Spectral Estimates of the $p$-Laplace Neumann operator and Brennan's Conjecture

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by  Vladimir Gol'dshtein

SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN
OPERATOR AND BRENNAN’S CONJECTURE
7
1
V.GOL’DSHTEIN,V.PCHELINTSEV,A.UKHLOV
0
2
Abstract. In this paper we obtain estimates for the first nontrivial eigen-
n
valueofthep-LaplaceNeumannoperatorinboundedsimplyconnectedplanar
a domains Ω ⊂ R2. This study is based on a quasiconformal version of the
J
universal weightedPoincaré-Sobolevinequalitiesobtainedinourprevious pa-
8
pers for conformal weights. The suggested weights in the present paper are
1
Jacobians of quasiconformal mappings. The main technical tool is the the-

ory of composition operators in relation with the Brennan’s Conjecture for
]
P (quasi)conformal mappings.
A
.
h
t 1. Introduction
a
m
In this paper we obtain lower estimates for the first nontrivial eigenvalue of the
[ p-Laplace Neumann operator with the Neumann boundary condition

1 div( up 2 u)=µ up 2u in Ω
− p −
v − |∇ | ∇ | |
3 (∂u =0 on ∂Ω,
∂n
4
in a bounded simply connected planar domain Ω R2. The weak statement of
1
⊂
5 this spectral problem is as follows: a function u solves the previous problem iff
0 u W1(Ω) and
∈ p
.
1
0 ˆ (|∇u(x)|p−2∇u(x)·∇v(x))dx =µpˆ |u(x)|p−2u(x)v(x)dx
7
Ω Ω
1
v: for all v ∈Wp1(Ω).
WedemonstratethatintegrabilityofJacobiansofquasiconformalmappingsper-
i
X mit us to obtain lowerestimates of the first non-trivialeigenvalue µ(1)(Ω) in terms
p
r of Sobolev norms of a quasiconformal mapping of the unit disc D onto Ω. So, we
a can conclude that µ(1)(Ω) depends on the quasiconformal geometry of Ω:
p
Theorem A.Let Ω R2 be a K-quasiconformal α-regular domain and ϕ:Ω
⊂ →
D be a K-quasiconformal mapping. Suppose that the Brennan’s Conjecture holds.
Then for every
4K α(2K 1)+2
p max , − ,2
∈ 2K+1 αK
(cid:18) (cid:26) (cid:27) (cid:19)
the following estimate
1
µ(p1)(Ω) ≤KkJϕ−1 |Lα2(D)kqin∈fInBpαα−p2,q(D)k|Dϕ−1|p−2 |Lp−qq(D)ko
0
Key words and phrases: ellipticequations, Sobolevspaces,quasiconformalmappings.
0
2010 Mathematics Subject Classification: 35P15,46E35, 30C65.
1
SPECTRAL PROPERTIES 2
holds.
Here I = [1,2p/(4K (2K 1)p)) and B (D) is the best constant in the
r,q
− −
corresponding(r,q)-Poincaré-Sobolevinequalityintheunitdisc Dforr =αp/(α
−
2).
Let us give few detailed comments to the theorem:
1.1 K-quasiconformal α-regular domains. Recall that a homeomorphism
ϕ : Ω Ω between planar domains is called K-quasiconformal if it preserves
→
orientation, belongs to the Sobolev class W1 (Ω) and its directional derivatives
2,loc
∂ satisfy tehe distortion inequality
α
max ∂ ϕ Kmin ∂ ϕ a.e. in Ω.
α α
α | |≤ α | |
The notionof conformalregulardomains wasintroducedin [7] andwasused for
study conformal spectral stability of the Laplace operator. In the present work we
introduce the more general notion of quasiconformal regular domains.
A simply connected planar domain Ω R2 is called a K-quasiconformal α-
⊂
regular domain if there exists a K-quasiconformalmapping ϕ:Ω D such that
→
J(y,ϕ−1) α2 dy < for some α>2.
ˆ | | ∞
D
The domain Ω R2 is called a K-quasiconformal regular domain if it is a K-
⊂
quasiconformalα-regular domain for some α>2.
Note that class of quasiconformal regular domain includes the class of Gehring
domains [2] and can be described in terms of quasihyperbolic geometry [20].
Remark 1.1. The notion of quasiconformal α-regular domain is more general then
the notion of conformal α-regular domain. Consider, for example, the unit square
Q R2. Then Q is a conformal α-regular domain for 2 < α 4 [16] and is a
⊂ ≤
quasiconformal α-regular domain for all 2 < α because the unit square Q is
≤ ∞
quasiisometrically equivalent to the unit disc D.
Remark 1.2. Because ϕ : Ω D is a quasiconformal mapping, then integrability
→
of the derivative is equivalent to integrability of the Jacobian:
J(y,ϕ−1) α2 dy Dϕ−1(y)α dy Kα2 J(y,ϕ−1) α2 dy.
ˆ | | ≤ˆ | | ≤ ˆ | |
D D D
1.2 Brennan’s Conjecture. WecanconcludefromthetheoremAthatBren-
nan’s Conjecture leads to the spectral estimates of the p-Laplace operator in qua-
siconformal α-regular domains Ω R2.
⊂
Generalized Brennan’s Conjecture for quasiconformalmappings [22] states
that
4K 2Kβ
(1.1) Dϕ(x)β dx<+ , for all <β < 0 .
ˆ | | ∞ 2K+1 (K 1)β +2
0
−
Ω
If K = 1 we have Brennan’s conjecture for conformal mappings [6] which is
proved for β (4/3,β ), where β =3.752 [18].
0 0
∈
1.3 Historical sketch. In 1961 G.Polya [25] obtained upper estimates for
eigenvaluesofNeumann-Laplaceoperatorinso-calledplane-coveringdomains. Namely,
SPECTRAL PROPERTIES 3
for the first eigenvalue:
µ(1)(Ω) 4π Ω 1.
2 ≤ | |−
The lowerestimates forthe µ(1)(Ω) wereknownbeforeonly forconvexdomains.
p
Inthe classicalwork[26] itwasprovedthat ifΩ is convexwithdiameter d(Ω) (see,
also [8, 9, 28]), then
π2
(1.2) µ(1)(Ω) .
2 ≥ d(Ω)2
In [8] was proved that if Ω Rn is a bounded convex domain having diameter
⊂
d then for p 2
≥ p
π
µ(1)(Ω) p
p ≥ d(Ω)
(cid:18) (cid:19)
where
1
(p 1)p
− dt (p 1)p1
π =2 =2π − .
p ˆ (1 tp/(p 1))p1 psin(π/p)
0 − −
In the case of non-convex domains in [17] was proved that if Ω R2 be a con-
⊂
formalα-regular domain then for every p (max 4/3,(α+2)/α ,2) the following
∈ { }
inequality holds
1
µ(1)(Ω) ≤
p
q∈[1,2ipn/f(4−p))nBpαα−p2,q(D)·kDϕ−1|p−2|Lp−qq(D)kokDϕ−1||Lα(D)k2.
The first non-trivial eigenvalue of the Neumann boundary problem for the p-
Laplaceoperatorµ(p1)(Ω)−p1 isequaltothebestconstantBp,p(Ω)(see,forexample,
[5]) in the p-Poincaré-Sobolev inequality
inf f c L (Ω) B (Ω) f L (Ω) , f W1(Ω).
c R|| − | p ||≤ p,p ||∇ | p || ∈ p
∈
1.4Methods. Thesuggestedmethodisbasedonthecompositionoperatorsthe-
ory in relationwith Brennan’s Conjecture that allows to obtain universalweighted
Poincaré-Sobolev inequalities in any simple connected domain Ω = R2 for quasi-
6
conformalweights h(x):= J(x,ϕ) generatedby quasiconformalhomeomorphisms
| |
ϕ : Ω D (Theorem 4.1). In quasiconformal regular domains these weighted in-
→
equalities imply non-weighted Poincaré-Sobolev inequalities. This method based
on the theory of composition operators[27, 32] and its applications to the Sobolev
type embedding theorems [11, 12].
The following diagram illustrates this idea:
W1(Ω) ϕ∗ W1(D)
p −→ q
↓ ↓
L (Ω) (ϕ−1)∗ L (D).
s r
←−
Heretheoperatorϕ definedbythecompositionruleϕ (f)=f ϕisabounded
∗ ∗
◦
composition operator on Sobolev spaces induced by a homeomorphism ϕ of Ω and
D and the operator (ϕ 1) defined by the composition rule (ϕ 1) (f) = f ϕ 1
− ∗ − ∗ −
◦
SPECTRAL PROPERTIES 4
is a bounded composition operator on Lebesgue spaces. This method allows to
transferPoincaré-Sobolevinequalitiesfromregulardomains(forexample,fromthe
unit disc D) to Ω.
Theorem B.Let Ω R2 be a K-quasiconformal α-regular domain and ϕ:Ω
⊂ →
D be a K-quasiconformal mapping. Suppose that the Brennan’s Conjecture holds.
Then for every
4K α(2K 1)+2
p max , − ,2
∈ 2K+1 αK
(cid:18) (cid:26) (cid:27) (cid:19)
the p-Poincaré-Sobolev inequality
inf f c L (Ω) B (Ω) f L (Ω) , f W1(Ω),
c R|| − | p ||≤ p,p ||∇ | p || ∈ p
∈
holds with the constant
Bpp,p(Ω)≤qin∈fI(cid:26)Bpαα−p2,q(D)·K |Dϕ−1|p−2 |Lp−qq(D) · Jϕ−1 |Lα2(D) (cid:27).
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
Remark 1.3. In the Introduction we formulate main results under the assump-
tions that the Brennanś conjecture holds true 4K/(2K+1) < β < 4K/(2K 1).
−
In the main part of the paper we proved main results for 4K/(2K +1) < β <
2Kβ /(β (K 1)+2) for β that is a recent value for which the Brennanś conjec-
0 0 0
−
ture is proved.
The next main theorem establish a connection between Brennan’s Conjecture
and composition operators on Sobolev spaces:
Theorem C.Let Ω R2 be a simply connected domain. Generalized Brennan’s
⊂
Conjecture holds for a number β (4K/(2K +1), 4K/(2K 1)) if and only if
∈ −
any K-quasiconformal homeomorphism ϕ : Ω D induces a bounded composition
→
operator
ϕ :L1(D) L1(Ω)
∗ p → q
for any p (2,+ ) and q =pβ/(p+β 2).
∈ ∞ −
IntherecentworkswestudycompositionoperatorsonSobolevspacesdefinedon
planar domains in connection with the conformal mappings theory [13]. This con-
nectionleadstoweightedSobolevembeddings[14,15]withtheuniversalconformal
weights. Another application of conformal composition operators was given in [7]
wherethespectralstabilityproblemforconformalregulardomainswasconsidered.
2. Composition operators and quasiconformal mappings
InthissectionwerecallbasicfactsaboutcompositionoperatorsonLebesgueand
Sobolev spaces and the quasiconformal mappings theory. Let Ω Rn, n 2, be
⊂ ≥
a n-dimensional Euclidean domain. For any 1 p < we consider the Lebesgue
≤ ∞
space L (Ω) of measurable functions f :Ω R equipped with the following norm:
p
→
1
p
f L (Ω) = f(x)pdx < .
k | p k ˆ | | ∞
(cid:18) (cid:19)
Ω
The following theorem about composition operators on Lebesgue spaces is well
known (see, for example [32]):
SPECTRAL PROPERTIES 5
Theorem 2.1. Let ϕ : Ω Ω be a weakly differentiable homeomorphism between
→
two domains Ω and Ω. Then the composition operator
e
ϕ :L (Ω) L (Ω), 1 s r < ,
∗ r s
e → ≤ ≤ ∞
is bounded, if and only if ϕ 1 possesses the Luzin N-property and
−
e
r s
r r−s
ˆ |J(y,ϕ−1)| r−s dy =K <∞, 1≤s<r <∞,
(cid:18)e (cid:19)
Ω (cid:0) (cid:1)
1
esssup J(y,ϕ 1) s =K < , 1 s=r < .
−
e | | ∞ ≤ ∞
y Ω
∈ (cid:0) (cid:1)
The norm of the composition operator ϕ =K.
∗
k k
We consider the Sobolev space W1(Ω), 1 p< , as a Banach space of locally
p ≤ ∞
integrable weakly differentiable functions f : Ω R equipped with the following
→
norm:
1 1
p p
f W1(Ω) = f(x)pdx + f(x)pdx .
k | p k ˆ | | ˆ |∇ |
(cid:18) (cid:19) (cid:18) (cid:19)
Ω Ω
Recall that the Sobolev space W1(Ω) coincides with the closer of the space of
p
smooth functions C (Ω) in the norm of W1(Ω).
∞ p
WeconsideralsothehomogeneousseminormedSobolevspaceL1(Ω),1 p< ,
p ≤ ∞
of locally integrable weakly differentiable functions f : Ω R equipped with the
→
following seminorm:
1
p
f L1(Ω) = f(x)pdx .
k | p k ˆ |∇ |
(cid:18) (cid:19)
Ω
Recall that the embedding operator i:L1(Ω) L (Ω) is bounded.
p → 1,loc
By the standard definition functions of L1(Ω) are defined only up to a set of
p
measure zero, but they can be redefined quasieverywhere i. e. up to a set of p-
capacity zero. Indeed, every function u L1(Ω) has a unique quasicontinuous
∈ p
representation u˜ L1(Ω). A function u˜ is termed quasicontinuous if for any ε>0
∈ p
there is an open set U such that the p-capacity of U is less then ε and on the set
ε ε
Ω U the function u˜ is continuous (see, for example [19, 24]).
ε
\
Let Ω and Ω be domains in Rn. We say that a homeomorphism ϕ : Ω Ω
→
induces a bounded composition operator
e e
ϕ :L1(Ω) L1(Ω), 1 q p ,
∗ p → q ≤ ≤ ≤∞
by the compositionrule ϕ (f)=f ϕ, if the compositionϕ (f) L1(Ω) is defined
∗ e ◦ ∗ ∈ q
quasi-everywherein Ω and there exists a constant K (Ω)< such that
p,q
∞
kϕ∗(f)|L1q(Ω)k≤Kp,q(Ω)kf |L1p(Ω)k
for any function f L1(Ω) [33].
∈ p e
Let Ω Rn be an open set. A mapping ϕ : Ω Rn belongs to L1 (Ω),
⊂ → p,loc
1 p , if its coordeinate functions ϕ belong to L1 (Ω), j = 1,...,n. In
≤ ≤ ∞ j p,loc
this case the formal Jacobi matrix Dϕ(x) = ∂ϕi(x) , i,j = 1,...,n, and its
∂xj
determinant (Jacobian) J(x,ϕ) = detDϕ(x) ar(cid:16)e well d(cid:17)efined at almost all points
SPECTRAL PROPERTIES 6
x Ω. The norm Dϕ(x) of the matrix Dϕ(x) is the norm of the corresponding
∈ | |
linear operator Dϕ(x):Rn Rn defined by the matrix Dϕ(x).
→
Let ϕ:Ω Ω be weakly differentiable in Ω. The mapping ϕ is the mapping of
→
finite distortion if Dϕ(z) =0 for almost all x Z = z Ω:J(x,ϕ)=0 .
| | ∈ { ∈ }
A mapping ϕe: Ω Rn possesses the Luzin N-property if a image of any set
→
of measure zero has measure zero. Mote that any Lipschitz mapping possesses the
Luzin N-property.
The following theorem gives the analytic description of composition operators
on Sobolev spaces:
Theorem 2.2. [27, 32] A homeomorphism ϕ:Ω Ω between two domains Ω and
→
Ω induces a bounded composition operator
e
e ϕ∗ :L1p(Ω)→L1q(Ω), 1≤q <p<∞,
if and only if ϕ W1 (Ω), haes finite distortion, and
∈ 1,loc
p q
q p−q
Dϕ(x)p p q
K (Ω)= | | − dx < .
p,q ˆ J(x,ϕ)  ∞
(cid:18)| |(cid:19)
Ω
 
Recall that a homeomorphismϕ:Ω Ω is called a K-quasiconformalmapping
→
if ϕ W1 (Ω) and there exists a constant 1 K < such that
∈ n,loc ≤ ∞
e
Dϕ(x)n K J(x,ϕ) for almost all x Ω.
| | ≤ | | ∈
Quasiconformalmappings havea finite distortion,i.e. Dϕ(x)=0 for almostall
points x that belongs to set Z = x Ω : J(x,ϕ) = 0 and any quasiconformal
{ ∈ }
mapping possesses Luzin N-property. A mapping which is inverse to a quasicon-
formal mapping is also quasiconformal.
If ϕ : Ω Ω is a K-quasiconformal mapping then ϕ is differentiable almost
→
everywhere in Ω and
e
ϕ(B(x,r))
J(x,ϕ) =J (x):= lim | | for almost all x Ω.
ϕ
| | r 0 B(x,r) ∈
→ | |
Note, that a homeomorphism ϕ:Ω Ω is a K-quasiconformal mapping if and
→
onlyifϕgeneratesbythecompositionruleϕ (f)=f ϕanisomorphismofSobolev
∗
◦
spaces L1(Ω) and L1(Ω): e
n n
K−n1kfe|L1n(Ω)k≤kϕ∗f |L1n(Ω)k≤Kn1kf |L1n(Ω)k
for any f L1(Ω) [29]. e e
∈ n
Compositions of Sobolev functions of the spaces L1(Ω), p = n, with quasicon-
p ′ 6
formal mappingsewas studied also in [21].
For any planar K-quasiconformal homeomorphism ϕ : Ω Ω, the following
→
sharp results is known: J(x,ϕ) L (Ω) for any α <K/(K 1) ([1]).
α ,loc ∗
∈ ∗ −
If K 1 then 1-quasiconformal homeomorphisms are conformael mappings and
≡
in the space Rn, n 3, are exhausted byeMöbius transformations.
≥
SPECTRAL PROPERTIES 7
3. Composition operators and Brennan’s Conjecture
Brennan’s Conjecture [6] is that if ϕ : Ω D is a conformal mappings of a
→
simply connected planar domain Ω, Ω=R2, onto the unit disc D then
6
4
(3.1) ϕ(x)β dx<+ , for all <β <4.
′
ˆ | | ∞ 3
Ω
For 4/3<s<3, it is a comparatively easy consequence of the Koebe distortion
theorem(see,forexample,[4]). J.Brennan[6](1973)extendedthis rangeto4/3<
s < 3+δ, where δ > 0, and conjectured it to hold for 4/3 < s < 4. The example
of Ω=C ( , 1/4] shows that this range of s cannot be extended.
\ −∞ −
Brennan’s Conjecture proved for β (4/3,β ), were β =3.752 [18]. Brennan’s
0 0
∈
Conjectureforquasiconformalmappingswasconsideredin[22]. In[22]wasproved,
that if ϕ:Ω D be a K-quasiconformalmapping, then
→
4K 2Kβ
(3.2) Dϕ(x)β dx<+ , for all <β < 0 .
ˆ | | ∞ 2K+1 (K 1)β +2
0
−
Ω
Here β is the proved upper bound for Brennan’s Conjecture.
0
Now we prove, that Generalized Brennan’s Conjecture leads to boundedness of
composition operators on Sobolev spaces generates by quasiconformal mappings.
Theorem C.Let Ω R2 be a simply connected domain. Generalized Brennan’s
⊂
Conjectureholds foranumberβ (4K/(2K+1), 2Kβ /(β (K 1)+2))ifandonly
0 0
∈ −
ifanyK-quasiconformalhomeomorphismϕ:Ω Dinducesaboundedcomposition
→
operator
ϕ :L1(D) L1(Ω)
∗ p → q
for any p (2,+ ) and q =pβ/(p+β 2).
∈ ∞ −
Proof. By the composition theorem [27, 32] a homeomorphism ϕ:Ω D induces
→
a bounded composition operator
ϕ :L1(D) L1(Ω), 1 q <p< .
∗ p → q ≤ ∞
if and only if ϕ W1 (Ω), has finite distortion and
∈ 1,loc
p q
q p−q
Dϕ(x)p p q
K (Ω)= | | − dx < .
p,q ˆ J(x,ϕ)  ∞
(cid:18)| |(cid:19)
Ω
 
Becauseϕisaquasiconformalmapping,thenϕ W1 (Ω)andJacobianJ(x,ϕ)=
∈ n,loc 6
0 for almost all x Ω. Hence the p-dilatation
∈
Dϕ(x)p
K (x)= | |
p
J(x,ϕ)
| |
is well defined for almost all x Ω and so ϕ is a mapping of finite distortion.
∈
By Brennan’s Conjecture
4K 2Kβ
Dϕ(x)β dx<+ , for all <β < 0 .
ˆ | | ∞ 2K+1 (K 1)β +2
0
−
Ω
SPECTRAL PROPERTIES 8
Then
q q
pq Dϕ(x)p p q Dϕ(x)2 p q
Kpp,−qq(Ω)=ˆ |J(x,ϕ)| − dx=ˆ |J(x,ϕ)| |Dϕ(x)|p−2 − dx
(cid:18)| |(cid:19) (cid:18)| | (cid:19)
Ω Ω
≤Kp−qq ˆ |Dϕ(x)|p−2 p−qq dx=Kp−qq ˆ |Dϕ(x)|β dx<∞,
Ω (cid:0) (cid:1) Ω
for β =(p 2)q/(p q). Hence we have a bounded composition operator
− −
ϕ :L1(D) L1(Ω)
∗ p → q
for any p (2,+ ) and q =pβ/(p+β 2).
∈ ∞ −
Let us check that q <p. Because p >2 we have that p+β 2>β > 0 and so
−
β/(p+β 2)<1. Hence we obtain q <p.
−
Now, let the composition operator
ϕ :L1(D) L1(Ω), q <p,
∗ p → q
isboundedforanyp (2,+ )andq =pβ/(p+β 2). Then,usingtheHadamard
∈ ∞ −
inequality:
J(x,ϕ) Dϕ(x)2 for almost all x Ω,
| |≤| | ∈
and Theorem 2.2, we have
q
Dϕ(x)β dx= Dϕ(x) (pp−2q)q dx |Dϕ(x)|p p−q dx<+ .
ˆ | | ˆ | | − ≤ˆ J(x,ϕ) ∞
(cid:18)| |(cid:19)
Ω Ω Ω
(cid:3)
The suggested approach to the Poincaré-Sobolev type inequalities in bounded
planar domains Ω R2 is based on translation of these inequalities from the unit
⊂
disc D to Ω. On this way we use the following duality [31]:
Theorem 3.1. Let a homeomorphism ϕ : Ω Ω, Ω,Ω R2, generates by the
′ ′
→ ⊂
composition rule ϕ (f)=f ϕ a bounded composition operator
∗
◦
ϕ :L1(Ω) L1(Ω), 1<q p< ,
∗ p ′ → q ≤ ∞
thentheinversemappingϕ 1 :Ω Ωgeneratesbythecompositionrule(ϕ 1) (g)=
− ′ − ∗
→
g ϕ 1 a bounded composition operator
−
◦
1 1 1 1
(ϕ−1)∗ :L1q′(Ω)→L1p′(Ω′), q + q =1,p + p =1.
′ ′
From Theorem C and Theorem 3.1 immediately follow Theorem 3.2:
Theorem 3.2. Let Ω R2 be a simply connected domain and ϕ : Ω D be a
⊂ →
K-quasiconformal homeomorphism. Suppose that p (2Kβ /((K+1)β 2),2).
0 0
∈ −
Then the inverse mapping ϕ 1 induces a bounded composition operator
−
(ϕ 1) :L1(Ω) L1(D)
− ∗ p → q
for any q such that
(2β 4)p 2p
0
1 q − < .
≤ ≤ 2Kβ ((K 1)β +2)p 4K (2K 1)p
0 0
− − − −
SPECTRAL PROPERTIES 9
The inequality
p q
p−q
(3.3) ||(ϕ−1)∗f|L1q(D)||≤Kp1 ˆ |Dϕ−1(y)|(pp−−2q)qdy ||f|L1p(Ω)||
D
 
holds for any function f L1(Ω).
∈ p
Proof. By Theorem C we have a bounded composition operator
ϕ :L1 (D) L1 (Ω)
∗ q′ → p′
for q (2,+ ) and p =q β/(q +β 2).
′ ′ ′ ′
∈ ∞ −
Then, by Brennan’s Conjecture, p (2,2Kβ /((K 1)β +2)). Now using
′ 0 0
∈ −
Theorem 3.1 we have
p 2Kβ
′ 0
p= ,2 .
p 1 ∈ (K+1)β 2
′− (cid:18) 0− (cid:19)
Since
p q β
′ ′
p= = ,
p 1 q β (q +β+2)
′− ′ − ′
we obtain by direct calculations that
(4 2β )p
0
q′ = − .
2Kβ ((K+1)β 2)p
0 0
− −
By Theorem 3.1 q =q /(q 1) and q p. By elementary calculations
′ ′
− ≤
(2β 4)p 2p
0
1 q − < .
≤ ≤ 2Kβ ((K 1)β +2)p 4K (2K 1)p
0 0
− − − −
Now we provethe inequality (3.3). Let f L1(Ω) C (Ω). Then the composi-
∈ p ∩ ∞
tion g =(ϕ 1) (f) L1 (D) [30]. Hence, using Theorem 2.2 we obtain
− ∗ ∈ 1,loc
p q
q p−q
Dϕ 1(y)p p q
g L1(D) | − | − dy f L1(Ω)
|| | q ||≤ˆ J(y,ϕ 1)  || | p ||
D (cid:18)| − |(cid:19)
  p q
q p−q
Dϕ 1(y)2 Dϕ 1(y)p 2 p q
= | − | ·| − | − − dy f L1(Ω)
ˆ J(y,ϕ 1)  || | p ||
D (cid:18) | − | (cid:19)
  p q
p−q
≤Kp1 ˆ |Dϕ−1(y)|(pp−−2q)qdy ||f|L1p(Ω)||.
D
 
Approximating an arbitrary function f L1(Ω) by smooth functions [31, 32],
we obtain the required inequality. ∈ p (cid:3)
4. Poincaré-Sobolev inequalities
Weighted Poincaré-Sobolev inequalities. Let Ω R2 be a planar domain
⊂
and let v : Ω R be a real valued function, v > 0 a. e. in Ω. We consider the
→
SPECTRAL PROPERTIES 10
weighted Lebesgue space L (Ω,v), 1 p < , of measurable functions f :Ω R
p
≤ ∞ →
with the finite norm
1
p
f L (Ω,v) := f(x)pv(x)dx < .
k | p k ˆ | |  ∞
Ω
 
It is a Banach space for the norm f L (Ω,v) .
p
k | k
Using Theorem 3.2 we prove
Theorem 4.1. Suppose that Ω R2 is a simply connected domain and h(x) =
⊂
J(x,ϕ) is the quasiconformal weight defined by a K-quasiconformal homeomor-
| |
phism ϕ:Ω D.Then for every p (2Kβ /((K+1)β 2),2) and every function
0 0
→ ∈ −
f W1(Ω), the inequality
∈ p
1 1
r p
inf f(x) crh(x)dx B (Ω,h) f(x)pdx
c Rˆ | − |  ≤ r,p ˆ |∇ | 
∈
Ω Ω
   
holds for any r such that
p 2β 4
0
1 r < −
≤ 2 p · Kβ
0
−
with the constant
B (Ω,h) inf K (D) B (D) .
r,p p,q r,q
≤q [1,2p/(4K (2K 1)p)){ · }
∈ − −
HereB (D)isthebestconstantinthe(non-weeight)Poincaré-Sobolevinequality
r,q
in the unit disc D R2 and
⊂
p q
p−q
Kp,q(D)=Kp1 ˆ |Dϕ−1(y)|(pp−−2q)qdy .
D
e  
Proof. By conditions of the theorem there exists a K-quasiconformal mapping ϕ:
Ω D such that
→
4K 2Kβ
Dϕ(x)β dx<+ , for all <β < 0 .
ˆ | | ∞ 2K+1 (K 1)β +2
0
−
Ω
By Theorem 3.2, if
(2β 4)p 2p
0
(4.1) 1 q − <
≤ ≤ 2Kβ ((K 1)β +2)p 4K (2K 1)p
0 0
− − − −
then the inequality
(4.2) (f ϕ 1) L (D) K (D) f L (Ω)
− q p,q p
||∇ ◦ | ||≤ ||∇ | ||
holds for every function f L1(Ω).
∈ p e
Let f L1(Ω) C1(Ω). Then the function g = f ϕ 1 is defined almost
∈ p ∩ ◦ −
everywhereinDandbelongstotheSobolevspaceL1(D)[30]. Hence,bytheSobolev
q
embedding theorem g =f ϕ 1 W1,q(D) [24] and the classicalPoincaré-Sobolev
−
◦ ∈
inequality,
(4.3) inf f ϕ 1 c L (D) B (D) (f ϕ 1) L (D)
− r r,q − q
c R|| ◦ − | ||≤ ||∇ ◦ | ||
∈