A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class

Abstract

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brézis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

1. Introduction and Main Results

In [1], a generalization of the Ogawa type inequality [2] to the parabolic framework has been shown. Ogawa inequality can be considered as a generalized version in the Lizorkin-Triebel spaces of the remarkable estimate of Brézis-Gallouët-Wainger [3, 4] that holds in a limiting case of the Sobolev embedding theorem. The inequality showed in [1, Theorem 1.1] provides an estimate of the 𝐿^∞ norm of a function in terms of its parabolic BMO norm, with the aid of the square root of the logarithmic dependency of a higher order Sobolev norm. More precisely, for any vector-valued function 𝑓=∇𝑔∈𝑊^2𝑚,𝑚₂(ℝ^𝑛+1), 𝑔∈𝐿²(ℝ^𝑛+1) with 𝑚,𝑛∈ℕ^∗, 2𝑚>(𝑛+2)/2, there exists a constant 𝐶=𝐶(𝑚,𝑛)>0 such that:‖𝑓‖_{𝐿∞(ℝ𝑛+1)}≤𝐶1+‖𝑓‖_{BMO(ℝ𝑛+1)}log⁺‖𝑓‖_𝑊^2𝑚,𝑚_2(ℝ𝑛+1)+‖𝑔‖_{𝐿∞(ℝ𝑛+1)}^1/2,(1.1) where 𝑊^2𝑚,𝑚₂ is the parabolic Sobolev space (we refer to [5] for the definition and further properties), and BMO is the parabolic bounded mean oscillation space (defined via parabolic balls instead of Euclidean ones [1, Definition 2.1]). The above inequality reflects a limiting case of Sobolev embeddings in the parabolic framework (see [6, 7] for similar type inequalities, and [2–4, 8–11] for various elliptic versions). By considering functions 𝑓∈𝑊^2𝑚,𝑚₂(Ω_𝑇) defined on the bounded domainΩ_𝑇=(0,1)^𝑛×(0,𝑇),𝑇>0,(1.2) we have the following estimate (see [1, Theorem 1.2]):‖𝑓‖_{𝐿∞(Ω𝑇)}≤𝐶1+‖𝑓‖_BMO(Ω𝑇)+‖𝑓‖_{𝐿1(Ω𝑇)}log⁺‖𝑓‖_𝑊^2𝑚,𝑚_2(Ω𝑇)^1/2.(1.3) The different norms of 𝑓 appearing in inequalities (1.1) and (1.3) are finite since𝑊^2𝑚,𝑚₂↪𝐶^𝛾,𝛾/2↪𝐿^∞↪BMO,forsome0<𝛾<1,(1.4) where 𝐶^𝛾,𝛾/2 is the parabolic Hölder space that will be defined later. Moreover, it is easy to check that 𝑔 is bounded and continuous.

The purpose of this paper is to show that the condition 𝑓=∇𝑔∈𝑊^2𝑚,𝑚₂ (vector-valued case), or 𝑓∈𝑊^2𝑚,𝑚₂ (scalar-valued case) can be relaxed. Indeed, inequalities (1.1) and (1.3) can be applied to a wider class of Hölder continuous functions 𝑓=∇𝑔∈𝐶^𝛾,𝛾/2, 0<𝛾<1 (vector-valued case), or 𝑓∈𝐶^𝛾,𝛾/2 (scalar-valued case). To be more precise, we now state the main results of this paper. Our first theorem is the following:

Theorem 1.1 (Logarithmic Hölder inequality on ℝ^𝑛+1). Let 0<𝛾<1. For any 𝑓=∇𝑔∈𝐶^𝛾,𝛾/2(ℝ^𝑛+1)∩𝐿²(ℝ^𝑛+1)   with 𝑔∈𝐿²(ℝ^𝑛+1), there exists a constant 𝐶=𝐶(𝛾,𝑛)>0  such that ‖𝑓‖_{𝐿∞(ℝ𝑛+1)}≤𝐶1+‖𝑓‖_{BMO(ℝ𝑛+1)}log⁺‖𝑓‖_{𝐶𝛾,𝛾/2(ℝ𝑛+1)}+‖𝑔‖_{𝐿∞(ℝ𝑛+1)}^1/2.(1.5)

The second theorem deals with functions defined on the bounded domain Ω_𝑇.

Theorem 1.2 (Logarithmic Hölder inequality on a bounded domain). Let 0<𝛾<1. For any 𝑓∈𝐶^𝛾,𝛾/2(Ω_𝑇), there exists a constant 𝐶=𝐶(𝛾,𝑛,𝑇)>0  such that ‖𝑓‖_{𝐿∞(Ω𝑇)}≤𝐶1+‖𝑓‖_BMO(Ω𝑇)+‖𝑓‖_{𝐿1(Ω𝑇)}log⁺‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)}^1/2.(1.6)

We notice that inequalities (1.5) and (1.6) directly imply (with the aid of the embeddings (1.4)), (1.1), and (1.3).

Remark 1.3. The same inequality (1.5) still holds for scalar-valued functions 𝑓=𝜕𝑔/𝜕𝑥_𝑖∈𝐶^𝛾,𝛾/2(ℝ^𝑛+1)∩𝐿²(ℝ^𝑛+1), 𝑖∈1,…,𝑛+1, with 𝑔∈𝐿^∞(ℝ^𝑛+1).
This paper is organized as follows: in Section 2, we give the definitions of some basic functional spaces used throughout this paper. Section 3 is devoted to the proofs of the main results.

2. Definitions

Let 𝒪 be an open subset of ℝ^𝑛+1. A generic element 𝑧∈ℝ^𝑛+1 has the form 𝑧=(𝑥,𝑡) with 𝑥=(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛. We begin by defining parabolic Hölder spaces 𝐶^𝛾,𝛾/2.

Definition 2.1 (Parabolic Hölder spaces). For 0<𝛾<1, we define the parabolic space of Hölder continuous functions of order 𝛾 in the following way: 𝐶^𝛾,𝛾/2(𝒪)=𝑓∈𝐶𝒪,‖𝑓‖_{𝐶𝛾,𝛾/2(𝒪)}<∞,(2.1) where ‖𝑓‖_{𝐶𝛾,𝛾/2(𝒪)}=‖𝑓‖_{𝐿∞(𝒪)}+⟨𝑓⟩^(𝛾)_𝑥,𝒪+⟨𝑓⟩^(𝛾/2)_𝑡,𝒪,(2.2) with ⟨𝑓⟩^(𝛾)_𝑥,𝒪=sup_{(𝑥,𝑡),(𝑥,𝑡)∈𝒪,𝑥≠𝑥}||𝑓(𝑥,𝑡)−𝑓𝑥^,𝑡||||𝑥−𝑥′||^𝛾,⟨𝑓⟩^(𝛾/2)_𝑡,𝒪=sup_{(𝑥,𝑡),(𝑥,𝑡)∈𝒪,𝑡≠𝑡}||𝑓(𝑥,𝑡)−𝑓𝑥,𝑡^||||𝑡−𝑡′||^𝛾/2.(2.3)

For a detailed study of parabolic Hölder spaces, we refer the reader to [5]. We now briefly recall some basic facts about Littlewood-Paley decomposition which are crucial in obtaining our logarithmic inequalities. Given the expansive (𝑛+1)×(𝑛+1) matrix 𝐴=diag{2,…,2,2²} (parabolic anisotropy), the corresponding Littlewood-Paley decomposition asserts that any tempered distribution 𝑓∈𝒮^(ℝ^𝑛+1) can be decomposed as𝑓=_𝑗∈ℤ𝜑_𝑗∗𝑓,where𝜑_𝑗(𝑧)=||det𝐴||^𝑗𝜑𝐴^𝑗𝑧,(2.4) with the convergence in 𝒮′/𝒫 (modulo polynomials). Here 𝜑∈𝒮(ℝ^𝑛+1) is a test function such that supp𝜑 is compact and bounded away from the origin, and ∑_𝑗∈ℤ𝜑(𝐴^𝑗𝑧)=1 for all 𝑧∈ℝ^𝑛+1⧵{0}, where 𝜑 is the Fourier transform of 𝜑. The sequence (𝜑_𝑗)_𝑗∈ℤ is mainly used to define parabolic homogeneous Lizorkin-Triebel, Hardy, and Besov spaces (see for instance [12, 13]). We only present here the spaces that are used throughout the analysis. For 1≤𝑝≤∞, we define the parabolic homogeneous Lizorkin-Triebel space 𝐹̇‌⁰_𝑝,2 as the space of functions 𝑓∈𝒮^(ℝ^𝑛+1) with finite quasinorms:‖𝑓‖_̇‌𝐹⁰_{𝑝,2(ℝ𝑛+1)}=‖‖‖‖⎛⎜⎜⎝_||𝑗_||<∞||𝜑_𝑗∗𝑓||²⎞⎟⎟⎠^1/2‖‖‖‖_{𝐿𝑝(ℝ𝑛+1)}<∞.(2.5) The space 𝐹̇‌⁰_𝑝,2 can be identified with the parabolic Hardy space 𝐻^𝑝, 1≤𝑝<∞ having the following square function characterization stated informally as:𝐻^𝑝ℝ^𝑛+1=⎧⎪⎨⎪⎩𝑓∈𝒮^ℝ^𝑛+1;⎛⎜⎜⎝_||𝑗_||<∞||𝜑_𝑗∗𝑓||²⎞⎟⎟⎠^1/2∈𝐿^𝑝⎫⎪⎬⎪⎭.(2.6) This identification (see Bownik [14]) can be stated as follows: for all 1≤𝑝<∞, we have𝐹̇‌⁰_𝑝,2ℝ^𝑛+1≃𝐻^𝑝ℝ^𝑛+1.(2.7) Now, for defining the inhomogeneous parabolic Besov space 𝐵^𝛾_∞,∞ used later in obtaining our results, we use a slightly different sequence. Indeed, let 𝜃∈𝐶^∞₀(ℝ^𝑛+1) be any cut-off function satisfying:𝜃(𝑧)=1,if|𝑧|_𝑝≤1,0,if|𝑧|_𝑝≥2,(2.8) where |⋅|_𝑝 is the parabolic quasinorm associated to the matrix 𝐴 (see [1]). Taking the new function (but keeping the same notation) 𝜑₀ defined via the relation𝜑₀=𝜃,(2.9) we can give the definition of the Besov space 𝐵^𝛾_∞,∞.

Definition 2.2 (Parabolic inhomogeneous Besov spaces). Take the smoothness parameter 0<𝛾<1. Let (𝜑_𝑗)_𝑗∈ℤ be the sequence such that 𝜑₀ is given by (2.9), while 𝜑_𝑗 is given by (2.4) for all 𝑗≥1. We define the parabolic inhomogeneous Besov space 𝐵^𝛾_∞,∞ as the space of all functions 𝑓∈𝒮^(ℝ^𝑛+1) with finite quasinorms ‖𝑓‖_𝐵^𝛾_∞,∞=sup_𝑗≥02^𝛾𝑗‖‖𝜑_𝑗∗𝑓‖‖_{𝐿∞(ℝ𝑛+1)}.(2.10)

3. Proofs of Theorems

The proof of Theorem 1.1 relies on the following two lemmas of different interests.

Lemma 3.1. Let 0<𝛾<1 and let 𝑁>0 be a positive integer. Then for any 𝑓=∇𝑔∈𝐶^𝛾,𝛾/2(ℝ^𝑛+1)∩𝐿²(ℝ^𝑛+1) with 𝑔∈𝐿²(ℝ^𝑛+1), there exists a constant 𝐶=𝐶(𝛾,𝑛)>0 such that ‖‖‖‖⎛⎜⎝_𝑗_<−𝑁2^−2𝛾𝑗||𝜑_𝑗∗𝑓||²⎞⎟⎠^1/2‖‖‖‖_𝐿∞≤𝐶‖𝑔‖_𝐿∞.(3.1)

Proof. We provide a proof of (3.1) in the general case 𝑁=1. We use the fact that 𝜕_𝑖𝑔=𝑓_𝑖 (for which we keep denoting it by 𝑓, i.e. 𝑓=𝑓_𝑖) for some 𝑖=1,…,𝑛+1, with 𝑔∈𝐿^∞(ℝ^𝑛+1). For 𝑧∈ℝ^𝑛+1, define Φ(𝑧)=𝜕_𝑖𝜑(𝑧),(3.2)Φ_𝑗(𝑧)=||det𝐴||^𝑗Φ𝐴^𝑗𝑧,∀𝑗≤−1.(3.3) Using (2.4) we obtain: 𝜕_𝑖𝜑_𝑗(𝑧)=2^𝑗Φ_𝑗(𝑧)if𝑖=1,…,𝑛,2^2𝑗Φ_𝑗(𝑧)if𝑖=𝑛+1.(3.4) We now compute (see (3.3) and (3.4)): ‖‖‖‖⎛⎜⎝_𝑗_≤−12^−2𝛾𝑗||𝜑_𝑗∗𝑓||²⎞⎟⎠^1/2‖‖‖‖_𝐿∞≤𝐶sup_𝑗≤−1‖‖Φ_𝑗∗𝑔‖‖_𝐿∞,(3.5) where the constant 𝐶 is given by: 𝐶²=⎧⎪⎨⎪⎩_𝑗_≤−12^{2𝑗(1−𝛾)},if𝑖=1,…,𝑛,_𝑗_≤−12^{2𝑗(2−𝛾)},if𝑖=𝑛+1,(3.6) which is finite 0<𝐶<+∞ under the choice 0<𝛾<1.(3.7) In order to terminate the proof, it suffices to show that ‖‖Φ_𝑗∗𝑔‖‖_𝐿∞≤𝐶‖𝑔‖_𝐿∞,(3.8) which can be deduced, by translation and dilation invariance, from the following estimate: ||(Φ∗𝑔)(0)||≤𝐶‖𝑔‖_𝐿∞.(3.9) Indeed, define the positive radial decreasing function ℎ(𝑟)=ℎ(‖𝑧‖) as follows: ℎ(𝑟)=sup_{‖𝑧‖≥𝑟}||Φ(𝑧)||.(3.10) From (3.2), we remark that the function Φ is the inverse Fourier transform of a compactly supported function. Hence, we have ℎ(0)=‖Φ‖_𝐿∞<+∞,(3.11) and the asymptotic behavior ℎ(𝑟)≤𝐶𝑟^𝑛+2,∀𝑟≥1.(3.12) We compute (taking 𝑆^𝑛_𝑟 as the 𝑛-dimensional sphere of radius 𝑟): ||(Φ∗𝑔)(0)||≤_ℝ𝑛+1||Φ(−𝑧)||||𝑔(𝑧)||𝑑𝑧≤^∞₀_𝑆𝑛𝑟||Φ(−𝑧)||||𝑔(𝑧)||𝑑𝜎(𝑧)𝑑𝑟≤𝐶^∞₀𝑟^𝑛ℎ(𝑟)𝑑𝑟‖𝑔‖_𝐿∞.(3.13) Using (3.11) and (3.12) we deduce that: ^∞₀𝑟^𝑛ℎ(𝑟)𝑑𝑟=¹₀𝑟^𝑛ℎ(𝑟)𝑑𝑟+^∞₁𝑟^𝑛ℎ(𝑟)𝑑𝑟≤𝐶¹₀ℎ(0)𝑑𝑟+^∞₁𝑟^𝑛𝑟^𝑛+2𝑑𝑟≤𝐶‖Φ‖_𝐿∞+1,(3.14) which, together with (3.13), directly implies (3.9). As a conclusion, we obtain (see (3.5)): ‖‖‖‖⎛⎜⎝_𝑗_≤−12^−2𝛾𝑗||𝜑_𝑗∗𝑓||²⎞⎟⎠^1/2‖‖‖‖_𝐿∞≤𝐶‖𝑔‖_𝐿∞,(3.15) and hence inequality (3.1) holds.

Lemma 3.2. Let 𝑁>0 be a positive integer. Then for any 𝑓∈BMO(ℝ^𝑛+1) there exists a constant 𝐶=𝐶(𝑛)>0 such that: ‖‖‖‖⎛⎜⎜⎝_||𝑗_||<𝑁||𝜑_𝑗∗𝑓||²⎞⎟⎟⎠^1/2‖‖‖‖_𝐿∞≤𝐶‖𝑓‖_BMO.(3.16)

Proof. The proof provides inequality (3.16) for all |𝑗|<∞ by showing that 𝐹̇‌⁰_∞,2≃BMO and then using (2.5). Before starting the proof, we remind the reader that: ‖𝑓‖_{BMO(ℝ𝑛+1)}=sup_{𝒬⊆ℝ𝑛+1}inf_𝑐∈ℝ⎛⎜⎝1||𝒬||_𝒬||𝑓−𝑐||⎞⎟⎠,(3.17) where 𝒬 denotes any arbitrary parabolic cube. Using the result of Bownik [15, Theorem 1.2], we have the following duality argument (that can be viewed as the parabolic extension of the well-known isotropic result of Triebel [12], and Frazier and Jawerth [16]): 𝐹̇‌⁰_1,2^≃𝐹̇‌⁰_∞,2,(3.18) where (𝐹̇‌⁰_1,2)′ stands for the dual space of 𝐹̇‌⁰_1,2. Applying (2.7) with 𝑝=1 we obtain: 𝐹̇‌⁰_1,2≃𝐻¹.(3.19) Using the description of the dual of parabolic Hardy spaces 𝐻^𝑝 for 0<𝑝≤1 (see Bownik [17, Theorem 8.3]), we get: (𝐻^𝑝)^=𝒞^𝑙_𝑞,𝑠(3.20) with the terms 𝑝,𝑙,𝑞,𝑠 chosen such that: 𝑙=1𝑝−1,1≤𝑞𝑞−1≤∞,𝑝<𝑞𝑞−1,𝑠∈ℕ,𝑠≥⌊𝑙⌋,⌊𝑙⌋=max{𝑛∈ℤ;𝑛≤𝑙}.(3.21) The function space 𝒞^𝑙_𝑞,𝑠, 𝑙≥0, 1≤𝑞<∞ and 𝑠∈ℕ (called the Campanato space), is the space of all 𝑓∈𝐿^𝑞_loc(ℝ^𝑛+1) (defined up to addition by 𝑃∈𝒫_𝑠; the set of all polynomials in (𝑛+1) variables of degree at most 𝑠) so that: ‖𝑓‖_𝒞^𝑙_{𝑠,𝑞(ℝ𝑛+1)}=sup_{𝒬⊆ℝ𝑛+1}inf_{𝑃∈𝒫𝑠}||𝒬||^𝑙⎛⎜⎝1||𝒬||_𝒬||𝑓−𝑃||^𝑞⎞⎟⎠^1/𝑞<∞.(3.22) Choosing 𝑝=1, 𝑙=0, 𝑞=1 and 𝑠=0, we can easily see that conditions (3.21) are all satisfied, and that (see (3.22) and (3.17)): 𝒞⁰_1,0≃BMO.(3.23) This identification, together with (3.20), finally gives 𝐻¹^≃BMO.(3.24) The proof then directly follows from (3.18), (3.19), and (3.24).

Proof of Theorem 1.1. Let 𝑁∈ℕ be any arbitrary integer. Using (2.4), we estimate ‖𝑓‖_𝐿∞ in the following way: ‖𝑓‖_𝐿∞≤‖‖‖‖_𝑗<−𝑁2^𝛾𝑗2^−𝛾𝑗||𝜑_𝑗∗𝑓||‖‖‖‖_𝐿∞+‖‖‖‖_||𝑗_||≤𝑁||𝜑_𝑗∗𝑓||‖‖‖‖_𝐿∞+‖‖‖‖_𝑗>𝑁2^−𝛾𝑗2^𝛾𝑗||𝜑_𝑗∗𝑓||‖‖‖‖_𝐿∞≤𝐶_𝛾2^{−𝛾𝑁𝐴1}‖‖‖‖⎛⎜⎝_𝑗_<−𝑁2^−2𝛾𝑗|𝜑_𝑗∗𝑓|²⎞⎟⎠^1/2‖‖‖‖_𝐿∞+(2𝑁+1)^{1/2𝐴2}‖‖‖‖⎛⎜⎝_|𝑗_|≤𝑁||𝜑_𝑗∗𝑓||²⎞⎟⎠^1/2‖‖‖‖_𝐿∞+𝐶^_𝛾2^{−𝛾𝑁𝐴3}sup_𝑗>𝑁2^𝛾𝑗‖‖𝜑_𝑗∗𝑓‖‖_𝐿∞,(3.25) where 𝐶_𝛾=12^2𝛾−1^1/2,𝐶^_𝛾=2^−𝛾1−2^−𝛾.(3.26) Using (3.1), we assert that: 𝐴₁≤𝐶‖𝑔‖_𝐿∞,(3.27) while (3.16) gives: 𝐴₂≤𝐶‖𝑓‖_BMO.(3.28) In order to estimate 𝐴₃, we proceed in the following way: 𝐴₃≤sup_𝑗≥12^𝛾𝑗‖‖𝜑_𝑗∗𝑓‖‖_𝐿∞≤sup_𝑗≥12^𝛾𝑗‖‖𝜑_𝑗∗𝑓‖‖_𝐿∞+‖‖𝜑₀∗𝑓‖‖_𝐿∞,𝜑₀isgivenby(2.9),(3.29) hence (see Definition 2.2) 𝐴₃≤‖𝑓‖_𝐵^𝛾_∞,∞.(3.30) Using the well-known result (see, e.g., [18]) 𝐵^𝛾_∞,∞=𝐶^𝛾,𝛾/2,(3.31) we finally obtain 𝐴₃≤‖𝑓‖_{𝐶𝛾,𝛾/2}.(3.32) Inequalities (3.25), (3.27), (3.28), and (3.32) imply ‖𝑓‖_𝐿∞≤𝐶(2𝑁+1)^1/2‖𝑓‖_BMO+2^−𝛾𝑁‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞.(3.33) We optimize (3.33) in 𝑁 by setting: 𝑁=1if‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞≤2^𝛾‖𝑓‖_BMO.(3.34) Then it is easy to check (using (3.33)) that ‖𝑓‖_∞≤𝐶‖𝑓‖_BMO1+log⁺‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞‖𝑓‖_BMO^1/2.(3.35) In the case where ‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞>2^𝛾‖𝑓‖_BMO, we take 1≤𝛽<2^𝛾 such that 𝑁=𝑁(𝛽)=log⁺_2𝛾𝛽‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞‖𝑓‖_BMO−12∈ℕ.(3.36) In fact this is valid since the function 𝑁(𝛽) varies continuously from 𝑁(1) to 𝑁(2^𝛾)=1+𝑁(1) on the interval [1,2^𝛾]. Using (3.33) with the above choice of 𝑁, we obtain: ‖𝑓‖_∞≤C2^1/2log⁺_2𝛾𝛽‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞‖𝑓‖_BMO^1/2‖𝑓‖_BMO+2^𝛾/2𝛽‖𝑓‖_BMO≤𝐶2(𝛾log2)^1/2log⁺‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞‖𝑓‖_BMO^1/2‖𝑓‖_BMO+2^𝛾/2𝛽‖𝑓‖_BMO,(3.37) where for the second line we have used the fact that log⁺𝛽<log⁺‖𝑓‖_{𝐶𝛾,𝛾/2}+‖𝑔‖_𝐿∞‖𝑓‖_BMO.(3.38) The above computations again imply (3.35). By using the inequality: 𝑥log𝑒+𝑦𝑥^1/2≤𝐶1+𝑥(log(𝑒+𝑦))^1/2,for0<𝑥≤1,𝐶𝑥(log(𝑒+𝑦))^1/2,for𝑥>1,(3.39) in (3.35), we directly arrive to our result.

We now present the proof of Theorem 1.2 that involves finer estimates on the Hölder norm.

Proof of Theorem 1.2. For the sake of simplifying the ideas of the proof, we only consider 1-spatial dimensions 𝑥=𝑥₁. The general 𝑛-dimensional case can be easily deduced. Following the same notations of [1], we let Ω_𝑇=(−1,2)×(−𝑇,2𝑇), 𝒵₁⊆𝒵₂⊆Ω_𝑇 such that 𝒵₁=(𝑥,𝑡);−14<𝑥<54,−𝑇4<𝑡<5𝑇4,𝒵₂=(𝑥,𝑡);−34<𝑥<74,−3𝑇4<𝑡<7𝑇4.(3.40) We also take the cut-off function Ψ∈𝐶^∞₀(ℝ²), 0≤Ψ≤1 satisfying: Ψ(𝑥,𝑡)=1,for(𝑥,𝑡)∈𝒵₁,0,for(𝑥,𝑡)∈ℝ²⧵𝒵₂.(3.41) The main idea of the proof consists in extending the function 𝑓 to a suitable function of the form Ψ𝑓̃‌, where 𝑓̃‌ is defined on Ω_𝑇. We then apply inequality (1.5) (the scalar-valued version with 𝑛=1) to Ψ𝑓̃‌, and we estimate the different norms in order to get the result. However, away from the complicated extension (Sobolev extension) of the function 𝑓̃‌ that was done in [1], we here consider a simpler symmetric extension. Indeed, we first take the spatial symmetry of the function 𝑓: 𝑓(̃‌𝑥,𝑡)=𝑓(−𝑥,𝑡),for−1<𝑥<0,0≤𝑡≤𝑇,𝑓(2−𝑥,𝑡),for1<𝑥<2,0≤𝑡≤𝑇,(3.42) and then the symmetry with respect to 𝑡: 𝑓(̃‌𝑥,𝑡)=𝑓(𝑥,−𝑡),for−1<𝑥<2,−𝑇<𝑡≤0,𝑓(𝑥,2𝑇−𝑡),for−1<𝑥<2,𝑇≤𝑡<2𝑇.(3.43) We claim that Ψ𝑓̃‌∈𝐶^𝛾,𝛾/2(ℝ²) with ‖‖Ψ𝑓‖‖̃‌_{𝐶𝛾,𝛾/2(ℝ2)}≤‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)}.(3.44) In this case, we apply the scalar-valued version of inequality (1.5) (see Remark 1.3) to the function Ψ𝑓̃‌ with 𝑖=1 and 𝑔(𝑥,𝑡)=∫^𝑥₀Ψ(𝑦,𝑡)𝑓(̃‌𝑦,𝑡)𝑑𝑦. This, together with the fact that Ψ=1 on Ω_𝑇, leads to the following estimate: ‖𝑓‖_{𝐿∞(Ω𝑇)}≤‖‖Ψ𝑓‖‖̃‌_{𝐿∞(ℝ2)}≤𝐶1+‖‖Ψ𝑓‖‖̃‌_BMO(ℝ2)log⁺‖‖Ψ𝑓‖‖̃‌_{𝐶𝛾,𝛾/2(ℝ2)}+‖𝑔‖_{𝐿∞(ℝ2)}^1/2.(3.45) It is worth noticing that choosing 𝑖=1 above is somehow restrictive. In fact, we could also have used the inequality with 𝑖=2 and 𝑔(𝑥,𝑡)=∫^𝑡₀Ψ(𝑥,𝑠)𝑓(̃‌𝑥,𝑠)𝑑𝑠.
In [7] it was shown that ‖Ψ𝑓‖̃‌_BMO(ℝ2)≤𝐶(‖𝑓‖_BMO(Ω𝑇)+‖𝑓‖_{𝐿1(Ω𝑇)}), while it is clear that ‖𝑔‖_{𝐿∞(ℝ2)}≤𝐶‖𝑓‖̃‌_𝐿∞(Ω_𝑇)≤𝐶‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)}. These arguments, along with (3.44) and (3.45), directly terminate the proof. The only point left is to show the claim (3.44). Recall the norm ‖‖Ψ𝑓‖‖̃‌_{𝐶𝛾,𝛾/2(ℝ2)}=‖‖Ψ𝑓‖‖̃‌_{𝐿∞(ℝ2)}+Ψ𝑓̃‌^(𝛾)_𝑥,ℝ2+Ψ𝑓̃‌^(𝛾/2)_𝑡,ℝ2.(3.46) It is evident that ‖‖Ψ𝑓‖‖̃‌_{𝐿∞(ℝ2)}≤𝐶‖𝑓‖_{𝐿∞(Ω𝑇)},(3.47) hence we only need to estimate the two terms ⟨Ψ𝑓⟩̃‌^(𝛾)_𝑥,ℝ2 and ⟨Ψ𝑓⟩̃‌^(𝛾/2)_𝑡,ℝ2. We only deal with ⟨Ψ𝑓⟩̃‌^(𝛾)_𝑥,ℝ2 since the second term can be treated similarly. We examine the different positions of (𝑥,𝑡),(𝑥′,𝑡)∈ℝ². If (𝑥,𝑡),(𝑥^,𝑡)∈ℝ²⧵𝒵₂, 𝑥≠𝑥′, then (since Ψ=0 over ℝ²⧵𝒵₂): ||Ψ𝑓̃‌(𝑥,𝑡)−Ψ𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾=0.(3.48) If both (𝑥,𝑡),(𝑥′,𝑡)∈Ω_𝑇, 𝑥≠𝑥′, then the special extension (3.42) and (3.43) of the function 𝑓 guarantees the existence of 𝑥,𝑡,𝑥′,𝑡∈Ω_𝑇,(3.49) such that: 𝑓(̃‌𝑥,𝑡)=𝑓𝑥,𝑡,𝑓(̃‌𝑥′,𝑡)=𝑓𝑥′,𝑡.(3.50) Two cases can be considered. Either 𝑥=𝑥′ (see Figure 1), then we forcedly have 𝑓(̃‌𝑥,𝑡)=𝑓(̃‌𝑥′,𝑡),(3.51) and therefore ||Ψ𝑓̃‌(𝑥,𝑡)−Ψ𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤⟨Ψ⟩^(𝛾)_𝑥,Ω_𝑇‖‖𝑓‖‖̃‌_𝐿∞(Ω_𝑇)≤𝐶‖𝑓‖_{𝐿∞(Ω𝑇)}≤𝐶‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)},(3.52) or 𝑥≠𝑥′, then we forcedly have (see Figure 2) ||𝑥−𝑥^||^𝛾≥||𝑥−𝑥^||^𝛾.(3.53) In this case, we compute ||Ψ𝑓̃‌(𝑥,𝑡)−Ψ𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤||𝑓(̃‌𝑥,𝑡)||||Ψ(𝑥,𝑡)−Ψ𝑥^,𝑡||||𝑥−𝑥^||^𝛾+||Ψ𝑥^,𝑡||||𝑓(̃‌𝑥,𝑡)−𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤‖‖𝑓‖‖̃‌_𝐿∞(Ω_𝑇)⟨Ψ⟩^(𝛾)_𝑥,Ω_𝑇+||𝑓(̃‌𝑥,𝑡)−𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾.(3.54) Using (3.50) and (3.53), we deduce that: ||𝑓(̃‌𝑥,𝑡)−𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾=||𝑓𝑥,𝑡−𝑓𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤||𝑓𝑥,𝑡−𝑓𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤⟨𝑓⟩^(𝛾)_𝑥,Ω𝑇,(3.55) therefore, by (3.54), we obtain ||Ψ𝑓̃‌(𝑥,𝑡)−Ψ𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤‖‖𝑓‖‖̃‌_𝐿∞(Ω_𝑇)⟨Ψ⟩^(𝛾)_𝑥,Ω_𝑇+⟨𝑓⟩^(𝛾)_𝑥,Ω𝑇≤𝐶‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)}.(3.56) The remaining case is when (𝑥,𝑡)∈𝒵₂ and (𝑥′,𝑡)∈ℝ²⧵Ω_𝑇 (see Figure 3). In this case, we have (Ψ𝑓)̃‌(𝑥^,𝑡)=0 and ||𝑥−𝑥^||^𝛾≥14^𝛾,(3.57) hence ||Ψ𝑓̃‌(𝑥,𝑡)−Ψ𝑓̃‌𝑥^,𝑡||||𝑥−𝑥^||^𝛾≤4^𝛾‖‖𝑓‖‖̃‌_{𝐿∞(𝒵2)}≤𝐶‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)}.(3.58) From (3.48), (3.52), (3.56), and (3.58), we finally deduce that Ψ𝑓̃‌^(𝛾)_𝑥,ℝ2≤𝐶‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)}.(3.59) Arguing in exactly the same way as above, we also find that: Ψ𝑓̃‌^(𝛾/2)_𝑡,ℝ2≤𝐶‖𝑓‖_{𝐶𝛾,𝛾/2(Ω𝑇)},(3.60) with a possibly different constant 𝐶 that depends on 𝑇. Indeed, the term 𝑇 enters in estimating ⟨Ψ𝑓⟩̃‌^(𝛾/2)_𝑡,ℝ2 since (3.57) is now replaced (see again Figure 3) by ||𝑡−𝑡^||^𝛾≥𝑇4^𝛾.(3.61) This shows the claim.

Figure 1 

Case (𝑥,𝑡),(𝑥,𝑡)∈Ω𝑇 with 𝑥=𝑥′.

(a)

(b)

Figure 2 

Case (𝑥,𝑡),(𝑥′,𝑡)∈Ω𝑇 with 𝑥≠𝑥′. (a): 𝑥′≠𝑥′. (b):𝑥′=𝑥′.

Figure 3 

Case (𝑥,𝑡)∈𝒵2 and (𝑥,𝑡)∈ℝ2⧵Ω𝑇.

Remark 3.3. In the case of multispatial coordinates 𝑥_𝑖, 𝑖=1,…,𝑛, we simultaneously apply the extension (3.42) to each spatial coordinate while fixing all other coordinates including 𝑡. Finally, by fixing the spatial variables, we make the extension with respect to 𝑡 as in (3.43).

Acknowledgments

The author is highly indebted to Professor Lars-Erik Persson and to an anonymous referee for their useful comments on earlier versions of the paper.