te-
ichmüller and Grunsky norms and Lakic’s result [19] on the density of
Strebel points (in arbitrary Teichmüller space).
Indeed, it suffices to establish the assertion of Theorem 3 for φ ∈
B2(D) with ∥φ∥ < c0. Both Teichmüller and Grunsky norms are contin-
uous on the universal Teichmüller space T modelled as a bounded domain
in B2 containing the origin. Hence, the equality (4) is preserving also for
the limit function f = lim rn of any sequence of rational functions in B2.
But this equality implies that any f ∈ B2 with sufficiently small ∥f∥
generates the Beltrami coefficient
µ(z) =
1
2
|z − z|2f ◦ gD(z),
where gD is a conformal map of H onto domain D (called harmonic), and
this coefficient is extremal in its equivalence class. On the other side, it
is not of Teichmüller type.
The latter is impossible in view of the indicated above openness re-
sults.
1.4. Generalization of Theorems 3 and 4
One can see from the proof of Theorem 4 in Section 5 that actually
its arguments are valid for an arbitrary meromorphic function φ on C
having poles of order two which are located on a quasicircle L passing
through the infinite point and accumulate to this point. This gives the
following extension of the above theorems.
Theorem 5. For any simply connected domain D ∈ Ĉ with quasicon-
formal boundary L passing through the infinite point whose conformal
mapping function satisfies ∥SgD∥B2 < 1/2, the subspace M2 in B2(D)
formed by meromorphic functions φ on C with poles of order two, which
are located on L and accumulate to ∞, is nonwhere dense in the space
B2(D).
If ∥φ∥B2(D) is sufficiently small, so that the Schwarzian equation
Sw = φ has a univalent solution w(z) on D, then
κD(w) = k(w) = ∥φ∥B2(D).
2. Proof of Theorem 1
First note that Ap(D) ⊂ BpD), and for any φ ∈ Ap(D),
∥φ∥Bp ≤ 4
π
∥φ∥Ap . (5)
186 Extremal quasiconformality vs bounded rational...
Indeed, since both these norms are conformally invariant, it suffices to
verify (5) for D = H. Then λD(z) = 1/(2y), and applying the mean
inequality for holomorphic functions, one gets
|f(z)| ≤ 1
πy2
∫∫
|z−ζ|≤y
|f(ζ)|dξdη ≤ (2y)2−p
πy2
∫∫
|z−ζ|≤y
(2η)2−pdξdη
(here ζ = ξ + iη, z ∈ D, η ≤ 2y), which yields (5). It follows also that
for any φ ∈ Ap(D),
lim
ρ→0
sup
δD(z)≥ρ
λpD(z)|φ(z)| = 0. (6)
Without loss of generality, one can assume that the boundary curve
L ∋ ∞ and 0 ∈ D. For any φ ∈ Bp(D),
|φ(z)| ≤ ∥φ∥BpλD(z)
−p ≍ ∥φ∥BpδD(z)
−p;
hence it belongs to Ap+1 and its integral
Iφ(z) =
∫ z
0
φ(ζ)dζ
belongs to Ap.
By the Bers approximation theorem [4], there exists a sequence of
rational functions r̃j(z) with simple poles on L and no other singularities,
such that1
lim
n→∞
∥r̃j − Iφ∥Ap = 0. (7)
Further, due to [5], for every ψ ∈ Bp(D) the following reproducing for-
mula is valid:
ψ(z) = −2p− 1
π
∫∫
D
(ζ − h(ζ)2p−2(∂h(ζ)/∂ζ))ψ(h(ζ))
(ζ − z)2p
dξdη; (8)
here ζ 7→ h(ζ) is a quasiconformal reflection with respect to the quasicircle
L = ∂D (i.e., orientation reversing quasiconformal automorphism of Ĉ
mapping D onto its complementary domain and leaving fixed all points
of L) which is uniformly bilipschizian on C, i.e., for all points z1, z2 ∈ C
the inequality
c−1
0 |z1 − z2| ≤ |h(z1)− h(z2)| ≤ c0|z1 − z2|
1This theorem is proved in [4] for the integrable holomorphic functions f ∈ A2; the
proof for the weighted spaces Ap can be done along the same lines, see [12].
S. L. Krushkal 187
holds with some constant c0 > 1. Moreover, the numerator
νψ(ζ) = −2p− 1
π
(ζ − h(ζ))2p−2∂h(ζ)
∂ζ
is estimated uniformly by
|νψ(ζ)| ≤ c(c0)∥ψ∥B2(D)λD(ζ)
2−p.
Applying (8) to ψ = r̃j−Iφ and differentiating both sides in z, one obtains
from (7) that rj = r̃′j are convergent to φ in Bp+1(D), completing the
proof of the theorem. �
3. Proof of Theorem 2
We apply the following result on integral approximation in the unit
disk given in [12] improving for the disk the Bers approximation theorem
mentioned above and also related to the theory of extremal quasiconfor-
mal maps.
Proposition 1. Let p and m be two integers such that p ≥ 2 and m ≥ 1.
Then for any ψ ∈ Ap(D), there exists a sequence of rational functions r̃j
which have only simple poles on the unit circle S1 and satisfy the condition
Im[ζmr̃j(ζ)] = 0 on S1 (outside of the poles of r̃j), such that
lim
j→∞
∥r̃j − ψ∥Ap = 0.
For even m and ζ = eiθ ∈ S1, we have
Im[r̃j(ζ)dζ
m] = (−1)m/2 Im[ζmr̃j(ζ)]dθ
m; (9)
thus the above proposition can be reformulated as follows.
Proposition 2. For any function ψ ∈ Ap(D), there exists a sequence
of rational functions r̃j which have only simple poles on the circle S1
and satisfy the condition Im[r̃j(ζ)dζ
2m] = 0 on S1, such that lim
j→∞
∥r̃j −
ψ∥Ap = 0.
Take m = p and, applying the fractional linear map σ(z) = (z −
i)/(z+ i) of the upper half-plane onto the unit disk, approximate similar
to Theorem 1 the integrated functions
σ∗Iψ = (Iψ ◦ σ)(σ′)4m−2
188 Extremal quasiconformality vs bounded rational...
by the corresponding rational functions
σ∗r̃j = (r̃j ◦ σ)(σ′)4m−2 ∈ Ap+1(H)
having real poles and coefficients in view of (9). Now applying the repro-
ducing formula for the upper half-plane,
ψ(z) =
2p− 1
π
∫∫
D
(ζ − ζ)2p−2ψ(ζ)
(ζ − z)2p
dξdη,
one straightforwardly obtains the conclusion of Theorem 2.
4. Backgrounds of Theorem 3
As was mentioned, Theorem 3 relies on some intrinsic features of ex-
tremal quasiconformal maps and the Grunsky operator. For convenience,
we briefly describe here these underlying results.
4.1. Extremal quasiconformality
Let L be a quasicircle passing through the points 0, 1,∞ which is
the common boundary of two domains D and D∗. Take the unit ball of
Beltrami coefficients supported on D∗,
Belt(D∗)1 = {µ ∈ L∞(C) : µ|D = 0 ∥µ∥∞ < 1}
and consider the corresponding quasiconformal automorphisms wµ(z) of
the sphere Ĉ satisfying on C the Beltrami equation ∂w = µ∂w preserv-
ing the points 0, 1,∞ fixed. We call the quantity k(w) = ∥µw∥∞ the
dilatation of the map w.
Take the equivalence classes [µ] and [wµ] letting the coefficients µ1
and µ2 from Belt(D∗)1 be equivalent if the corresponding maps wµ1 and
wµ2 coincide on L (and hence on D). These classes are in one-to-one
correspondence with the Schwarzians Swµ on D which fill a bounded
domain in the space B2(D) modelling the universal Teichmüller space
T = T(D) with the base point D. The quotient map
ϕT : Belt(D∗)1 → T, ϕT(µ) = Swµ
is holomorphic (as the map from L∞(D∗) to B2(D)). Its intrinsic Te-
ichmüller metric is defined by
τT(ϕT(µ), ϕT(ν))=
1
2
inf
{
logK
(
wµ∗◦
(
wν∗
)−1)
: µ∗ ∈ ϕT(µ), ν∗ ∈ ϕT(ν)
}
,
S. L. Krushkal 189
It is the integral form of the infinitesimal Finsler metric
FT(ϕT(µ), ϕ
′
T(µ)ν) = inf{∥ν∗/(1− |µ|2)∥∞ : ϕ′T(µ)ν∗ = ϕ′T(µ)ν}
on the tangent bundle T T of T, which is locally Lipschitzian.
We call the Beltrami coefficient µ ∈ Belt(D∗)1 extremal (in its class)
if
∥µ∥∞ = inf{∥ν∥∞ : ϕT(ν) = ϕT(µ)}
and call µ infinitesimally extremal if
∥µ∥∞ = inf{∥ν∥∞ : ν ∈ L∞(D∗), ϕ′T(0)ν = ϕ′T(0)µ}.
Any infinitesimally extremal Beltrami coefficient µ is globally extremal
(and vice versa), and by the basic Hamilton–Krushkal–Reich–Strebel the-
orem the extremality of µ is equivalent to the equality
∥µ∥∞ = inf{| < µ,ψ >D∗ | : ψ ∈ A2(D
∗) : ∥ψ∥ = 1}
(where A2(D
∗) is the subspace of L1(D
∗) formed by holomorphic func-
tions on D∗) and the pairing
⟨µ, ψ⟩D∗ =
∫∫
D∗
µ(z)ψ(z)dxdy, µ ∈ L∞(D∗), ψ ∈ L1(D
∗) (z = x+iy).
Let w0 := wµ0 be an extremal representative of its class [w0] with
dilatation
k(w0) = ∥µ0∥∞ = inf{k(wµ) : wµ|L = w0|L},
and assume that there exists in this class a quasiconformal map w1 whose
Beltrami coefficient µA1 satisfies the inequality ess supAr
|µw1(z)| < k(w0)
in some ring domain R = D∗ \G complement to a domain G ⊃ D∗. Any
such w1 is called the frame map for the class [w0], and the corresponding
point in the universal Teichmüller space T is called the Strebel point.
These points have the following important properties.
Proposition 3. (i) If a class [f ] has a frame map, then the extremal map
f0 in this class (minimizing the dilatation ∥µ∥∞) is unique and either a
conformal or a Teichmüller map with Beltrami coefficient µ0 = k|ψ0|/ψ0
on D∗, defined by an integrable holomorphic quadratic differential ψ0 on
D∗ and a constant k ∈ (0, 1) [23].
(ii) The set of Strebel points is open and dense in T [8, 19].
190 Extremal quasiconformality vs bounded rational...
The first assertion holds, for example, for asymptotically conformal
(hence for all smooth) curves L. Similar results hold also for arbitrary
Riemann surfaces (cf. [7, 8]).
The boundary dilatation H(f) admits also a local version Hp(f) in-
volving the Beltrami coefficients supported in the neighborhoods of a
boundary point p ∈ ∂D. Moreover (see, e.g., [8, Ch. 17]), H(f) =
supp∈∂DHp(f), and the points with Hp(f) = H(f) are called substan-
tial for f and for its equivalence class.
4.2. The Grunsky–Milin inequalities
Let D∗ ∋ ∞ be a simply connected domain with quasiconformal
boundary and Σ0(D∗) denote the class of univalent Ĉ-holomorphic func-
tions in D∗ with expansions f(z) = z + b0 + b1z
−1 + . . . near z = ∞
admitting quasiconformal extensions to Ĉ. Their Grunsky–Milin coeffi-
cients αmn are defined from the expansion
− log
f(z)− f(ζ)
z − ζ
=
∞∑
m,n=1
αmn
χ(z)m χ(ζ)n
, (10)
choosing the branch of the logarithmic function which vanishes as z =
ζ → ∞. Here χ denotes a conformal map of D∗ onto the disk D∗ so that
χ(∞) = ∞, χ′(∞) > 0.
Each coefficient αmn(f) in (10) is a polynomial of a finite number
of the initial coefficients b1, b2, . . . , bm+n−1 of f ; hence it depends holo-
morphically on Beltrami coefficients of extensions of f as well as on the
Schwarzian derivatives Sf ∈ B2(D
∗).
A theorem of Milin extending the Grunsky univalence criterion for the
disk D∗ states that a holomorphic function f(z) = z+const+O(z−1) in
a neighborhood of z = ∞ can be continued to a univalent function in the
whole domain D∗ if and only if the coefficients αmn satisfy the inequality
∣∣∣ ∞∑
m,n=1
√
mn αmnxmxn
∣∣∣ ≤ 1
for any point x = (xn) from the unit sphere S(l2) of the Hilbert space
of sequences x = (xn) with ∥x∥2 =
∞∑
1
|xn|2 (cf. [10, 20, 22]). We call the
quantity
κD∗(f) = sup
{∣∣∣ ∞∑
m,n=1
√
mn αmn xmxn
∣∣∣ : x = (xn) ∈ S(l2)
}
S. L. Krushkal 191
the Grunsky norm of f . The inequality κD∗(f) ≤ 1 is necessary and
sufficient for univalence of f in D∗ (see [10,20,22]). In the canonical case
D∗ = D∗, we have the classical Grunsky coefficients.
Consider the set
A2
2(D) = {ψ ∈ A2(D) : ψ = ω2}
consisting of the integrable holomorphic functions on D having only
zeros of even order and put
αD(f) = sup {|⟨µ0, ψ⟩D| : ψ ∈ A2
2, ∥ψ∥A2(D) = 1}.
The following proposition from [16] completely describes the relation be-
tween the Grunsky and Teichmüller norms (more special results were
obtained in [13,18]).
Proposition 4. For all f ∈ Σ0(D∗),
κD∗(f) ≤ k
k + αD(f)
1 + αD(f)k
, k = k(f),
and κD∗(f) < k unless
αD(f) = ∥µ0∥∞, (11)
where µ0 is an extremal Beltrami coefficient in the equivalence class [f ].
The last equality is equivalent to κD∗(f) = k(f).
If κD∗(f) = k(f) and the class of [f ] is a Strebel point, then µ0 is
necessarily of the form
µ0 = ∥µ0∥∞|ψ0|/ψ0 with ψ0 ∈ A2
2(D).
Note that geometrically (11) means the equality of the Carathéodory
and Teichmüller distances on the geodesic disk {ϕT(tµ0/∥µ0∥) : t ∈ D}
in the universal Teichmüller space T.
5. Proof of Theorem 4
We first prove this theorem for D = H (and hence for the disk). In
this canonical case, one gets a somewhat stronger result; moreover, the
arguments are simpler and illustrate all underlying features. Now the
poles of rn are real, and λH(z) = 1/|z − z| = 1/(2y).
The assertion of the theorem follows from the next two lemmas. The
first lemma ensures the existence for any rn ∈ B2(D) of a sequence of
points zn ∈ D convergent to a boundary point a0 on which the supremum
192 Extremal quasiconformality vs bounded rational...
of λ−2
D |rn| is attained (such a0 can be distinct from the poles of rn). The
second lemma yields that this a0 must be an essential point for rn, and
therefore this function represents a non-Strebel point.
Of course, all this is valid to much more general functions from B2(D).
A special case (the convex hull of fractions 1/(z − a)2 with real a) was
considered in [15].
Lemma 1. For any simply connected domain D ⊂ Ĉ with quasiconformal
boundary L and any rational function rn with poles of order two on L of
the form (3),
∥rn∥B2(D) = lim sup
z→L
λD(z)
−2|rn(z)|. (12)
So, there is a boundary point z0 at which the maximal value of
λD(z)
−2|rn(z)| on D is attained.
Proof. Consider first the case D = D, and let rn ∈ B2(D) satisfy
lim sup
|z|→1
(1− |z|2)2|rn(z)| < sup
z∈D
(1− |z|2)2|rn(z)|, (13)
i.e., the polyanalytic function
F (z) = (1− zz)2|rn(z)|
with F (0) = rn(0) attains its maximal value on D at some inner point
z0 ∈ D. Applying, if needed, the conformal automorphism
z 7→ (z − z0)/(1− z0z)
of D, one reduces the proof to the case z0 = 0.
If r′n(0) = a ̸= 0, then rn(z) = rn(0)+az+. . . , and hence, for z = ρeiθ
and small ρ > 0,
max
θ
|rn(ρeiθ)| = |rn(0)|+ |a|ρ+O(ρ2).
This yields
max
θ
|F (ρeiθ)| = |F (0)|+ |a|ρ+O(ρ2) > |F (0)|, ρ→ 0,
which contradicts the maximality of |F (z)| at z = 0. So, for such rational
functions rn, the inequality (13) can never occur, and
lim sup
|z|→1
(1− |z|2)2|rn(z)| = sup
z∈D
(1− |z|2)2|rn(z)| = ∥rn∥B2(D).
S. L. Krushkal 193
If r′n(0) = 0, we approximate this function by rational rn,ε with the
same poles ak ∈ ∂D, replacing one of the coefficients ck by ck + ε so that
r′n,ε(0) ̸= 0.
Since at the point z0, where the function
Fε(z) = (1− zz)2|rn,ε(z)|
attains its maximal value, this value is positive, one can define in a neigh-
borhood of z0 a single valued branch gn,ε(z) =
√
rn,ε(z), and in this
neighborhood
Fε(z) = (1− zz)2gn,ε(z)gn,ε(z).
Noting that both partial derivatives ∂zFε(z), ∂zFε(z) vanish at z0 and
∂zFε(z) = −2(1− zz)zgn,ε(z)gn,ε(z) + (1− zz)2gn,ε(z)g
′
n,ε(z),
one obtains
−2z0gn,ε(z0) + (1− z0z0)g
′
n,ε(z0) = 0,
and therefore,
g′n,ε(z0) = − 2z0
1− z0z0
gn,ε(z) ̸= 0.
This yields, in the same manner as above, that every such rn,ε satisfies
the equality (12). Since this equality remains valid in the limit as ε→ 0,
the assertion of Lemma for D = D is established.
The case of the generic quasidiskD is reduced to the above one, taking
a conformal map χD function of D onto the disk D with χD(z0) = 0 and
applying the above arguments to functions rn ◦ χ, which have the same
properties as rn. This completes the proof of the lemma. �
Note that rn(z) = O(1/z2) as z → ∞, so the quadratic differential
rn(z)dz
2 has at the infinite point a pole of the second order. If the
boundary of domain D contains z = ∞, then the maximal value in (12)
can be obtained at this point (and accordingly, (1 − |ζ|2)2|rn(ζ)χD(ζ)|
can attain its maximum at ζ = χD(∞)).
Lemma 2. Let D be a simply connected domain on Ĉ with quasiconfor-
mal boundary L and such that ∥SgD∥B2 < 1/2. There exists a constant
c0 > 0 such that for any rational function rn with poles of order two on
L of the form (3) and with
∥rn∥B2(D) < c0
194 Extremal quasiconformality vs bounded rational...
the boundary points of D at which the maximal value in (12) is attained
are substantial for extremal quasiconformal extensions of conformal im-
mersions f : D → Ĉ generated by the Schwarzian equation Sf = rn on
D.
Proof. It is sufficient to prove the lemma for domains with boundaries
containing ∞. We first consider the canonical case D = H for which a
somewhat stronger result will be obtained.
The equation Sf (z) = φ(z) defines the conformal immersion fφ : H →
Ĉ determined uniquely by the requirement to preserve the points 0, 1,∞.
By the Ahlfors–Weill theorem [1], every φ ∈ B2(H) with ∥φ∥ < 1/2
is the Schwarzian derivative Sf of a univalent function f in H, and f has
a quasiconformal extension onto the lower half-plane H∗ with Betrami
coefficient of the form
µφ(z) = −2y2φ(z), φ = Sf (z = x+ iy ∈ H) (14)
called the harmonic Beltrami coefficient (in the spirit of the Kodaira–
Spencer deformation theory).
Our aim is to show that for every rn with real poles aj of order two
and ∥rn∥B2(H) < 1/2 the corresponding harmonic Beltrami coefficient µrn
in H is extremal in its class, and
κ(frn ◦ σ) = k(frn) = ∥rn∥B2(H∗), (15)
where σ is the appropriate Moebius map of D∗ onto H∗.
It suffices to establish the relations (15) for rn with sufficiently small
norm.
Pick the point a0 ∈ R at which the equality in (12) is attained, and
two points x′, x′′ located on R in the left to all poles aj (so, µrn(z) = 0
on [x′, x′′]). We now establish that
sup
∥ψ∥A2(H)=1
|⟨µrn , ψ⟩H| = sup
∥ψ∥
A2
2(H)
=1
|⟨µrn , ψ⟩H| = Ha0(f), (16)
which implies the equalities (15) and extremality of µrn in its class.
Using the conformal map z = g(ζ) of the half-strip
Π+ = {ζ = ξ + iη : ξ > 0, 0 < η < 1}
onto H with g(x′) = 0, g(x′′) = 1, g(a0) = ∞, we pull-back µrn/b(a0)
(where b(a0) is the local boundary dilatation at the point a0) to the
Beltrami coefficient
µ∗(ζ) :=
1
Ha0
g∗(µrn)(ζ) =
1
Ha0
(µrn ◦ g)(ζ) g′(ζ)/g′(ζ)
S. L. Krushkal 195
on Π+, which satisfies lim
ξ→∞
|µ∗(ξ + iη)| = ∥µ∗∥∞ = 1 and has the limit
function
µ∗(ζ0) = lim
ζ→ζ0∈∂Π+
µ∗(ζ)
with
µ∗(iη) = 0. (17)
We claim that the sequence
ωm(ζ) =
1
m
e−ζ/m, m = 1, 2, . . . (ζ ∈ Π+),
is degenerating for ν∗. First of all, these ωm belong to A2
2(Π+); ωm(ζ) →
0 uniformly on Π+ ∩ {|ζ| < M} for any M < ∞, and ∥ωm∥A2(Π+) = 1.
Further,
⟨µ∗, ωm⟩Π+ =
1
m
∫∫
Π+
µ∗(ζ)ωm(ζ)dξdη
=
1∫
0
e−iη/mdη
( 1
m
∞∫
0
µ∗(ξ + iη)e−ξ/mdξ
)
. (18)
The inner integral can be evaluated using the Laplace transform of µ∗ in
ξ. Integrating by parts and applying (17), one obtains
∞∫
0
∂µ∗(ξ + iη)
∂ξ
e−ξ/mdξ =
1
m
∞∫
0
µ∗(ξ + iη)e−ξ/mdξ.
On the other hand, Abel’s theorem for the Laplace transform yields that
the nontangential limit
lim
s→0
∞∫
0
∂µ∗(ξ + iη)
∂ξ
e−sξdξ =
∞∫
0
∂µ∗(ξ + iη)
∂ξ
dξ = µ∗(∞)− µ∗(iη);
hence,
lim
m→∞
1
m
∞∫
0
µ∗(ξ + iη)e−ξ/mdξ = µ∗(∞).
By Lebesgue’s theorem on dominated convergence, the iterated integral
in (18) is estimated as follows
lim
m→∞
|⟨ν∗, ωm⟩Π+ | =
∣∣∣ 1∫
0
dη lim
m→∞
1
m
∞∫
0
µ∗(ξ + iη)e−ξ/mdξ
∣∣∣ = 1. (19)
196 Extremal quasiconformality vs bounded rational...
Since by (12), the left-hand side equals to ∥µ∗∥∞ and all functions ωm
belong to A2
2(Π+), this proves our claim.
Now, applying the inverse conformal map ζ = g−1(z) : Π+ → H, one
obtains the degenerating sequence
{ψm = (ωm ◦ g−1)(g′)−2} ⊂ A2
2(H),
for the initial Beltrami coefficient µrm on H. By (19),
lim
m→∞
|⟨µrm , ψm⟩H| = ∥µrm∥∞ = 1,
which implies, together with Lemma 1, the assertion of Theorem 4 for
the half-plane.
Let now D be the generic simply connected domain bounded by qua-
sicircle L passing through 0, 1,∞ and such that the Schwarzian of the
conformal map gD of H∗ onto D preserving the points 0, 1,∞ satisfies
∥SgD∥B2(H∗) < 1/2.
Given a rational function rn(z) of the form (3) with poles on L, we
consider the univalent solution w = fn(z) of the equation
Sw(z) = trn(z), z ∈ D
and its composition with gD, taking t > 0 so small that
Sfn◦gD = (Sf ◦ gD)(g′D)2 + SgD (20)
also satisfies
∥Sfn◦gD∥B2(H∗) < 1/2. (21)
The Beltrami coefficients of arbitrary quasiconformal extensions ĝD and
f̂n of gD and fn, respectively, across the boundaries of their domains to
Ĉ are related by
µ
f̂n
◦ ĝD = µ
f̂n◦ĝ−1
D
◦ ĝD =
µ
f̂n◦ĝD − µĝD
1− µĝDµf̂n◦ĝD
∂ζ ĝD
∂ζ ĝD
.
In particular, using their Ahlfors–Weill extensions (14), one gets, in view
of (20),
µ
f̂n
◦ ĝD = −2tη2rn ◦ gD(η)
∂ζ ĝD
∂ζ ĝD
+O(t2) = −2tλ−2
D rn +O(t2),
S. L. Krushkal 197
or equivalently, for z = gD(ζ),
µ
f̂
(z) = −2tλ−2
D (z)rn(z) +O(t2) as t→ 0.
The remainders in the last two equalities are uniformly bounded in L∞
norm for all t for which the bound (21) is valid.
We establish now that the harmonic Beltrami coefficient
νtrn(z) = tλ−2
D (z)rn(z) (22)
is infinitesimally extremal in its equivalence class.
Indeed, taking again the point a0 ∈ L on which the upper limit (12)
for chosen rn is attained and mapping the domain D conformally onto
the half-strip Π+ so that a0 is going to ∞, one can repeat for fn the above
arguments and derive from (16) and (19) that the boundary dilatation at
a0 is equal to ∥rn∥B2(D). This yields that the Beltrami coefficient (22) is
infinitesimally extremal in Belt(D)1 and, moreover, its norm is attained
on functions from A2
2(D).
As was mentioned in Section 4.1, such a Beltrami coefficient must be
simultaneously globally extremal in its equivalence class. This implies
the assertions of Lemma 2 and of Theorem 4, completing their proofs. �
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Contact information
Samuel L. Krushkal Department of Mathematics, Bar-Ilan Uni-
versity, Israel,
Department of Mathematics, University of
Virginia, Charlottesville, USA
E-Mail: krushkal@math.biu.ac.il
|
| id | nasplib_isofts_kiev_ua-123456789-169439 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-07T13:32:42Z |
| publishDate | 2019 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Krushkal, S.L. 2020-06-13T10:16:25Z 2020-06-13T10:16:25Z 2019 Extremal quasiconformality vs bounded rational approximation / S.L. Krushkal // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 181-199. — Бібліогр.: 26 назв. — англ. 1810-3200 2000 MSC. Primary: 30C62, 30C75, 30E10; Secondary: 30F45, 30F60, 32G15 https://nasplib.isofts.kiev.ua/handle/123456789/169439 We show that, on most of the hyperbolic simply connected domains, the weighty bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to the integral approximation). The obstructions are caused by the features of extremal quasiconformality. The paper is devoted to the 100th anniversary of Georgii Dmitrievich Suvorov, my first university adviser and teacher. He was an outstanding mathematician and a widely talented, extremely great human being. en Інститут прикладної математики і механіки НАН України Український математичний вісник Extremal quasiconformality vs bounded rational approximation Article published earlier |
| spellingShingle | Extremal quasiconformality vs bounded rational approximation Krushkal, S.L. |
| title | Extremal quasiconformality vs bounded rational approximation |
| title_full | Extremal quasiconformality vs bounded rational approximation |
| title_fullStr | Extremal quasiconformality vs bounded rational approximation |
| title_full_unstemmed | Extremal quasiconformality vs bounded rational approximation |
| title_short | Extremal quasiconformality vs bounded rational approximation |
| title_sort | extremal quasiconformality vs bounded rational approximation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169439 |
| work_keys_str_mv | AT krushkalsl extremalquasiconformalityvsboundedrationalapproximation |