báo cáo hóa học:" Research Article ´ ¨ Sharp Constants of Brezis-Gallouet-Wainger Type Inequalities with a Double Logarithmic Term on Bounded Domains in Besov and Triebel-Lizorkin Spaces" pdf

Volume 2010, Article ID 584521, 38 pagesdoi:10.1155/2010/584521 Research Article Sharp Constants of Br ´ezis-Gallou ¨et-Wainger Type Inequalities with a Double Logarithmic Term on Bounde

Volume 2010, Article ID 584521, 38 pages

doi:10.1155/2010/584521

Research Article

Sharp Constants of Br ´ezis-Gallou ¨et-Wainger

Type Inequalities with a Double Logarithmic Term

on Bounded Domains in Besov and

Triebel-Lizorkin Spaces

1 Heian Jogakuin St Agnes’ School, 172-2, Gochomecho, Kamigyo-ku, Kyoto 602-8013, Japan

2 Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

3 Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

4 Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku,

Osaka 558-8585, Japan

Received 4 October 2009; Revised 15 September 2010; Accepted 12 October 2010

Academic Editor: Veli B Shakhmurov

Commons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

The present paper concerns the Sobolev embedding in the endpoint case It is known that the

quantified why this embedding fails by means of the H ¨older-Zygmund norm In the present paper

we will give a complete quantification of their results and clarify the sharp constants for thecoefficients of the logarithmic terms in Besov and Triebel-Lizorkin spaces

1 Introduction and Known Results

We establish sharp Br´ezis-Gallou¨et-Wainger type inequalities in Besov and Triebel-Lizorkinspaces as well as fractional Sobolev spaces on a bounded domainΩ ⊂ Rn Throughout thepresent paper, we place ourselves in the setting ofRn with n≥ 2 We treat only real-valuedfunctions

First we recall the Sobolev embedding theorem in the critical case For 1 < q <∞, it is

well known that the embedding W n/q,qRn  → L rRn  holds for any q ≤ r < ∞, and does not hold for r  ∞, that is, one cannot estimate the L∞-norm by the W n/q,q-norm However, the

Br´ezis-Gallou¨et-Wainger inequality states that the L∞-norm can be estimated by the W n/q,q

-norm with the partial aid of the W s,p -norm with s > n/p and 1 ≤ p ≤ ∞ as follows:

u q/ q−1

L∞Rn ≤ λ1 log1 u W s,pRn



1.1

holds whenever u ∈ W n/q,qRn  ∩ W s,pRn  satisfies u W n/q,qRn  1, where 1 ≤ p ≤ ∞,

1 < q < ∞, and s > n/p Inequality 1.1 for the case n  p  q  s  2 dates back to

Br´ezis-Gallou¨et1 Later on, Br´ezis and Wainger 2 obtained 1.1 for the general case, and

remarked that the power q/q − 1 in 1.1 is maximal; equation 1.1 fails for any largerpower Ozawa 3 proved 1.1 with the Sobolev norm u W s,pRn in 1.1 replaced by thehomogeneous Sobolev normu W˙s,pRn An attempt of replacingu W s,pRn with the othernorms has been made in several papers For instance, Kozono et al.4 generalized 1.1 with

both of W n/q,qRn  and W s,pRn replaced by the Besov spaces and applied it to the regularityproblem for the Navier-Stokes equation and the Euler equation Moreover, Ogawa5 proved

1.1 in terms of Triebel-Lizorkin spaces for the purpose to investigate the regularity to thegradient flow of the harmonic map into a sphere We also mention that1.1 was obtained inthe Besov-Morrey spaces in6

In what follows, we concentrate on the case q  n and replace the function space

W n/q,qRn  by W 1,n

0 Ω with a bounded domain Ω in Rn Note that the norm of W01,nΩ isequivalent to∇u L nΩbecause of the Poincar´e inequality When the differential order s  m

is an integer with 1≤ m ≤ n, and n/m < p ≤ n/m − 1, the first, second and fourth authors

7 generalized the inequality corresponding to 1.1 and discussed how optimal the constant

λ is To describe the sharpness of the constant λ, they made a formulation more precise as

Here for the sake of definiteness, define

∇u L nΩ |∇u| L nΩ, |∇u| 

Then they proved the following theorem, which gives the sharp constants for λ1and

λ2in1.2 Here and below, Λ1andΛ2are constants defined by

Λ1 1

ω n 1/n−1−1 , Λ2 Λ1

where ω n−1 2π n/2 / Γn/2 is the surface area of S n−1 {x ∈ R n; |x|  1} SeeDefinition 2.5

below for the definition of the strong local Lipschitz condition for a domainΩ

Theorem 1.1 7, Theorem 1.2  Let n ≥ 2, 0 < α < 1, m ∈ {1, 2, , n}, and, Ω be a bounded

domain inRn satisfying the strong local Lipschitz condition.

i Assume that either

I λ1 > Λ1

α , λ2∈ R or II λ1 Λ1

α , λ2 ≥ Λ2

holds Then there exists a constant C such that inequality 1.2 with s  m and p 

n/ m − α holds for all u ∈ W 1,n

We note that the differential order m of the higher order Sobolev space inTheorem 1.1

had to be an integer The primary aim of the present paper is to pass Theorem 1.1 tothose which include Sobolev spaces of fractional differential order Meanwhile, higher-orderSobolev spaces are continuously embedded into corresponding H ¨older spaces Standing

on such a viewpoint, the first, second, and fourth authors 8 proved a result similar

to Theorem 1.1 for the homogeneous H ¨older space ˙C 0,αΩ instead of the Sobolev space

W m,n/ m−α Ω Furthermore, it is known that the H¨older space C 0,αΩ is expressed as the

marginal case of the Besov space B α, ∞,∞ Ω provided that 0 < α < 1, which allows us to

extendTheorem 1.1with the same sharp constants in Besov spaces

In general, we set up the following problem in a fixed function space XΩ, which is contained in L∞Ω

Fix a function space X Ω For given constants λ1 > 0 and λ2 ∈ R, does there exist a

constant C such that

holds for all u ∈ W 1,n

0 Ω ∩ XΩ under the normalization ∇u L nΩ 1?

We call W s,pRn  an auxiliary space of 1.7 First we state the following proposition,which is an immediate consequence of an elementary inequality,

log1  st ≤ logs  st  log1  t  log s for t ≥ 0, s ≥ 1 1.8

Proposition 1.2 Let Ω be a domain in R n , and let X1Ω, X2Ω be function spaces satisfying

u X1Ω≤ Mu X2Ω for u ∈ X2Ω 1.9

with some constant M ≥ 1.

i If inequality 1.7 holds in XΩ  X1Ω with a constant C, then so does 1.7 in XΩ 

X2Ω with another constant C,

or equivalently,

i if inequality 1.7 fails in XΩ  X2Ω with any constant C, then so does 1.7 in

X Ω  X1Ω with any constant C.

From the proposition above, the sharp constants for λ1and λ2in1.7 are independent

of the choice of the equivalent norms of the auxiliary space XΩ On the other hand, note that

these sharp constants may depend on the definition of∇u L nΩ; there are several manners

to define∇u L nΩ In what follows, we choose1.3 as the definition of ∇u L nΩ

In the present paper we will include Besov and Triebel-Lizorkin spaces as an auxiliary

space XΩ To describe the definition of Besov and Triebel-Lizorkin spaces, we denote by B R

the open ball inRn centered at the origin with radius R > 0, that is, B R  {x ∈ R n; |x| < R}.

Define the Fourier transformF and its inverse F−1by

Next, we fix functions ψ0, ϕ0 ∈ C∞

c Rn  which are supported in the ball B4, in the annulus

B4\ B1, respectively, and satisfying

where we set ϕ0k  ϕ0·/2 k  Here, χ E is the characteristic function of a set E and C∞c Ω

denotes the class of compactly supported C∞-functions onΩ We also denote by CcΩ theclass of compactly supported continuous functions onΩ

Definition 1.3 Take ψ0, ϕ0satisfying1.12, and let u ∈ S Rn

i Let 0 < s < ∞, 0 < p ≤ ∞, and 0 < q ≤ ∞ The Besov space B s,p,qRn is normed by

with the obvious modification when q ∞.

ii Let 0 < s < ∞, 0 < p < ∞, and 0 < q ≤ ∞ The Triebel-Lizorkin space F s,p,qRn isnormed by

u F s,p,qRn ψ0Du 

L pRn

L pRn

1.14

with the obvious modification when q  ∞; one excludes the case p  ∞.

Different choices of ψ0 and ϕ0 satisfying1.12 yield equivalent norms in 1.13 and

1.14 We refer to 9 for exhaustive details of this fact Here and below, we denote by A s,p,q

the spaces B s,p,q with 0 < s < ∞, 0 < p ≤ ∞, 0 < q ≤ ∞, or F s,p,q with 0 < s < ∞, 0 < p < ∞, 0 <

q ≤ ∞ Unless otherwise stated, the letter A means the same scale throughout the statement.

As in9,10 , we adopt a traditional method of defining function spaces on a domain

Ω ⊂ Rn

Definition 1.4 Let 0 < s < ∞ and 0 < p, q ≤ ∞.

i The function space A s,p,qΩ is defined as the subset of D Ω obtained by restricting

elements in A s,p,qRn to Ω, and the norm is given by

u A s,p,qΩ infv A s,p,qRn; v ∈ A s,p,qRn , v|Ω  u in D Ω. 1.15

ii The function space A s,p,q

0 Ω is defined as the closure of C∞

c Ω in the norm of

A s,p,qΩ

iii The potential space H s,p Ω stands for F s,p,2Ω

Now we state our main result, which claims that the sharp constants in1.7 are given

by the same ones as inTheorem 1.1when XΩ  A s,p α,s ,q Ω or A s,p α,s ,q

0 Ω, where in whatfollows we denote

A s,p α,s ,q Ω ⊂ L∞Ω and the formulation of Theorem 1.5 remains unchanged no matter

what equivalent norms we choose for the norm of the function space A s,p α,s ,qΩ Indeed,

Proposition 1.2i resp., ii shows that the condition on λ1and λ2for which inequality1.7holdsresp., fails remains unchanged if we replace the definition of the norm  · A s,pα,s,qΩ

with any equivalent norm

In the case 0 < α < 1, we can determine the condition completely.

Theorem 1.5 Let n ≥ 2, 0 < α < 1, s ≥ α, 0 < q ≤ ∞, and let Ω be a bounded domain in R n and

X Ω  A s,p α,s ,q Ω.

i Assume that either (I) or (II) holds Then there exists a constant C such that inequality 1.7

holds for all u ∈ W 1,n

HenceTheorem 1.5impliesTheorem 1.1

In order to state our results in the case α ≥ 1 for a general bounded domain Ω, wereplace assumptionII by the slightly stronger one

i Assume that either (I) or II holds Then there exists a constant C such that inequality

1.7 holds for all u ∈ W 1,n

0 Ω ∩ A s,p α,s ,q Ω with ∇u L nΩ 1.

ii Assume that either (III) or (IV) holds Then for any constant C, the inequality 1.7 fails

for some u ∈ C∞

c Ω with ∇u L nΩ 1.

Remark 1.8 We have to impose the strong local Lipschitz condition inTheorem 1.7, because

we use the universal extension theorem obtained by Rychkov12, Theorem 2.2

However, in the case 1 < α < 2, we can also determine the condition completely as in the case 0 < α < 1 provided that we restrict the functions to C cΩ

Theorem 1.9 Let n ≥ 2, 1 < α < 2, s ≥ α, 0 < q ≤ ∞, let Ω be a bounded domain in R n , and

X Ω  A s,p α,s ,q Ω.

i Assume that either (I) or (II) holds Then there exists a constant C such that inequality 1.7

holds for all u ∈ W 1,n

Remark 1.11. i The assertion inCorollary 1.10corresponding toTheorem 1.7still holds even

if we do not impose the strong local Lipschitz condition, because there is a trivial extension

operator from A s,p,q0 Ω into A s,p,qRn

ii If ∂Ω is smooth, then we can see that

u ∈ CΩ, u  0 on ∂Ω for u ∈ W 1,n

0 Ω ∩ A s,p α,s ,q Ω. 1.18

However, W01,n Ω ∩ A s,p α,s ,q Ω is not contained in A s,p α,s ,q

0 Ω, in general

Remark 1.12 The power n/ n − 1 on the left-hand side of 1.7 is optimal in the sense that

r  n/n − 1 is the largest power for which there exist λ1and C such that

u r

L∞ Ω≤ λ1log

1 u XΩ C 1.19

can hold for all u ∈ W 1,n

0 Ω ∩ XΩ with ∇u L nΩ  1 Here, XΩ is as in Theorems1.5,

1.7, and1.9andCorollary 1.10 Indeed, if r > n/n − 1, then for any λ1> 0 and any constant

C,1.19 does not hold for some u ∈ W 1,n

0 Ω ∩ XΩ with ∇u L nΩ 1, which is shown bycarrying out a similar calculation to the proof of Theorems1.5,1.7, and1.9ii; seeRemark 3.9

below for the details To the contrary, if 1 ≤ r < n/n − 1, then for any λ1 > 0, there exists

a constant C such that1.19 holds for all u ∈ W 1,n

0 Ω ∩ XΩ with ∇u L nΩ  1 This factfollows from the embedding described below and the same assertion concerning the Br´ezis-Gallou¨et-Wainger type inequality in the H ¨older space, which is shown in8, Remark 3.5 for

0 < α < 1 andRemark 4.3for α ≥ 1.

Finally let us describe the organization of the present paper In Section 2, weintroduce some notation of function spaces and state embedding theorems Section 3 isdevoted to proving the negative assertions of Theorems 1.5–1.9 Section 4 describes theaffirmative assertions of Theorems1.5and 1.7.Section 5concerns the affirmative assertion

ofTheorem 1.9 In the appendix, we prove elementary calculus which we stated inSection 5

2 Preliminaries

First we provide a brief view of H ¨older and H ¨older-Zygmund spaces Throughout the present

paper, C denotes a constant which may vary from line to line.

For 0 < α ≤ 1, ˙C 0,αRn  denotes the homogeneous H¨older space of order α endowed

with the seminorm

and C 0,αRn  denotes the nonhomogeneous H¨older space of order α endowed with the norm

u C 0,αRn u L∞ Rn u C˙0,αRn. 2.2Define also

uC˙0,αRn;Rn sup

x,y∈Rn

x /  y

x − yα 2.3

for an Rn-valued function u For 1 ≤ α ≤ 2, ˙C 1,α−1Rn denotes the homogeneous

H¨older-Zygmund space of order α, the set of all continuous functions u endowed with the seminorm

u C 1,α−1Rn u L∞ Rn u C˙1,α−1Rn. 2.5

Note that ˙C 0,1Rn  is a proper subset of ˙C 1,0Rn  We remark that, in defining ˙C 1,α−1Rn, it

is necessary that we assume the functions continuous Here we will exhibit an example of

a discontinuous function u satisfying u C˙1,α−1Rn  0 in the appendix We will not need todefine the H ¨older-Zygmund space of the higher order We need an auxiliary function space;

for 1 < α ≤ 2, let ˙C∇1,α−1Rn  denote the analogue of ˙C 1,α−1Rn endowed with the seminorm

attains the infimum definingu C˙0,αΩsee 13, Theorem 3.1.1  Moreover, we also observe

since the zero-extended function v of u onRn \Ω attains the infimum defining ∇u C˙0,α−1Ω;Rn

An elementary relation between these spaces and B α, ∞,∞Rn is as follows

Lemma 2.1 Taibleson, 14, Theorem 4  Let 0 < α < 2 Then one has the norm equivalence

B α, ∞,∞Rn   C α,α−αRn , 2.11

where α denotes the integer part of α; α  max{k ∈ N ∪ {0}; k ≤ α}.

We remark that Lemma 2.1is still valid for α ≥ 2 after defining the function space

C α,α−αRn appropriately However, we do not go into detail, since we will use the space

in the sense of continuous embedding.

Proof We accept all the embeddings whenΩ  Rn; see9 for instance The case when Ω hassmooth boundary is covered in9 However, as the proof below shows, the results are stillvalid even when the boundary ofΩ is not smooth For the sake of convenience, let us prove

the second one To this end we take u ∈ B s,p,q Ω Then by the definition of B s,p,qΩ and its

norm, we can find v ∈ B s,p,qRn so that

v|Ω  u in D Ω, u B s,p,qΩ≤ v B s,p,qRn≤ 2u B s,p,qΩ. 2.13Now that we acceptv B s −n1/p−1/p,p,qRn≤ C s,p, p,q v B s,p,qRn, we have

u B s −n1/p−1/p,p,qΩ≤ v B s −n1/p−1/p,p,qRn≤ C s,p, p,q v B s,p,qRn. 2.14Combining these observations, we see that the second embedding holds

We need the following proposition later, which claims that ˙C 1,α−1Rn  → ˙C∇1,α−1Rn for

1 < α < 2 in the sense of continuous embedding.

Proposition 2.3 Let 1 < α < 2 Then there exists C α > 0 such that

u C˙1,α−1

∇ Rn≤ C α u C˙1,α−1Rn for u ∈ ˙C 1,α−1Rn . 2.15

The proof is somehow well knownsee 15, Chapter 0 when n  1 Here for the sake

of convenience we include it in the appendix We will show that this fact is also valid on adomainΩ ⊂ Rn

Proposition 2.4 Let 1 < α < 2 and Ω be a domain in R n Then there exists C α > 0 such that

and obtain the desired result

Let us establish the following proposition Here, unlike a bounded domainΩ, for thewhole spaceRn we adopt the following definition of the norm of W 1,nRn:

u W 1,nRn u L nRn ∇u L nRn. 2.19

Definition 2.5 One says that a bounded domain Ω satisfies the strong local Lipschitz condition

if Ω has a locally Lipschitz boundary, that is, each point x on the boundary of Ω has a neighborhood U x whose intersection with the boundary of Ω is the graph of a Lipschitzcontinuous function

The definition for a general domain is more complicated; see11 for details

Proposition 2.6 Let 0 < γ < α Then one has

u B γ, ∞,∞Rn≤ C γ u γ/α

B α, ∞,∞Rnu1−γ/αW 1,nRn for u ∈ W 1,nRn  ∩ B α, ∞,∞Rn . 2.20

Furthermore, let Ω be a bounded domain in R n satisfying the strong local Lipschitz condition Then one has

u B γ, ∞,∞Ω≤ C γ u γ/α B α, ∞,∞Ω∇u1−γ/αL nΩ for u ∈ W 1,n

0 Ω ∩ B α, ∞,∞ Ω. 2.21

Proposition 2.6can be obtained directly from a theory of interpolation However, theproof being simple, we include it for the sake of reader’s convenience

Proof of Proposition 2.6 Let us take ζ ∈ C∞

c Rn  so that ζ  1 on B4\ B1and supp ζ ⊂ B8\ B 1/2.Set

L∞ Rn

 1

2π n2k

∗ u 

L∞Rn

≤ Cu L nRn≤ Cu W 1,nRn.

2.25

u B β, ∞,∞Ω≤ Eu B β, ∞,∞Rn≤ C β u B β, ∞,∞Ω for u ∈ B β, ∞,∞ Ω,

∇u L nΩ≤ Eu W 1,nRn≤ C∇u L nΩ for u ∈ W 1,n

0 Ω 2.28

for all γ ≤ β < ∞ Then 2.21 is an immediate consequence of 2.20

3 Counterexample for the Inequality

In this section, we will give the proof of assertionii of Theorems1.5–1.9.Lemma 2.2showsthat

B s,p,min {p,q} Ω → F s,p,q Ω, 3.1

and hence it suffices to consider the case As,p α,s ,q Ω  B s,p α,s ,qΩ in view ofProposition 1.2

i Furthermore,Lemma 2.2also shows that

B s,p α, s ,min {p α, s ,q}Ω → B s,p α,s ,qΩ for s > s, 3.2

and hence we have only to consider the case 0 < q ≤ p α,s  n/s − α ≤ 1 Therefore, it suffices

to show the following theorem for the proof ofii of Theorems1.5–1.9

Theorem 3.1 Let n ≥ 2, α > 0, s ≥ n  α, 0 < q ≤ p α,s , and let Ω be a bounded domain in R n and

X Ω  B s,p α,s ,q Ω Assume that either (III) or (IV) holds Then for any constant C, inequality 1.7

fails for some u ∈ C∞

c Ω with ∇u L nΩ 1.

Here and below, we use the notation

 s  log1  s for s ≥ 0 3.3

for short, and then  ◦ s  log1  log1  s for s ≥ 0 We note that inequality 1.7

with XΩ  B s,p α,s ,q Ω holds for all u ∈ W 1,n

0 Ω ∩ B s,p α,s ,q Ω with ∇u L nΩ  1 if and

only if there exists a constant C independent of u such that F α,s,q u; λ1, λ2 ≤ C holds for all

that F α,s,q u j ; λ1, λ2 → ∞ as j → ∞ under assumption III or IV In the case that Ω  R n

and that all the functions are supported in B1, we can choose such a sequence

Lemma 3.2 Let n ≥ 2, α > 0, s ≥ n  α, 0 < q ≤ p α,s , andΩ  Rn Then there exists a family of functions {u j}∞j1⊂ C∞

c Rn  \ {0} with supp u j ⊂ B1for all j ∈ N such that

F α,s,q

u j ; λ1, λ2



−→ ∞ as j −→ ∞ 3.5

under assumption (III) or (IV) of Theorem 3.1

We can now proveTheorem 3.1once we acceptLemma 3.2

Proof of Theorem 3.1 Examining1.7 fails, so we may assume that λ1, λ2 ≥ 0 Fix z0 ∈ Ω and

The first and the second equalities are immediate, while the third inequality is a direct

consequence of the fact that the dilation u → uR0· is an isomorphism over B s,p α,s ,qRn.Using1.8 and the fact that λ1, λ2≥ 0, we have

from which we conclude that F α,s,q v j ; λ1, λ2 → ∞ as j → ∞.

We now concentrate on the proof ofLemma 3.2, and we first prepare several lemmas.Let ϕ0 ∈ C∞

c 0, ∞ be a smooth function that is nonnegative, supported on the

interval1, 4 and satisfies

Observe that3.10 forces ϕ02  1

Proposition 3.3 i It holds

Proof i In view of the size of the support of ϕ0, we easily obtain3.11

ii If we integrate both the sides of inequality 3.11, then we have

t dt. 3.13

As a consequence, it follows that

log 2 1−1

We also note that supp u j ⊂ B 1/2 since supp w l ⊂ B 1/2 l

When we are going to specify the best constant,3.19 is the heart of the matter

Lemma 3.4 Let n ≥ 2 and 0 < p < ∞ Then one has

Thus, we obtain3.17 by applying 3.12

We next verify3.18 Recall that Λ1 is defined byΛ1  1/ω 1/n−1 n−1 If we insert thedefinitions3.15 and 3.16, then we have

u j p

L pRn 1

Λn−1 1

log 2 j p

Using3.11, we have

u j p

L pRn≤ 1

Λn−1 1

log 2 j p

log 2 j p

,

u jp

L pRn≥ 1

Λn−1 1

log 2 j p

log 2 j p

∇u j n

L nRn 1

Λn−1 1

log 2 j n

log 2 j n

log 2 j n

Lemma 3.5 Let n ≥ 2, α > 0, s ≥ n  α, and 0 < q ≤ p α,s Then one has

Proposition 3.6 Take ψ0 satisfying1.12 Let n ≥ 2, α > 0, and s ≥ n  α Then there exists a

constant C α,s such that

and estimateφ m ∗ χ B RL pRncrudely, where 0 < p ≤ ∞ and R > 0 Let sdenote the positive

part of s ∈ R, that is, s max{s, 0}.

Proposition 3.8 If m > n1/p − 1, 0 < p ≤ ∞, and R > 0, then there exists a constant C p,m > 0 such that

Let us turn to the estimate outside B 2R Since

n

φ m

L pRnR n

3.33

Thus we have proved the assertion

We first proveProposition 3.6 We abbreviate χ B

Let us turn to provingProposition 3.7

Proof of Proposition 3.7 Since ϕ k does not contain the origin as its support, we can define

A direct calculation shows that

Furthermore, let Ω be a bounded domain in R n satisfying the strong local Lipschitz condition Then one has

u... not contain the origin as its support, we can define

As in 9,10 , we adopt a traditional method of defining function spaces on a domain

Ω ⊂ Rn