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

Volume 2010, Article ID 584521,38pages 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

Kei Morii,

Tokushi Sato,

Yoshihiro Sawano,

and Hidemitsu Wadade

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

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

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

4Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

Correspondence should be addressed to Yoshihiro Sawano,yoshihiro-sawano@celery.ocn.ne.jp Received 4 October 2009; Revised 15 September 2010; Accepted 12 October 2010

Academic Editor: Veli B. Shakhmurov

Copyrightq2010 Kei Morii et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present paper concerns the Sobolev embedding in the endpoint case. It is known that the embeddingW^1,nRⁿ→L^∞Rⁿfails forn≥2. Br´ezis-Gallou¨et-Wainger and some other authors 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 the coefficients 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-Lizorkin spaces as well as fractional Sobolev spaces on a bounded domainΩ ⊂ Rⁿ. Throughout the present paper, we place ourselves in the setting ofRⁿwithn≥ 2. We treat only real-valued functions.

First we recall the Sobolev embedding theorem in the critical case. For 1< q <∞, it is well known that the embeddingW^n/q,qRⁿ→L^rRⁿholds for anyq≤r <∞, and does not hold forr ∞, that is, one cannot estimate theL^∞-norm by theW^n/q,q-norm. However, the

Br´ezis-Gallou¨et-Wainger inequality states that theL^∞-norm can be estimated by theW^n/q,q- norm with the partial aid of theW^s,p-norm withs > n/pand 1≤p≤ ∞as follows:

u^q/q−1_L∞Rⁿ ≤λ

1log

1u_W^s,p_Rn

1.1

holds whenever u ∈ W^n/q,qRⁿ∩W^s,pRⁿsatisfies u_W^n/q,q_Rⁿ 1, where 1 ≤ p ≤ ∞, 1 < q < ∞, ands > n/p. Inequality 1.1 for the casen p q s 2 dates back to Br´ezis-Gallou¨et1 . Later on, Br´ezis and Wainger2 obtained1.1for the general case, and remarked that the power q/q−1in 1.1is maximal; equation 1.1 fails for any larger power. Ozawa 3 proved 1.1 with the Sobolev normuW^s,pRⁿ in 1.1 replaced by the homogeneous Sobolev normuW˙^s,pRⁿ. An attempt of replacinguW^s,pRⁿ with the other norms has been made in several papers. For instance, Kozono et al.4 generalized1.1with both ofW^n/q,qRⁿandW^s,pRⁿreplaced by the Besov spaces and applied it to the regularity problem for the Navier-Stokes equation and the Euler equation. Moreover, Ogawa5 proved 1.1in terms of Triebel-Lizorkin spaces for the purpose to investigate the regularity to the gradient flow of the harmonic map into a sphere. We also mention that1.1was obtained in the Besov-Morrey spaces in6 .

In what follows, we concentrate on the case q n and replace the function space W^n/q,qRⁿbyW₀^1,nΩwith a bounded domainΩinRⁿ. Note that the norm ofW₀^1,nΩis equivalent to∇uLⁿΩbecause of the Poincar´e inequality. When the differential ordersm is an integer with 1≤m≤n, andn/m < p≤n/m−1, the first, second and fourth authors 7 generalized the inequality corresponding to1.1and discussed how optimal the constant λis. To describe the sharpness of the constantλ, they made a formulation more precise as follows:

For given constantsλ₁>0 andλ₂ ∈R, does there exist a constantCsuch that u^n/n−1_L∞Ω ≤λ1log

1u_W^s,p_Ω λ2log

1log

1u_W^s,p_Ω C

1.2

holds for allu∈W₀^1,nΩ∩W^s,pΩwith∇uLⁿΩ1?

Here for the sake of definiteness, define

∇u_LnΩ|∇u|_LnΩ, |∇u|

_n

i1

∂u

∂x_i ²

1/2

. 1.3

We call the first term and the second term of the right-hand side of 1.2 the single logarithmic term and the double logarithmic term, respectively. We remark that the double logarithmic term grows weaker than the single one asuW^s,pΩ → ∞.

Then they proved the following theorem, which gives the sharp constants forλ₁and λ2in1.2. Here and below,Λ1andΛ2are constants defined by

Λ1 1

ω_n−1^1/n−1, Λ2 Λ1

n 1

nω^1/n−1_n−1 , 1.4

whereω_n−12π^n/2/Γn/2is the surface area ofSⁿ⁻¹ {x∈Rⁿ; |x|1}. SeeDefinition 2.5 below for the definition of the strong local Lipschitz condition for a domainΩ.

Theorem 1.17, Theorem 1.2 . Letn ≥ 2, 0 < α <1,m ∈ {1,2, . . . , n}, and,Ωbe a bounded domain inRⁿsatisfying the strong local Lipschitz condition.

iAssume that either

Iλ₁ > Λ1

α , λ₂∈R or IIλ₁ Λ1

α , λ₂ ≥ Λ2

α 1.5

holds. Then there exists a constant Csuch that inequality 1.2 with s m and p n/m−αholds for allu∈W₀^1,nΩ∩W^m,n/m−αΩwith∇uLⁿΩ1.

iiAssume that either

IIIλ₁ < Λ1

α , λ₂∈R or IVλ₁ Λ1

α , λ₂< Λ2

α 1.6

holds. Then for any constantC, inequality1.2withs mandp n/m−αfails for someu∈W₀^1,nΩ∩W₀^m,n/m−αΩwith∇uLⁿΩ1.

We note that the differential ordermof 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 to those which include Sobolev spaces of fractional differential order. Meanwhile, higher-order Sobolev 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−αΩ. Furthermore, it is known that the H ¨older spaceC^0,αΩ is expressed as the marginal case of the Besov spaceB^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 spaceXΩ, which is contained inL^∞Ω.

Fix a function space XΩ. For given constants λ₁ > 0 andλ₂ ∈ R, does there exist a constantCsuch that

u^n/n−1_L∞Ω ≤λ1log

1u_XΩ λ2log

1log

1u_XΩ

C

1.7

holds for allu∈W₀^1,nΩ∩XΩunder the normalization∇u_LnΩ1?

We callW^s,pRⁿan auxiliary space of1.7. First we state the following proposition, which is an immediate consequence of an elementary inequality,

log1st≤logsst log1t logs fort≥0, s≥1. 1.8

Proposition 1.2. LetΩbe a domain inRⁿ, and letX1Ω,X2Ωbe function spaces satisfying u_X1Ω≤Mu_X2Ω foru∈X2Ω 1.9

with some constantM≥1.

iIf inequality1.7holds inXΩ X₁Ωwith a constantC, then so does1.7inXΩ X2Ωwith another constantC,

or equivalently,

iif inequality 1.7 fails in XΩ X₂Ω with any constant C, then so does 1.7 in XΩ X1Ωwith any constantC.

From the proposition above, the sharp constants forλ₁andλ₂in1.7are independent of the choice of the equivalent norms of the auxiliary spaceXΩ. On the other hand, note that these sharp constants may depend on the definition of∇uLⁿΩ; there are several manners to define∇uLⁿΩ. In what follows, we choose1.3as the definition of∇uLⁿΩ.

In the present paper we will include Besov and Triebel-Lizorkin spaces as an auxiliary spaceXΩ. To describe the definition of Besov and Triebel-Lizorkin spaces, we denote byBR

the open ball inRⁿcentered at the origin with radiusR >0, that is,B_R {x∈Rⁿ; |x|< R}.

Define the Fourier transformFand its inverseF⁻¹by Fuξ 1

2π^n/2

Rⁿe^{−√−1x·ξ}uxdx, F⁻¹ux 1 2π^n/2

Rⁿe^√−1x·ξuξdξ 1.10 for u ∈ SRⁿ, respectively, and they are also extended on SRⁿ by the usual way. For ϕ∈ SRⁿ, define an operatorϕDby

ϕDuF⁻¹ ϕFu 1

2π^n/2

F⁻¹ϕ

∗u. 1.11

Next, we fix functionsψ⁰, ϕ⁰ ∈ C^∞_c Rⁿwhich are supported in the ballB4, in the annulus B₄\B₁, respectively, and satisfying

∞ k−∞

ϕ⁰_kx χ_Rⁿ_\{0}x, ψ⁰x 1−^∞

k0

ϕ⁰_kx forx∈Rⁿ, 1.12

where we set ϕ⁰_k ϕ⁰·/2^k. Here,χE is the characteristic function of a setE and C^∞_c Ω denotes the class of compactly supportedC^∞-functions onΩ. We also denote byCcΩthe class of compactly supported continuous functions onΩ.

Definition 1.3. Takeψ⁰, ϕ⁰satisfying1.12, and letu∈ SRⁿ.

iLet 0< s <∞, 0< p≤ ∞, and 0< q≤ ∞. The Besov spaceB^s,p,qRⁿis normed by

u_Bs,p,qRⁿψ⁰Du

L^pRⁿ _∞

k0

2^skϕ⁰_kDu

L^pRⁿ

_q1/q

1.13

with the obvious modification whenq∞.

iiLet 0 < s <∞, 0< p < ∞, and 0 < q ≤ ∞. The Triebel-Lizorkin spaceF^s,p,qRⁿis normed by

u_F^s,p,q_Rⁿψ⁰Du

L^pRⁿ

_∞

k0

2^skϕ⁰_kDu_q1/q

L^pRⁿ

1.14

with the obvious modification whenq∞; one excludes the casep∞.

Different choices of ψ⁰ andϕ⁰ satisfying1.12yield equivalent norms in1.13and 1.14. We refer to9 for exhaustive details of this fact. Here and below, we denote byA^s,p,q the spacesB^s,p,qwith 0< s <∞, 0< p≤ ∞, 0< q≤ ∞, orF^s,p,qwith 0< s <∞, 0< p <∞, 0<

q≤ ∞. Unless otherwise stated, the letterAmeans 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≤ ∞.

iThe function spaceA^s,p,qΩis defined as the subset ofDΩobtained by restricting elements inA^s,p,qRⁿtoΩ, and the norm is given by

u_A^s,p,q_Ωinf

v_A^s,p,q_Rⁿ;v∈A^s,p,qRⁿ, v|_Ω uinDΩ

. 1.15

iiThe function space A^s,p,q₀ Ω is defined as the closure of C_c^∞Ω in the norm of A^s,p,qΩ.

iiiThe potential spaceH^s,pΩstands forF^s,p,2Ω.

Now we state our main result, which claims that the sharp constants in1.7are given by the same ones as inTheorem 1.1whenXΩ A^s,pα,s,qΩorA^s,p₀ ^α,s,qΩ, where in what follows we denote

p_α,s

⎧⎨

⎩ n

s−α fors > α,

∞ forsα.

1.16

Here, conditionsI–IVare the same as inTheorem 1.1. We should remark thatA^s,p₀ ^α,s,qΩ⊂ A^s,pα,s,qΩ ⊂ L^∞Ω and the formulation of Theorem 1.5 remains unchanged no matter what equivalent norms we choose for the norm of the function spaceA^s,pα,s,qΩ. Indeed, Proposition 1.2i resp.,iishows that the condition onλ₁andλ₂for which inequality1.7 holdsresp., failsremains 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. Letn≥ 2, 0< α < 1,s ≥α, 0< q≤ ∞, and letΩbe a bounded domain inRⁿand XΩ A^s,pα,s,qΩ.

iAssume that either (I) or (II) holds. Then there exists a constantCsuch that inequality1.7 holds for allu∈W₀^1,nΩ∩A^s,pα,s,qΩwith∇uLⁿΩ1.

iiAssume that either (III) or (IV) holds. Then for any constantC, the inequality1.7fails for someu∈C^∞_c Ωwith∇uLⁿΩ1.

Remark 1.6. If Ω has a Lipschitz boundary, then the Stein total extension theorem 11, Theorem 5.24 implies thatW^m,pΩ H^m,pΩ F^m,p,2Ωfor m ∈ Nand 1 < p < ∞.

HenceTheorem 1.5impliesTheorem 1.1.

In order to state our results in the caseα ≥ 1 for a general bounded domain Ω, we replace assumptionIIby the slightly stronger one

IIλ₁ Λ1

α , λ₂ >Λ2. 1.17

Unfortunately, we do not know whether the result in this case corresponding to the caseα≥1 inTheorem 1.5holds.

Theorem 1.7. Letn≥2,α≥1,s ≥α, 0 < q≤ ∞, letΩbe a bounded domain inRⁿsatisfying the strong local Lipschitz condition andXΩ A^s,pα,s,qΩ.

iAssume that either (I) orIIholds. Then there exists a constantCsuch that inequality 1.7holds for allu∈W₀^1,nΩ∩A^s,pα,s,qΩwith∇uLⁿΩ1.

iiAssume that either (III) or (IV) holds. Then for any constantC, the inequality1.7fails for someu∈C^∞_c Ωwith∇uLⁿΩ1.

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 toC_cΩ.

Theorem 1.9. Letn ≥ 2, 1 < α < 2,s ≥ α, 0 < q ≤ ∞, letΩbe a bounded domain inRⁿ, and XΩ A^s,pα,s,qΩ.

iAssume that either (I) or (II) holds. Then there exists a constantCsuch that inequality1.7 holds for allu∈W₀^1,nΩ∩A^s,pα,s,qΩ∩CcΩwith∇uLⁿΩ1.

iiAssume that either (III) or (IV) holds. Then for any constantC, inequality1.7fails for someu∈C_c^∞Ωwith∇uLⁿΩ1.

We also obtain the following corollary becauseC^∞_c Ω⊂A^s,p,q₀ Ω⊂A^s,p,qΩ.

Corollary 1.10. Theorems1.5,1.7, and1.9still hold true if one replacesA^s,pα,s,qΩbyA^s,p₀ ^α,s,qΩ.

Remark 1.11. iThe 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 fromA^s,p,q₀ ΩintoA^s,p,qRⁿ.

iiIf∂Ωis smooth, then we can see that u∈C

Ω

, u0 on∂Ω foru∈W₀^1,nΩ∩A^s,pα,s,qΩ. 1.18

However,W₀^1,nΩ∩A^s,pα,s,qΩis not contained inA^s,p₀ ^α,s,qΩ, in general.

Remark 1.12. The powern/n−1on the left-hand side of1.7is optimal in the sense that rn/n−1is the largest power for which there existλ₁andCsuch that

u^r_L∞Ω≤λ₁log

1u_XΩ

C 1.19

can hold for allu∈ W₀^1,nΩ∩XΩwith∇uLⁿΩ 1. Here,XΩis as in Theorems1.5, 1.7, and1.9andCorollary 1.10. Indeed, ifr > n/n−1, then for anyλ1>0 and any constant C,1.19does not hold for someu∈W₀^1,nΩ∩XΩwith∇uLⁿΩ1, which is shown by carrying 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 anyλ₁ > 0, there exists a constantCsuch that1.19holds for allu∈W₀^1,nΩ∩XΩwith∇uLⁿΩ 1. This fact follows 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 <1andRemark 4.3forα≥1.

Finally let us describe the organization of the present paper. In Section 2, we introduce some notation of function spaces and state embedding theorems. Section 3 is devoted to proving the negative assertions of Theorems 1.5–1.9. Section 4 describes the affirmative 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,Cdenotes a constant which may vary from line to line.

For 0 < α ≤ 1, ˙C^0,αRⁿdenotes the homogeneous H ¨older space of orderαendowed with the seminorm

uC˙^0,αRⁿ sup

x,y∈Rⁿ x /y

ux−u y

x−y^α , 2.1

andC^0,αRⁿdenotes the nonhomogeneous H ¨older space of orderαendowed with the norm u_C0,αRⁿu_L∞RⁿuC˙^0,αRⁿ. 2.2

Define also

uC˙^0,αRⁿ;Rⁿ sup

x,y∈Rⁿ x /y

ux−u y

x−y^α 2.3

for an Rⁿ-valued function u. For 1 ≤ α ≤ 2, ˙C^1,α−1Rⁿ denotes the homogeneous H ¨older- Zygmund space of orderα, the set of all continuous functionsuendowed with the seminorm

uC˙^1,α−1Rⁿ sup

x,y∈Rⁿ x /y

ux−2u xy

/2

u y

x−y^α , 2.4

andC^1,α−1Rⁿdenotes the nonhomogeneous H ¨older-Zygmund space of orderα, the set of all continuous functionsuendowed with the norm

u_C1,α−1Rⁿu_L∞RⁿuC˙^1,α−1Rⁿ. 2.5

Note that ˙C^0,1Rⁿis 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 functionusatisfyinguC˙^1,α−1Rⁿ 0 in the appendix. We will not need to define the H ¨older-Zygmund space of the higher order. We need an auxiliary function space;

for 1< α≤2, let ˙C_∇^1,α−1Rⁿdenote the analogue of ˙C^1,α−1Rⁿendowed with the seminorm

uC˙^1,α−1_∇ Rⁿ∇uC˙^0,α−1Rⁿ;Rⁿ sup

x,y∈Rⁿ x /y

∇ux− ∇u y

x−y^α−1 . 2.6

The other function spaces on a domain Ω ⊂ Rⁿ are made analogously to A^s,p,qΩ. For example, define

uC˙^0,αΩinf

vC˙^0,αRⁿ;v∈C˙^0,αRⁿ, v|_ΩuinDΩ , uC˙^1,α−1Ωinf

vC˙^1,α−1Rⁿ;v∈C˙^1,α−1Rⁿ, v|_Ωuin DΩ , uC˙^1,α−1_∇ Ωinf

∇vC˙^0,α−1Rⁿ;Rⁿ;v∈C˙^1,α−1_∇ Rⁿ, ∇v|_Ω∇uinDΩ .

2.7

A moment’s reflection shows that for 0< α≤1,uC˙^0,αΩcan be written as

uC˙^0,αΩ sup

x,y∈Ω x /y

ux−u y

x−y^α foru∈C˙^0,αΩ 2.8

since the function

vx inf

y∈Ω

u y

uC˙^0,αΩx−y^α

forx∈Rⁿ 2.9

attains the infimum defininguC˙^0,αΩsee13, Theorem 3.1.1 . Moreover, we also observe that

uC˙^1,α−1_∇ Ω∇uC˙^0,α−1Ω;Rⁿ sup

x,y∈Ω x /y

∇ux− ∇u y

x−y^α−1 foru∈C˙^1,α−1_∇ Ω∩C_cΩ 2.10 since the zero-extended functionvofuonRⁿ\Ωattains the infimum defining∇uC˙^0,α−1Ω;Rⁿ.

An elementary relation between these spaces andB^α,∞,∞Rⁿis as follows.

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

B^α,∞,∞RⁿC^α,α−αRⁿ, 2.11

whereαdenotes the integer part ofα;αmax{k∈N∪ {0}; k≤α}.

We remark that Lemma 2.1is still valid for α ≥ 2 after defining the function space C^α,α−αRⁿappropriately. However, we do not go into detail, since we will use the space C^α,α−αRⁿonly with 0< α <2.

We will invoke the following fact on the Sobolev type embedding for Besov and Triebel-Lizorkin spaces:

Lemma 2.2. Let 0< s <∞, 0< p <p≤ ∞, 0< q <q≤ ∞, and letΩbe a domain inRⁿ. Then

B^s,p,qΩ→B^s,p,qΩ, B^s,p,qΩ→B^{s−n1/p−1/p,p,q}Ω, Bs,p,min{p,q}Ω→F^s,p,qΩ→Bs,p,max{p,q}Ω

2.12

in the sense of continuous embedding.

Proof. We accept all the embeddings whenΩ Rⁿ; see9 for instance. The case whenΩhas smooth boundary is covered in9 . However, as the proof below shows, the results are still valid even when the boundary ofΩis not smooth. For the sake of convenience, let us prove the second one. To this end we takeu∈B^s,p,qΩ. Then by the definition ofB^s,p,qΩand its norm, we can findv∈B^s,p,qRⁿso that

v|_Ω u inDΩ, u_B^s,p,q_Ω≤ v_B^s,p,q_Rⁿ≤2u_B^s,p,q_Ω. 2.13

Now that we acceptv_B^{s−n1/p−1/p,p,q}_Rⁿ≤C_s,p,p,qvB^s,p,qRⁿ, we have

u_Bs−n1/p−1/p,p,qΩ≤ v_Bs−n1/p−1/p,p,qRⁿ≤C_s,p,p,qv_B^s,p,q_Rⁿ. 2.14

Combining 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−1Rⁿfor 1< α <2 in the sense of continuous embedding.

Proposition 2.3. Let 1< α <2. Then there existsCα>0 such that

uC˙^1,α−1_∇ Rⁿ≤C_αuC˙^1,α−1Rⁿ foru∈C˙^1,α−1Rⁿ. 2.15

The proof is somehow well knownsee15, Chapter 0 whenn1. Here for the sake of convenience we include it in the appendix. We will show that this fact is also valid on a domainΩ⊂Rⁿ.

Proposition 2.4. Let 1< α <2 andΩbe a domain inRⁿ. Then there existsC_α>0 such that

uC˙^1,α−1_∇ Ω≤C_αuC˙^1,α−1Ω foru∈C˙^1,α−1Ω. 2.16

Proof. For anyu∈C˙^1,α−1Ω, there exists an extensionv_u∈C˙^1,α−1RⁿofuonRⁿsuch that

v_u|_Ω u inDΩ, uC˙^1,α−1Ω≤ v_uC˙^1,α−1Rⁿ≤2uC˙^1,α−1Ω. 2.17

In particular,∇vu|Ω∇uinDΩ. By applyingProposition 2.3, we have uC˙^1,α−1_∇ Ω;Rⁿ

inf

∇vC˙^0,α−1Rⁿ;Rⁿ;v∈C˙_∇^1,α−1Rⁿ, ∇v|_Ω∇uinDΩ

≤ ∇v_uC˙^0,α−1Rⁿ;RⁿvuC˙_∇^1,α−1Rⁿ;Rⁿ

≤C_αvuC˙^1,α−1Rⁿ ≤2C_αuC˙^1,α−1Ω

2.18

and obtain the desired result.

Let us establish the following proposition. Here, unlike a bounded domainΩ, for the whole spaceRⁿwe adopt the following definition of the norm ofW^1,nRⁿ:

u_W^1,n_Rⁿu_Lⁿ_Rⁿ∇u_Lⁿ_Rⁿ. 2.19

Definition 2.5. One says that a bounded domainΩsatisfies 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 Ux whose intersection with the boundary of Ω is the graph of a Lipschitz continuous function.

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

Proposition 2.6. Let 0< γ < α. Then one has

u_Bγ,∞,∞Rⁿ≤C_γu^γ/α_Bα,∞,∞Rⁿu^1−γ/α_W1,nRⁿ foru∈W^1,nRⁿ∩B^α,∞,∞Rⁿ. 2.20

Furthermore, letΩbe a bounded domain inRⁿ satisfying the strong local Lipschitz condition. Then one has

u_B^γ,∞,∞_Ω≤Cγu^γ/α_Bα,∞,∞Ω∇u^1−γ/α_LnΩ foru∈W₀^1,nΩ∩B^α,∞,∞Ω. 2.21

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

Proof ofProposition 2.6. Let us takeζ∈C^∞_c Rⁿso thatζ1 onB4\B1and suppζ⊂B8\B1/2. Set

ζiξ ξi

|ξ|²ζξ, ζi,kξ ζi

ξ 2^k

2^kξi

|ξ|²ζ ξ

2^k

fork∈N∪ {0}, i∈ {1, . . . , n}. 2.22

Recall thatϕ⁰_kis supported onB₂k2\B₂k, and observe that

ϕ⁰_kξ 1 2^k

n i1

ξ_iζ_i,kξϕ⁰_kξ. 2.23

Hence we have

ϕ⁰_kDu

L^∞Rⁿ

1 2^k

n i1

ζ_i,kDϕ⁰_kD∂u

∂xi

L^∞Rⁿ

1 2πⁿ2^k

n i1

F⁻¹ζ_i,k∗F⁻¹ϕ⁰_k∗ ∂u

∂xi

L^∞Rⁿ

≤ 1 2πⁿ2^k

F⁻¹ϕ⁰_k

L¹Rⁿ

n i1

F⁻¹ζ_i,k

L^n/n−1Rⁿ

∂u

∂xi

LⁿRⁿ

1

2πⁿF⁻¹ϕ⁰

L¹Rⁿ

n i1

F⁻¹ζi

L^n/n−1Rⁿ

∂u

∂xi

LⁿRⁿ

≤C∇u_Lⁿ_Rⁿ≤Cu_W^1,n_Rⁿ.

2.24

A similar estimate forψ⁰is also available:

ψ⁰Du

L^∞Rⁿ 1 2π^n/2

F⁻¹ψ⁰

∗u

L^∞Rⁿ

≤Cu_LnRⁿ≤Cu_W1,nRⁿ.

2.25

Hence we have

u_Bγ,∞,∞Rⁿψ⁰Du

L^∞Rⁿ sup

k∈N∪{0}

2^γlϕ⁰_kDu

L^∞Rⁿ

≤Cγ sup

k∈N∪{0}min 1

2^α−γku_Bα,∞,∞Rⁿ,2^γku_W^1,n_Rⁿ

2.26

since 2^αkϕ⁰_kDuL^∞Rⁿ≤ uB^α,∞,∞Rⁿ. Hence we have

u_B^γ,∞,∞_Rⁿ≤Cγsup

t>0

min 1

t^α−γu_B^α,∞,∞_Rⁿ, t^γu_W^1,n_Rⁿ

Cγu^γ/α_Bα,∞,∞Rⁿu^1−γ/α_W1,nRⁿ.

2.27

It remains to prove2.21. The universal extension theorem obtained by Rychkov12, Theorem 2.2 yields that there exists a common extension operatorE:W₀^1,nΩB^γ,∞,∞Ω → W^1,nRⁿ B^γ,∞,∞Rⁿsuch that

u_Bβ,∞,∞Ω≤ Eu_Bβ,∞,∞Rⁿ≤C_βu_Bβ,∞,∞Ω foru∈B^β,∞,∞Ω,

∇u_LnΩ≤ Eu_W1,nRⁿ≤C∇u_LnΩ foru∈W₀^1,nΩ 2.28

for allγ≤β <∞. Then2.21is an immediate consequence of2.20.

3. Counterexample for the Inequality

In this section, we will give the proof of assertioniiof Theorems1.5–1.9.Lemma 2.2shows that

Bs,p,min{p,q}Ω→F^s,p,qΩ, 3.1

and hence it suffices to consider the caseA^s,pα,s,qΩ B^s,pα,s,qΩin view ofProposition 1.2 i. Furthermore,Lemma 2.2also shows that

B^{s,p α,s,min{pα,s,q}}Ω→B^s,pα,s,qΩ for s > s, 3.2 and hence we have only to consider the case 0< q≤pα,sn/s−α≤1. Therefore, it suffices to show the following theorem for the proof ofiiof Theorems1.5–1.9.

Theorem 3.1. Letn≥2,α >0,s≥nα, 0< q≤p_α,s, and letΩbe a bounded domain inRⁿand XΩ B^s,pα,s,qΩ. Assume that either (III) or (IV) holds. Then for any constantC, inequality1.7 fails for someu∈C^∞_c Ωwith∇uLⁿΩ1.

Here and below, we use the notation

s log1s fors≥0 3.3

for short, and then ◦s log1 log1s for s ≥ 0. We note that inequality 1.7 with XΩ B^s,pα,s,qΩ holds for allu ∈ W₀^1,nΩ∩B^s,pα,s,qΩwith ∇uLⁿΩ 1 if and only if there exists a constantCindependent ofusuch thatF^α,s,qu;λ1, λ2 ≤ Cholds for all u∈W₀^1,nΩ∩B^s,pα,s,qΩ\ {0}, where

F^α,s,qu;λ₁, λ₂

u_L∞Ω

∇u_LnΩ

_n/n−1

−λ₁

u_Bs,pα,s,qΩ

∇u_LnΩ

−λ₂◦

u_Bs,pα,s,qΩ

∇u_LnΩ

foru∈W₀^1,nΩ∩B^s,pα,s,qΩ\ {0}.

3.4

For the proof ofTheorem 3.1, we have to find a sequence{uj}^∞_j1 ⊂ C^∞_c Ω\ {0}such thatF^α,s,quj;λ₁, λ₂ → ∞asj → ∞under assumptionIIIorIV. In the case thatΩ Rⁿ and that all the functions are supported inB1, we can choose such a sequence.

Lemma 3.2. Letn≥ 2,α > 0,s≥ nα, 0< q ≤ p_α,s, andΩ Rⁿ. Then there exists a family of functions{uj}^∞_j1⊂C_c^∞Rⁿ\ {0}with suppu_j⊂B₁for allj∈Nsuch that

F^α,s,q

uj;λ1, λ2

−→ ∞ asj−→ ∞ 3.5

under assumption (III) or (IV) ofTheorem 3.1.

We can now proveTheorem 3.1once we acceptLemma 3.2.

Proof ofTheorem 3.1. Examining1.7fails, so we may assume thatλ₁, λ₂ ≥0. Fixz₀ ∈Ωand R₀ ≥1 such that

B

x∈Rⁿ;|x−z₀|< 1 R0

⊂Ω. 3.6

Let{uj}^∞_j1be a family of functions as inLemma 3.2. If we set

vjx

⎧⎨

⎩

u_jR0x−z₀ forx∈B,

0 forx∈Ω\B, 3.7

thenvj∈C^∞_cΩ, and there exists a constantCα,s,R0≥1 such that v_j

L^∞Ωu_j

L^∞Rⁿ, ∇vj

LⁿΩ∇uj

LⁿRⁿ, vj

B^s,pα,s,qΩ≤Cα,s,R0uj

B^s,pα,s,qRⁿ.

3.8

The first and the second equalities are immediate, while the third inequality is a direct consequence of the fact that the dilationu → uR0· is an isomorphism over B^s,pα,s,qRⁿ. Using1.8and the fact thatλ₁, λ₂≥0, we have

F^α,s,q

uj;λ1, λ2

≤ u_j

L^∞Rⁿ

∇uj

LⁿRⁿ

n/n−1

−λ1 1

C_α,s,R0 u_j

B^s,pα,s,qRⁿ

∇uj

LⁿRⁿ

−λ2◦ 1

Cα,s,R0

u_j

B^s,pα,s,qRⁿ

∇uj

LⁿRⁿ

≤ v_j

L^∞Ω

∇vj

LⁿΩ

n/n−1

−λ₁ v_j

B^s,pα,s,qΩ

∇vj

LⁿΩ

−λ2◦ v_j

B^s,pα,s,qΩ

∇vj

LⁿΩ

Cα,s,R0,λ1,λ2

F^α,s,q

vj;λ1, λ2

Cα,s,R0,λ1,λ2,

3.9

from which we conclude thatF^α,s,qvj;λ1, λ2 → ∞asj → ∞.

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

∞ l−∞

ϕ⁰

2^l2t

1 fort >0. 3.10

Observe that3.10forcesϕ⁰2 1.

Proposition 3.3. iIt holds

χ_1/2^j1_,1/4 t≤^j

l1

ϕ⁰

2^l2t

≤χ_1/2^j2_,1/2 t forj∈N. 3.11

iiIt holds

_∞

0

ϕ⁰t

t dtlog 2. 3.12

Proof. iIn view of the size of the support ofϕ⁰, we easily obtain3.11.

iiIf we integrate both the sides of inequality3.11, then we have 1

j _1/4

1/2^j1

1 tdt≤ 1

j j

l1

_∞

0

ϕ⁰ 2^l2t

t dt

_∞

0

ϕ⁰t

t dt≤ 1 j

_1/2

1/2^j2

1

tdt. 3.13

As a consequence, it follows that

log 2 1−1

j

≤ _∞

0

ϕ⁰t

t dt≤ log 2 11

j

. 3.14

A passage to the limit asj → ∞therefore yields3.12.

Define

wlx _∞

|x|

ϕ⁰ 2^l2t

t dt forx∈Rⁿ, l∈N. 3.15

Note thatw_lw₁2^l−1·. Set

ujx 1 log 2

j j

l1

wlx forx∈Rⁿ, j∈N. 3.16

We also note that suppu_j ⊂B_1/2since suppw_l⊂B_1/2^l.

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

Lemma 3.4. Letn≥2 and 0< p <∞. Then one has uj

L^∞Rⁿ1 forj∈N, 3.17

1−1/jp

nΛⁿ⁻¹₁ 2^nj1 ≤uj^p

L^pRⁿ ≤ 11/jp

nΛⁿ⁻¹₁ forj∈N, 3.18

1−1/j log 2

Λ1

_n−1

jⁿ⁻¹ ≤∇ujⁿ

LⁿRⁿ ≤ 11/j log 2

Λ1

_n−1

jⁿ⁻¹ forj ∈N. 3.19

Proof. It is not so hard to prove3.17. Indeed, a change of variables yields

uj

L^∞Rⁿuj0 1 log 2

j _∞

0

j l1

ϕ⁰ 2^l2t

t dt 1

log 2 _∞

0

ϕ⁰t

t dt. 3.20

Thus, we obtain3.17by applying3.12.

We next verify3.18. Recall that Λ1 is defined byΛ1 1/ω^1/n−1_n−1 . If we insert the definitions3.15and3.16, then we have

uj^p

L^pRⁿ 1

Λⁿ⁻¹₁ log 2

j_{p 1}

0

_∞

r

1 t

j l1

ϕ⁰

2^l2t dt

p

rⁿ⁻¹dr. 3.21