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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 148706, 15 pages
Research Article

A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class

1Mathematics Department, School of Arts and Sciences, Lebanese International University (LIU), Beirut Campus, Al-Mouseitbeh, P.O. Box 14-6404, Beirut, Lebanon
2Mathematics Department, Faculty of Sciences, Lebanese University, Hadeth, Beirut, Lebanon

Received 6 December 2010; Accepted 14 December 2010

Academic Editor: Lars Erik Persson

Copyright © 2012 H. Ibrahim. 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.


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=𝑔𝑊2𝑚,𝑚2(𝑛+1), 𝑔𝐿2(𝑛+1) with 𝑚,𝑛, 2𝑚>(𝑛+2)/2, there exists a constant 𝐶=𝐶(𝑚,𝑛)>0 such that:𝑓𝐿(𝑛+1)𝐶1+𝑓BMO(𝑛+1)log+𝑓𝑊2𝑚,𝑚2(𝑛+1)+𝑔𝐿(𝑛+1)1/2,(1.1) where 𝑊2𝑚,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 [24, 811] for various elliptic versions). By considering functions 𝑓𝑊2𝑚,𝑚2(Ω𝑇) defined on the bounded domainΩ𝑇=(0,1)𝑛×(0,𝑇),𝑇>0,(1.2) we have the following estimate (see [1, Theorem 1.2]):𝑓𝐿(Ω𝑇)𝐶1+𝑓BMO(Ω𝑇)+𝑓𝐿1(Ω𝑇)log+𝑓𝑊2𝑚,𝑚2(Ω𝑇)1/2.(1.3) The different norms of 𝑓 appearing in inequalities (1.1) and (1.3) are finite since𝑊2𝑚,𝑚2𝐶𝛾,𝛾/2𝐿BMO,forsome0<𝛾<1,(1.4) where 𝐶𝛾,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=𝑔𝑊2𝑚,𝑚2 (vector-valued case), or 𝑓𝑊2𝑚,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=𝑔𝐶𝛾,𝛾/2, 0<𝛾<1 (vector-valued case), or 𝑓𝐶𝛾,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 𝑛+1). Let 0<𝛾<1. For any 𝑓=𝑔𝐶𝛾,𝛾/2(𝑛+1)𝐿2(𝑛+1)   with 𝑔𝐿2(𝑛+1), there exists a constant 𝐶=𝐶(𝛾,𝑛)>0  such that 𝑓𝐿(𝑛+1)𝐶1+𝑓BMO(𝑛+1)log+𝑓𝐶𝛾,𝛾/2(𝑛+1)+𝑔𝐿(𝑛+1)1/2.(1.5)

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

Theorem 1.2 (Logarithmic Hölder inequality on a bounded domain). Let 0<𝛾<1. For any 𝑓𝐶𝛾,𝛾/2(Ω𝑇), there exists a constant 𝐶=𝐶(𝛾,𝑛,𝑇)>0  such that 𝑓𝐿(Ω𝑇)𝐶1+𝑓BMO(Ω𝑇)+𝑓𝐿1(Ω𝑇)log+𝑓𝐶𝛾,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/𝜕𝑥𝑖𝐶𝛾,𝛾/2(𝑛+1)𝐿2(𝑛+1), 𝑖1,,𝑛+1, with 𝑔𝐿(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 𝑛+1. A generic element 𝑧𝑛+1 has the form 𝑧=(𝑥,𝑡) with 𝑥=(𝑥1,,𝑥𝑛)𝑛. We begin by defining parabolic Hölder spaces 𝐶𝛾,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: 𝐶𝛾,𝛾/2(𝒪)=𝑓𝐶𝒪,𝑓𝐶𝛾,𝛾/2(𝒪)<,(2.1) where 𝑓𝐶𝛾,𝛾/2(𝒪)=𝑓𝐿(𝒪)+𝑓(𝛾)𝑥,𝒪+𝑓(𝛾/2)𝑡,𝒪,(2.2) with 𝑓(𝛾)𝑥,𝒪=sup(𝑥,𝑡),(𝑥,𝑡)𝒪,𝑥𝑥||𝑓(𝑥,𝑡)𝑓𝑥,𝑡||||𝑥𝑥||𝛾,𝑓(𝛾/2)𝑡,𝒪=sup(𝑥,𝑡),(𝑥,𝑡)𝒪,𝑡𝑡||𝑓(𝑥,𝑡)𝑓𝑥,𝑡||||𝑡𝑡||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 𝑓𝒮(𝑛+1) can be decomposed as𝑓=𝑗𝜑𝑗𝑓,where𝜑𝑗(𝑧)=||det𝐴||𝑗𝜑𝐴𝑗𝑧,(2.4) with the convergence in 𝒮/𝒫 (modulo polynomials). Here 𝜑𝒮(𝑛+1) is a test function such that supp𝜑 is compact and bounded away from the origin, and 𝑗𝜑(𝐴𝑗𝑧)=1 for all 𝑧𝑛+1{0}, where 𝜑 is the Fourier transform of 𝜑. The sequence (𝜑𝑗)𝑗 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 1𝑝, we define the parabolic homogeneous Lizorkin-Triebel space ̇𝐹0𝑝,2 as the space of functions 𝑓𝒮(𝑛+1) with finite quasinorms:𝑓̇𝐹0𝑝,2(𝑛+1)=||𝑗||<||𝜑𝑗𝑓||21/2𝐿𝑝(𝑛+1)<.(2.5) The space ̇𝐹0𝑝,2 can be identified with the parabolic Hardy space 𝐻𝑝, 1𝑝< having the following square function characterization stated informally as:𝐻𝑝𝑛+1=𝑓𝒮𝑛+1;||𝑗||<||𝜑𝑗𝑓||21/2𝐿𝑝.(2.6) This identification (see Bownik [14]) can be stated as follows: for all 1𝑝<, we havė𝐹0𝑝,2𝑛+1𝐻𝑝𝑛+1.(2.7) Now, for defining the inhomogeneous parabolic Besov space 𝐵𝛾, used later in obtaining our results, we use a slightly different sequence. Indeed, let 𝜃𝐶0(𝑛+1) be any cut-off function satisfying:𝜃(𝑧)=1,if|𝑧|𝑝1,0,if|𝑧|𝑝2,(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 relation𝜑0=𝜃,(2.9) we can give the definition of the Besov space 𝐵𝛾,.

Definition 2.2 (Parabolic inhomogeneous Besov spaces). Take the smoothness parameter 0<𝛾<1. Let (𝜑𝑗)𝑗 be the sequence such that 𝜑0 is given by (2.9), while 𝜑𝑗 is given by (2.4) for all 𝑗1. We define the parabolic inhomogeneous Besov space 𝐵𝛾, as the space of all functions 𝑓𝒮(𝑛+1) with finite quasinorms 𝑓𝐵𝛾,=sup𝑗02𝛾𝑗𝜑𝑗𝑓𝐿(𝑛+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=𝑔𝐶𝛾,𝛾/2(𝑛+1)𝐿2(𝑛+1) with 𝑔𝐿2(𝑛+1), there exists a constant 𝐶=𝐶(𝛾,𝑛)>0 such that 𝑗<𝑁22𝛾𝑗||𝜑𝑗𝑓||21/2𝐿𝐶𝑔𝐿.(3.1)

Proof. We provide a proof of (3.1) in the general case 𝑁=1. We use the fact that 𝜕𝑖𝑔=𝑓𝑖 (for which we keep denoting it by 𝑓, i.e. 𝑓=𝑓𝑖) for some 𝑖=1,,𝑛+1, with 𝑔𝐿(𝑛+1). For 𝑧𝑛+1, define Φ(𝑧)=𝜕𝑖𝜑(𝑧),(3.2)Φ𝑗(𝑧)=||det𝐴||𝑗Φ𝐴𝑗𝑧,𝑗1.(3.3) Using (2.4) we obtain: 𝜕𝑖𝜑𝑗(𝑧)=2𝑗Φ𝑗(𝑧)if𝑖=1,,𝑛,22𝑗Φ𝑗(𝑧)if𝑖=𝑛+1.(3.4) We now compute (see (3.3) and (3.4)): 𝑗122𝛾𝑗||𝜑𝑗𝑓||21/2𝐿𝐶sup𝑗1Φ𝑗𝑔𝐿,(3.5) where the constant 𝐶 is given by: 𝐶2=𝑗122𝑗(1𝛾),if𝑖=1,,𝑛,𝑗122𝑗(2𝛾),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 Φ𝑗𝑔𝐿𝐶𝑔𝐿,(3.8) which can be deduced, by translation and dilation invariance, from the following estimate: ||(Φ𝑔)(0)||𝐶𝑔𝐿.(3.9) Indeed, define the positive radial decreasing function (𝑟)=(𝑧) as follows: (𝑟)=sup𝑧𝑟||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 (0)=Φ𝐿<+,(3.11) and the asymptotic behavior (𝑟)𝐶𝑟𝑛+2,𝑟1.(3.12) We compute (taking 𝑆𝑛𝑟 as the 𝑛-dimensional sphere of radius 𝑟): ||(Φ𝑔)(0)||𝑛+1||Φ(𝑧)||||𝑔(𝑧)||𝑑𝑧0𝑆𝑛𝑟||Φ(𝑧)||||𝑔(𝑧)||𝑑𝜎(𝑧)𝑑𝑟𝐶0𝑟𝑛(𝑟)𝑑𝑟𝑔𝐿.(3.13) Using (3.11) and (3.12) we deduce that: 0𝑟𝑛(𝑟)𝑑𝑟=10𝑟𝑛(𝑟)𝑑𝑟+1𝑟𝑛(𝑟)𝑑𝑟𝐶10(0)𝑑𝑟+1𝑟𝑛𝑟𝑛+2𝑑𝑟𝐶Φ𝐿+1,(3.14) which, together with (3.13), directly implies (3.9). As a conclusion, we obtain (see (3.5)): 𝑗122𝛾𝑗||𝜑𝑗𝑓||21/2𝐿𝐶𝑔𝐿,(3.15) and hence inequality (3.1) holds.

Lemma 3.2. Let 𝑁>0 be a positive integer. Then for any 𝑓BMO(𝑛+1) there exists a constant 𝐶=𝐶(𝑛)>0 such that: ||𝑗||<𝑁||𝜑𝑗𝑓||21/2𝐿𝐶𝑓BMO.(3.16)

Proof. The proof provides inequality (3.16) for all |𝑗|< by showing that ̇𝐹0,2BMO and then using (2.5). Before starting the proof, we remind the reader that: 𝑓BMO(𝑛+1)=sup𝒬𝑛+1inf𝑐1||𝒬||𝒬||𝑓𝑐||,(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]): ̇𝐹01,2̇𝐹0,2,(3.18) where (̇𝐹01,2) stands for the dual space of ̇𝐹01,2. Applying (2.7) with 𝑝=1 we obtain: ̇𝐹01,2𝐻1.(3.19) Using the description of the dual of parabolic Hardy spaces 𝐻𝑝 for 0<𝑝1 (see Bownik [17, Theorem 8.3]), we get: (𝐻𝑝)=𝒞𝑙𝑞,𝑠(3.20) with the terms 𝑝,𝑙,𝑞,𝑠 chosen such that: 𝑙=1𝑝1,1𝑞𝑞1,𝑝<𝑞𝑞1,𝑠,𝑠𝑙,𝑙=max{𝑛;𝑛𝑙}.(3.21) The function space 𝒞𝑙𝑞,𝑠, 𝑙0, 1𝑞< and 𝑠 (called the Campanato space), is the space of all 𝑓𝐿𝑞loc(𝑛+1) (defined up to addition by 𝑃𝒫𝑠; the set of all polynomials in (𝑛+1) variables of degree at most 𝑠) so that: 𝑓𝒞𝑙𝑠,𝑞(𝑛+1)=sup𝒬𝑛+1inf𝑃𝒫𝑠||𝒬||𝑙1||𝒬||𝒬||𝑓𝑃||𝑞1/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)): 𝒞01,0BMO.(3.23) This identification, together with (3.20), finally gives 𝐻1BMO.(3.24) The proof then directly follows from (3.18), (3.19), and (3.24).

Proof of Theorem 1.1. Let 𝑁 be any arbitrary integer. Using (2.4), we estimate 𝑓𝐿 in the following way: 𝑓𝐿𝑗<𝑁2𝛾𝑗2𝛾𝑗||𝜑𝑗𝑓||𝐿+||𝑗||𝑁||𝜑𝑗𝑓||𝐿+𝑗>𝑁2𝛾𝑗2𝛾𝑗||𝜑𝑗𝑓||𝐿𝐶𝛾2𝛾𝑁𝐴1𝑗<𝑁22𝛾𝑗|𝜑𝑗𝑓|21/2𝐿+(2𝑁+1)1/2𝐴2|𝑗|𝑁||𝜑𝑗𝑓||21/2𝐿+𝐶𝛾2𝛾𝑁𝐴3sup𝑗>𝑁2𝛾𝑗𝜑𝑗𝑓𝐿,(3.25) where 𝐶𝛾=122𝛾11/2,𝐶𝛾=2𝛾12𝛾.(3.26) Using (3.1), we assert that: 𝐴1𝐶𝑔𝐿,(3.27) while (3.16) gives: 𝐴2𝐶𝑓BMO.(3.28) In order to estimate 𝐴3, we proceed in the following way: 𝐴3sup𝑗12𝛾𝑗𝜑𝑗𝑓𝐿sup𝑗12𝛾𝑗𝜑𝑗𝑓𝐿+𝜑0𝑓𝐿,𝜑0isgivenby(2.9),(3.29) hence (see Definition 2.2) 𝐴3𝑓𝐵𝛾,.(3.30) Using the well-known result (see, e.g., [18]) 𝐵𝛾,=𝐶𝛾,𝛾/2,(3.31) we finally obtain 𝐴3𝑓𝐶𝛾,𝛾/2.(3.32) Inequalities (3.25), (3.27), (3.28), and (3.32) imply 𝑓𝐿𝐶(2𝑁+1)1/2𝑓BMO+2𝛾𝑁𝑓𝐶𝛾,𝛾/2+𝑔𝐿.(3.33) We optimize (3.33) in 𝑁 by setting: 𝑁=1if𝑓𝐶𝛾,𝛾/2+𝑔𝐿2𝛾𝑓BMO.(3.34) Then it is easy to check (using (3.33)) that 𝑓𝐶𝑓BMO1+log+𝑓𝐶𝛾,𝛾/2+𝑔𝐿𝑓BMO1/2.(3.35) In the case where 𝑓𝐶𝛾,𝛾/2+𝑔𝐿>2𝛾𝑓BMO, we take 1𝛽<2𝛾 such that 𝑁=𝑁(𝛽)=log+2𝛾𝛽𝑓𝐶𝛾,𝛾/2+𝑔𝐿𝑓BMO12.(3.36) In fact this is valid since the function 𝑁(𝛽) varies continuously from 𝑁(1) to 𝑁(2𝛾)=1+𝑁(1) on the interval [1,2𝛾]. Using (3.33) with the above choice of 𝑁, we obtain: 𝑓C21/2log+2𝛾𝛽𝑓𝐶𝛾,𝛾/2+𝑔𝐿𝑓BMO1/2𝑓BMO+2𝛾/2𝛽𝑓BMO𝐶2(𝛾log2)1/2log+𝑓𝐶𝛾,𝛾/2+𝑔𝐿𝑓BMO1/2𝑓BMO+2𝛾/2𝛽𝑓BMO,(3.37) where for the second line we have used the fact that log+𝛽<log+𝑓𝐶𝛾,𝛾/2+𝑔𝐿𝑓BMO.(3.38) The above computations again imply (3.35). By using the inequality: 𝑥log𝑒+𝑦𝑥1/2𝐶1+𝑥(log(𝑒+𝑦))1/2,for0<𝑥1,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 Ω𝑇=(1,2)×(𝑇,2𝑇), 𝒵1𝒵2Ω𝑇 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 Ψ𝐶0(2), 0Ψ1 satisfying: Ψ(𝑥,𝑡)=1,for(𝑥,𝑡)𝒵1,0,for(𝑥,𝑡)2𝒵2.(3.41) The main idea of the proof consists in extending the function 𝑓 to a suitable function of the form Ψ̃𝑓, where ̃𝑓 is defined on Ω𝑇. We then apply inequality (1.5) (the scalar-valued version with 𝑛=1) to Ψ̃𝑓, and we estimate the different norms in order to get the result. However, away from the complicated extension (Sobolev extension) of the function ̃𝑓 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),for1<𝑥<0,0𝑡𝑇,𝑓(2𝑥,𝑡),for1<𝑥<2,0𝑡𝑇,(3.42) and then the symmetry with respect to 𝑡: ̃𝑓(𝑥,𝑡)=𝑓(𝑥,𝑡),for1<𝑥<2,𝑇<𝑡0,𝑓(𝑥,2𝑇𝑡),for1<𝑥<2,𝑇𝑡<2𝑇.(3.43) We claim that Ψ̃𝑓𝐶𝛾,𝛾/2(2) with Ψ̃𝑓𝐶𝛾,𝛾/2(2)𝑓𝐶𝛾,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 Ψ̃𝑓 with 𝑖=1 and 𝑔(𝑥,𝑡)=𝑥0Ψ(𝑦,𝑡)̃𝑓(𝑦,𝑡)𝑑𝑦. This, together with the fact that Ψ=1 on Ω𝑇, leads to the following estimate: 𝑓𝐿(Ω𝑇)Ψ̃𝑓𝐿(2)𝐶1+Ψ̃𝑓BMO(2)log+Ψ̃𝑓𝐶𝛾,𝛾/2(2)+𝑔𝐿(2)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 Ψ̃𝑓BMO(2)𝐶(𝑓BMO(Ω𝑇)+𝑓𝐿1(Ω𝑇)), while it is clear that 𝑔𝐿(2)𝐶̃𝑓𝐿(Ω𝑇)𝐶𝑓𝐶𝛾,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 Ψ̃𝑓𝐶𝛾,𝛾/2(2)=Ψ̃𝑓𝐿(2)+Ψ̃𝑓(𝛾)𝑥,2+Ψ̃𝑓(𝛾/2)𝑡,2.(3.46) It is evident that Ψ̃𝑓𝐿(2)𝐶𝑓𝐿(Ω𝑇),(3.47) hence we only need to estimate the two terms Ψ̃𝑓(𝛾)𝑥,2 and Ψ̃𝑓(𝛾/2)𝑡,2. We only deal with Ψ̃𝑓(𝛾)𝑥,2 since the second term can be treated similarly. We examine the different positions of (𝑥,𝑡),(𝑥,𝑡)2. If (𝑥,𝑡),(𝑥,𝑡)2𝒵2, 𝑥𝑥, then (since Ψ=0 over 2𝒵2): ||Ψ̃𝑓(𝑥,𝑡)Ψ̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾=0.(3.48) If both (𝑥,𝑡),(𝑥,𝑡)Ω𝑇, 𝑥𝑥, then the special extension (3.42) and (3.43) of the function 𝑓 guarantees the existence of 𝑥,𝑡,𝑥,𝑡Ω𝑇,(3.49) such that: ̃𝑓(𝑥,𝑡)=𝑓𝑥,𝑡,̃𝑓(𝑥,𝑡)=𝑓𝑥,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 ||Ψ̃𝑓(𝑥,𝑡)Ψ̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾Ψ(𝛾)𝑥,Ω𝑇̃𝑓𝐿(Ω𝑇)𝐶𝑓𝐿(Ω𝑇)𝐶𝑓𝐶𝛾,𝛾/2(Ω𝑇),(3.52) or 𝑥𝑥, then we forcedly have (see Figure 2) ||𝑥𝑥||𝛾||𝑥𝑥||𝛾.(3.53) In this case, we compute ||Ψ̃𝑓(𝑥,𝑡)Ψ̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾||̃𝑓(𝑥,𝑡)||||Ψ(𝑥,𝑡)Ψ𝑥,𝑡||||𝑥𝑥||𝛾+||Ψ𝑥,𝑡||||̃𝑓(𝑥,𝑡)̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾̃𝑓𝐿(Ω𝑇)Ψ(𝛾)𝑥,Ω𝑇+||̃𝑓(𝑥,𝑡)̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾.(3.54) Using (3.50) and (3.53), we deduce that: ||̃𝑓(𝑥,𝑡)̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾=||𝑓𝑥,𝑡𝑓𝑥,𝑡||||𝑥𝑥||𝛾||𝑓𝑥,𝑡𝑓𝑥,𝑡||||𝑥𝑥||𝛾𝑓(𝛾)𝑥,Ω𝑇,(3.55) therefore, by (3.54), we obtain ||Ψ̃𝑓(𝑥,𝑡)Ψ̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾̃𝑓𝐿(Ω𝑇)Ψ(𝛾)𝑥,Ω𝑇+𝑓(𝛾)𝑥,Ω𝑇𝐶𝑓𝐶𝛾,𝛾/2(Ω𝑇).(3.56) The remaining case is when (𝑥,𝑡)𝒵2 and (𝑥,𝑡)2Ω𝑇 (see Figure 3). In this case, we have (Ψ̃𝑓)(𝑥,𝑡)=0 and ||𝑥𝑥||𝛾14𝛾,(3.57) hence ||Ψ̃𝑓(𝑥,𝑡)Ψ̃𝑓𝑥,𝑡||||𝑥𝑥||𝛾4𝛾̃𝑓𝐿(𝒵2)𝐶𝑓𝐶𝛾,𝛾/2(Ω𝑇).(3.58) From (3.48), (3.52), (3.56), and (3.58), we finally deduce that Ψ̃𝑓(𝛾)𝑥,2𝐶𝑓𝐶𝛾,𝛾/2(Ω𝑇).(3.59) Arguing in exactly the same way as above, we also find that: Ψ̃𝑓(𝛾/2)𝑡,2𝐶𝑓𝐶𝛾,𝛾/2(Ω𝑇),(3.60) with a possibly different constant 𝐶 that depends on 𝑇. Indeed, the term 𝑇 enters in estimating Ψ̃𝑓(𝛾/2)𝑡,2 since (3.57) is now replaced (see again Figure 3) by ||𝑡𝑡||𝛾𝑇4𝛾.(3.61) This shows the claim.

Figure 1: Case (𝑥,𝑡),(𝑥,𝑡)Ω𝑇 with 𝑥=𝑥.
Figure 2: Case (𝑥,𝑡),(𝑥,𝑡)Ω𝑇 with 𝑥𝑥. (a): 𝑥𝑥. (b):𝑥=𝑥.
Figure 3: Case (𝑥,𝑡)𝒵2 and (𝑥,𝑡)2Ω𝑇.

Remark 3.3. In the case of multispatial coordinates 𝑥𝑖, 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).


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.


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