Research Article | Open Access
A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class
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 , a generalization of the Ogawa type inequality  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 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 , with , , there exists a constant such that: where is the parabolic Sobolev space (we refer to  for the definition and further properties), and 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 defined on the bounded domain we have the following estimate (see [1, Theorem 1.2]): The different norms of appearing in inequalities (1.1) and (1.3) are finite since where is the parabolic Hölder space that will be defined later. Moreover, it is easy to check that is bounded and continuous.
The purpose of this paper is to show that the condition (vector-valued case), or (scalar-valued case) can be relaxed. Indeed, inequalities (1.1) and (1.3) can be applied to a wider class of Hölder continuous functions , (vector-valued case), or (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 ). Let . For any with , there exists a constant such that
The second theorem deals with functions defined on the bounded domain .
Theorem 1.2 (Logarithmic Hölder inequality on a bounded domain). Let . For any , there exists a constant such that
Remark 1.3. The same inequality (1.5) still holds for scalar-valued functions , , with .
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.
Let be an open subset of . A generic element has the form with . We begin by defining parabolic Hölder spaces .
Definition 2.1 (Parabolic Hölder spaces). For , we define the parabolic space of Hölder continuous functions of order in the following way: where with
For a detailed study of parabolic Hölder spaces, we refer the reader to . We now briefly recall some basic facts about Littlewood-Paley decomposition which are crucial in obtaining our logarithmic inequalities. Given the expansive matrix (parabolic anisotropy), the corresponding Littlewood-Paley decomposition asserts that any tempered distribution can be decomposed as with the convergence in (modulo polynomials). Here is a test function such that is compact and bounded away from the origin, and for all , where is the Fourier transform of . The sequence 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 , we define the parabolic homogeneous Lizorkin-Triebel space as the space of functions with finite quasinorms: The space can be identified with the parabolic Hardy space , having the following square function characterization stated informally as: This identification (see Bownik ) can be stated as follows: for all , we have Now, for defining the inhomogeneous parabolic Besov space used later in obtaining our results, we use a slightly different sequence. Indeed, let be any cut-off function satisfying: where is the parabolic quasinorm associated to the matrix (see ). Taking the new function (but keeping the same notation) defined via the relation we can give the definition of the Besov space .
Definition 2.2 (Parabolic inhomogeneous Besov spaces). Take the smoothness parameter . Let be the sequence such that is given by (2.9), while is given by (2.4) for all . We define the parabolic inhomogeneous Besov space as the space of all functions with finite quasinorms
3. Proofs of Theorems
The proof of Theorem 1.1 relies on the following two lemmas of different interests.
Lemma 3.1. Let and let be a positive integer. Then for any with , there exists a constant such that
Proof. We provide a proof of (3.1) in the general case . We use the fact that (for which we keep denoting it by , i.e. ) for some , with . For , define Using (2.4) we obtain: We now compute (see (3.3) and (3.4)): where the constant is given by: which is finite under the choice In order to terminate the proof, it suffices to show that which can be deduced, by translation and dilation invariance, from the following estimate: Indeed, define the positive radial decreasing function as follows: From (3.2), we remark that the function is the inverse Fourier transform of a compactly supported function. Hence, we have and the asymptotic behavior We compute (taking as the -dimensional sphere of radius ): Using (3.11) and (3.12) we deduce that: which, together with (3.13), directly implies (3.9). As a conclusion, we obtain (see (3.5)): and hence inequality (3.1) holds.
Lemma 3.2. Let be a positive integer. Then for any there exists a constant such that:
Proof. The proof provides inequality (3.16) for all by showing that and then using (2.5). Before starting the proof, we remind the reader that: where 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 , and Frazier and Jawerth ): where stands for the dual space of . Applying (2.7) with we obtain: Using the description of the dual of parabolic Hardy spaces for (see Bownik [17, Theorem 8.3]), we get: with the terms chosen such that: The function space , , and (called the Campanato space), is the space of all (defined up to addition by ; the set of all polynomials in variables of degree at most ) so that: Choosing , , and , we can easily see that conditions (3.21) are all satisfied, and that (see (3.22) and (3.17)): This identification, together with (3.20), finally gives The proof then directly follows from (3.18), (3.19), and (3.24).
Proof of Theorem 1.1. Let be any arbitrary integer. Using (2.4), we estimate in the following way: where Using (3.1), we assert that: while (3.16) gives: In order to estimate , we proceed in the following way: hence (see Definition 2.2) Using the well-known result (see, e.g., ) we finally obtain Inequalities (3.25), (3.27), (3.28), and (3.32) imply We optimize (3.33) in by setting: Then it is easy to check (using (3.33)) that In the case where , we take such that In fact this is valid since the function varies continuously from to on the interval . Using (3.33) with the above choice of , we obtain: where for the second line we have used the fact that The above computations again imply (3.35). By using the inequality: 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 . The general -dimensional case can be easily deduced. Following the same notations of , we let , such that
We also take the cut-off function , satisfying:
The main idea of the proof consists in extending the function to a suitable function of the form , where is defined on . We then apply inequality (1.5) (the scalar-valued version with ) to , and we estimate the different norms in order to get the result. However, away from the complicated extension (Sobolev extension) of the function that was done in , we here consider a simpler symmetric extension. Indeed, we first take the spatial symmetry of the function :
and then the symmetry with respect to :
We claim that with
In this case, we apply the scalar-valued version of inequality (1.5) (see Remark 1.3) to the function with and . This, together with the fact that on , leads to the following estimate:
It is worth noticing that choosing above is somehow restrictive. In fact, we could also have used the inequality with and .
In  it was shown that , while it is clear that . 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 It is evident that hence we only need to estimate the two terms and . We only deal with since the second term can be treated similarly. We examine the different positions of . If , , then (since over ): If both , , then the special extension (3.42) and (3.43) of the function guarantees the existence of such that: Two cases can be considered. Either (see Figure 1), then we forcedly have and therefore or , then we forcedly have (see Figure 2) In this case, we compute Using (3.50) and (3.53), we deduce that: therefore, by (3.54), we obtain The remaining case is when and (see Figure 3). In this case, we have and hence From (3.48), (3.52), (3.56), and (3.58), we finally deduce that Arguing in exactly the same way as above, we also find that: with a possibly different constant that depends on . Indeed, the term enters in estimating since (3.57) is now replaced (see again Figure 3) by This shows the claim.
Remark 3.3. In the case of multispatial coordinates , , we simultaneously apply the extension (3.42) to each spatial coordinate while fixing all other coordinates including . Finally, by fixing the spatial variables, we make the extension with respect to 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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