A remark on a generalization of a logarithmic Sobolev inequality to the Holder 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 Brezis-Gallouet-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 the wider class of Holder continuous functions.


Introduction and main results
In [5], a generalization of the Ogawa type inequality [12] 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 [1,2] that holds in a limiting case of the Sobolev embedding theorem. The inequality showed in [5,Theorem 1.1] provides an estimate of the L ∞ norm of a function in terms of its parabolic BM O 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,m 2 (R n+1 ), g ∈ L 2 (R n+1 ) with m, n ∈ N * , 2m > n+2 2 , there exists a constant C = C(m, n) > 0 such that: where W 2m,m 2 is the parabolic Sobolev space (we refer to [11] for the definition and further properties), and BM O is the parabolic bounded mean oscillation space (defined via parabolic balls instead of Euclidean ones [5, 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 [1,2,3,8,9,10,12] for various elliptic versions). By considering functions f ∈ W 2m,m 2 (Ø T ) defined on the bounded domain Ø T = (0, 1) n × (0, T ), T > 0, we have the following estimate (see [5,Theorem 1.2]): (1. 2) The different norms of f appearing in inequalities (1.1) and (1.2) are finite since where C γ,γ/2 is the parabolic Hölder space that will be defined later. Moreover, it is easy to check that g bounded and continuous. The purpose of this paper to show that the condition f = ∇g ∈ W 2m,m 2 (vector-valued case), or f ∈ W 2m,m 2 (scalar-valued case) can be relaxed. Indeed, inequalities (1.1) and (1.2) 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: (1.4) The second theorem deals with functions defined on the bounded domain Ø T .

Remark 1.3 The same inequality (1.4) still holds for scalar-valued functions
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.

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 = (x 1 , . . . , 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: For a detailed study of parabolic Hölder spaces, we refer the reader to [11]. 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, 2 2 } (parabolic anisotropy), the corresponding Littlewood-Paley decomposition asserts that any tempered distribution f ∈ S ′ (R n+1 ) can be decomposed as 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 j z) = 1 for all z ∈ R n+1 \ {0}, whereφ is the Fourier transform of ϕ. The sequence (ϕ j ) j∈Z is mainly used to define homogeneous Lizorkin-Triebel and Besov spaces (see for instance [13,14]). However, 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: where | · | p is the parabolic quasi-norm associated to the matrix A (see [5]). Taking the new function (but keeping the same notation) ϕ 0 defined via the relation we can give the definition of the Besov space B γ ∞,∞ .
We define the parabolic inhomogeneous Besov space B γ ∞,∞ as the space of all functions f ∈ S ′ (R n+1 ) with finite quasi-norms

Proofs of theorems
We begin with the proof of Theorem 1.1 that strongly relies on the results obtained in [5].
Proof of Theorem 1.1. Let N ∈ N be any arbitrary integer. Using (2.2), we estimate f L ∞ in the following way: Step 2 of the proof of [5, Theorem 1.1] asserts that: while [5, Lemma 3.1] gives: In order to estimate A 3 , we proceed in the following way:

hence (see Definition 2.2)
A 3 ≤ f B γ ∞,∞ . Using the well known result (see for instance [4]) we finally obtain Optimizing the above inequality with respect to the variable N (see Step 2 of the proof of [5, Lemma 3.2]), we directly arrive into the result. We now present the proof of Theorem 1.2 that involve finer estimates on the Hölder norm.
We also take the cut-off function Ψ ∈ C ∞ 0 (R 2 ), 0 ≤ Ψ ≤ 1 satisfying: The main idea of the proof consists in extending the function f to a suitable function of the form Ψf wheref is defined on Ø T . We then apply inequality (1.4) (the scalar-valued version with n = 1) to Ψf and we estimate the different norms in order to get the result. However, away from the complicated extension (Sobolev extension) of the functionf that was done in [5], we here consider a simpler symmetric extension. Indeed, we first take the spatial symmetry of the function f : and then the symmetry with respect to t: We claim that Ψf ∈ C γ,γ/2 (R 2 ) with In this case, we apply the scalar-valued version of inequality (1.4) (see Remark 1.3) to the function Ψf with i = 1 and g(x, t) = x 0 Ψ(y, t)f (y, t)dy. This, together with the fact that Ψ = 1 on Ø T , lead to the following estimate: 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) = t 0 Ψ(x, s)f (x, s)ds. In [7] it was shown that . These arguments, along with (3.8) and (3.9), directly terminate the proof. The only point left is to show the claim (3.8). Recall the norm hence we only need to estimate the two terms Ψf (γ) x,R 2 and Ψf (γ/2) t,R 2 . We only deal with Ψf (γ) x,R 2 since the second term can be treated similarly. We examine the different positions of (x, t), (x ′ , t) ∈ R 2 .