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∈W22m,m(ℝn+1), g∈L2(ℝn+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‖W22m,m(Rn+1)+‖g‖L∞(Rn+1)))1/2),
where W22m,m 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∈W22m,m(ΩT) defined on the bounded domainΩT=(0,1)n×(0,T),T>0,
we have the following estimate (see [1, Theorem 1.2]):‖f‖L∞(ΩT)≤C(1+(‖f‖BMO(ΩT)+‖f‖L1(ΩT))(log+‖f‖W22m,m(ΩT))1/2).
The different norms of f appearing in inequalities (1.1) and (1.3) are finite sinceW22m,m↪Cγ,γ/2↪L∞↪BMO,forsome0<γ<1,
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=∇g∈W22m,m (vector-valued case), or f∈W22m,m (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∈Cγ,γ/2, 0<γ<1 (vector-valued case), or f∈Cγ,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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M22"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>).

Let 0<γ<1. For any f=∇g∈Cγ,γ/2(ℝn+1)∩L2(ℝn+1) with g∈L2(ℝn+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).

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 f∈Cγ,γ/2(ΩT), 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).

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/∂xi∈Cγ,γ/2(ℝn+1)∩L2(ℝn+1), i∈1,…,n+1, with g∈L∞(ℝn+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 ℝn+1. A generic element z∈ℝn+1 has the form z=(x,t) with x=(x1,…,xn)∈ℝn. 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)={f∈C(O¯),‖f‖Cγ,γ/2(O)<∞},
where
‖f‖Cγ,γ/2(O)=‖f‖L∞(O)+〈f〉x,O(γ)+〈f〉t,O(γ/2),
with
〈f〉x,O(γ)=sup(x,t),(x′,t)∈O,x≠x′|f(x,t)-f(x′,t)||x-x′|γ,〈f〉t,O(γ/2)=sup(x,t),(x,t′)∈O,t≠t′|f(x,t)-f(x,t′)||t-t′|γ/2.

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′(ℝn+1) can be decomposed asf=∑j∈Zφj*f,whereφj(z)=|detA|jφ(Ajz),
with the convergence in 𝒮′/𝒫 (modulo polynomials). Here φ∈𝒮(ℝn+1) is a test function such that suppφ̂ is compact and bounded away from the origin, and ∑j∈ℤφ̂(Ajz)=1 for all z∈ℝn+1∖{0}, where φ̂ is the Fourier transform of φ. The sequence (φj)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 Ḟp,20 as the space of functions f∈𝒮′(ℝn+1) with finite quasinorms:‖f‖Ḟp,20(Rn+1)=‖(∑|j|<∞|φj*f|2)1/2‖Lp(Rn+1)<∞.
The space Ḟp,20 can be identified with the parabolic Hardy space Hp, 1≤p<∞ having the following square function characterization stated informally as:Hp(Rn+1)={f∈S′(Rn+1);(∑|j|<∞|φj*f|2)1/2∈Lp}.
This identification (see Bownik [14]) can be stated as follows: for all 1≤p<∞, we haveḞp,20(Rn+1)≃Hp(Rn+1).
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∈C0∞(ℝn+1) be any cut-off function satisfying:θ(z)={1,if|z|p≤1,0,if|z|p≥2,
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 relationφ̂0=θ,
we can give the definition of the Besov space B∞,∞γ.

Take the smoothness parameter 0<γ<1. Let (φj)j∈ℤ be the sequence such that φ0 is given by (2.9), while φj 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′(ℝn+1) with finite quasinorms
‖f‖B∞,∞γ=supj≥02γj‖φj*f‖L∞(Rn+1).

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∈Cγ,γ/2(ℝn+1)∩L2(ℝn+1) with g∈L2(ℝn+1), there exists a constant C=C(γ,n)>0 such that
‖(∑j<-N2-2γj|φj*f|2)1/2‖L∞≤C‖g‖L∞.

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 g∈L∞(ℝn+1). For z∈ℝn+1, define
Φ(z)=(∂iφ)(z),Φj(z)=|detA|jΦ(Ajz),∀j≤-1.
Using (2.4) we obtain:
(∂iφj)(z)={2jΦj(z)ifi=1,…,n,22jΦj(z)ifi=n+1.
We now compute (see (3.3) and (3.4)):
‖(∑j≤-12-2γj|φj*f|2)1/2‖L∞≤Csupj≤-1‖Φj*g‖L∞,
where the constant C is given by:
C2={∑j≤-122j(1-γ),ifi=1,…,n,∑j≤-122j(2-γ),ifi=n+1,
which is finite 0<C<+∞ under the choice
0<γ<1.
In order to terminate the proof, it suffices to show that
‖Φj*g‖L∞≤C‖g‖L∞,
which can be deduced, by translation and dilation invariance, from the following estimate:
|(Φ*g)(0)|≤C‖g‖L∞.
Indeed, define the positive radial decreasing function h(r)=h(∥z∥) as follows:
h(r)=sup‖z‖≥r|Φ(z)|.
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∞<+∞,
and the asymptotic behavior
h(r)≤Crn+2,∀r≥1.
We compute (taking Srn as the n-dimensional sphere of radius r):
|(Φ*g)(0)|≤∫Rn+1|Φ(-z)||g(z)|dz≤∫0∞(∫Srn|Φ(-z)||g(z)|dσ(z))dr≤C(∫0∞rnh(r)dr)‖g‖L∞.
Using (3.11) and (3.12) we deduce that:
∫0∞rnh(r)dr=∫01rnh(r)dr+∫1∞rnh(r)dr≤C(∫01h(0)dr+∫1∞rnrn+2dr)≤C(‖Φ‖L∞+1),
which, together with (3.13), directly implies (3.9). As a conclusion, we obtain (see (3.5)):
‖(∑j≤-12-2γj|φj*f|2)1/2‖L∞≤C‖g‖L∞,
and hence inequality (3.1) holds.

Lemma 3.2.

Let N>0 be a positive integer. Then for any f∈BMO(ℝn+1) there exists a constant C=C(n)>0 such that:
‖(∑|j|<N|φj*f|2)1/2‖L∞≤C‖f‖BMO.

Proof.

The proof provides inequality (3.16) for all |j|<∞ by showing that Ḟ∞,20≃BMO and then using (2.5). Before starting the proof, we remind the reader that:
‖f‖BMO(Rn+1)=supQ⊆Rn+1infc∈R(1|Q|∫Q|f-c|),
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]):
(Ḟ1,20)′≃Ḟ∞,20,
where (Ḟ1,20)′ stands for the dual space of Ḟ1,20. Applying (2.7) with p=1 we obtain:
Ḟ1,20≃H1.
Using the description of the dual of parabolic Hardy spaces Hp for 0<p≤1 (see Bownik [17, Theorem 8.3]), we get:
(Hp)′=Cq,sl
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}.
The function space 𝒞q,sl, l≥0, 1≤q<∞ and s∈ℕ (called the Campanato space), is the space of all f∈Llocq(ℝn+1) (defined up to addition by P∈𝒫s; the set of all polynomials in (n+1) variables of degree at most s) so that:
‖f‖Cs,ql(Rn+1)=supQ⊆Rn+1infP∈Ps|Q|l(1|Q|∫Q|f-P|q)1/q<∞.
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)):
C1,00≃BMO.
This identification, together with (3.20), finally gives
(H1)′≃BMO.
The proof then directly follows from (3.18), (3.19), and (3.24).

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

Let N∈ℕ be any arbitrary integer. Using (2.4), we estimate ∥f∥L∞ in the following way:
‖f‖L∞≤‖∑j<-N2γj2-γj|φj*f|‖L∞+‖∑|j|≤N|φj*f|‖L∞+‖∑j>N2-γj2γj|φj*f|‖L∞≤Cγ2-γN‖(∑j<-N2-2γj|φj*f|2)1/2‖L∞︷A1+(2N+1)1/2‖(∑|j|≤N|φj*f|2)1/2‖L∞︷A2+Cγ′2-γN(supj>N2γj‖φj*f‖L∞)︷A3,
where
Cγ=(122γ-1)1/2,Cγ′=2-γ1-2-γ.
Using (3.1), we assert that:
A1≤C‖g‖L∞,
while (3.16) gives:
A2≤C‖f‖BMO.
In order to estimate A3, we proceed in the following way:
A3≤supj≥12γj‖φj*f‖L∞≤supj≥12γj‖φj*f‖L∞+‖φ0*f‖L∞,φ0isgivenby(2.9),
hence (see Definition 2.2)
A3≤‖f‖B∞,∞γ.
Using the well-known result (see, e.g., [18])
B∞,∞γ=Cγ,γ/2,
we finally obtain
A3≤‖f‖Cγ,γ/2.
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∞)).
We optimize (3.33) in N by setting:
N=1if‖f‖Cγ,γ/2+‖g‖L∞≤2γ‖f‖BMO.
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).
In the case where ∥f∥Cγ,γ/2+∥g∥L∞>2γ∥f∥BMO, we take 1≤β<2γ such that
N=N(β)=log2γ+(β‖f‖Cγ,γ/2+‖g‖L∞‖f‖BMO)-12∈N.
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:
‖f‖∞≤C[21/2(log2γ+(β‖f‖Cγ,γ/2+‖g‖L∞‖f‖BMO))1/2‖f‖BMO+2γ/2β‖f‖BMO]≤C[2(γlog2)1/2(log+(‖f‖Cγ,γ/2+‖g‖L∞‖f‖BMO))1/2‖f‖BMO+2γ/2β‖f‖BMO],
where for the second line we have used the fact that
log+β<log+‖f‖Cγ,γ/2+‖g‖L∞‖f‖BMO.
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,
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 <xref ref-type="statement" rid="thm1.2">1.2</xref>.

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 Ω̃T=(-1,2)×(-T,2T), 𝒵1⊆𝒵2⊆Ω̃T such that
Z1={(x,t);-14<x<54,-T4<t<5T4},Z2={(x,t);-34<x<74,-3T4<t<7T4}.
We also take the cut-off function Ψ∈C0∞(ℝ2), 0≤Ψ≤1 satisfying:
Ψ(x,t)={1,for(x,t)∈Z1,0,for(x,t)∈R2∖Z2.
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 Ω̃T. 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:
f̃(x,t)={f(-x,t),for-1<x<0,0≤t≤T,f(2-x,t),for1<x<2,0≤t≤T,
and then the symmetry with respect to t:
f̃(x,t)={f(x,-t),for-1<x<2,-T<t≤0,f(x,2T-t),for-1<x<2,T≤t<2T.
We claim that Ψf̃∈Cγ,γ/2(ℝ2) with
‖Ψf̃‖Cγ,γ/2(R2)≤‖f‖Cγ,γ/2(ΩT).
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)=∫0xΨ(y,t)f̃(y,t)dy. This, together with the fact that Ψ=1 on ΩT, leads to the following estimate:
‖f‖L∞(ΩT)≤‖Ψf̃‖L∞(R2)≤C(1+‖Ψf̃‖BMO(R2)(log+(‖Ψf̃‖Cγ,γ/2(R2)+‖g‖L∞(R2)))1/2).
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)=∫0tΨ(x,s)f̃(x,s)ds.

In [7] it was shown that ∥Ψf̃∥BMO(ℝ2)≤C(∥f∥BMO(ΩT)+∥f∥L1(ΩT)), while it is clear that ∥g∥L∞(ℝ2)≤C∥f̃∥L∞(Ω̃T)≤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
‖Ψf̃‖Cγ,γ/2(R2)=‖Ψf̃‖L∞(R2)+〈Ψf̃〉x,R2(γ)+〈Ψf̃〉t,R2(γ/2).
It is evident that
‖Ψf̃‖L∞(R2)≤C‖f‖L∞(ΩT),
hence we only need to estimate the two terms 〈Ψf̃〉x,ℝ2(γ) and 〈Ψf̃〉t,ℝ2(γ/2). We only deal with 〈Ψf̃〉x,ℝ2(γ) 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):
|(Ψf̃)(x,t)-(Ψf̃)(x′,t)||x-x′|γ=0.
If both (x,t),(x′,t)∈Ω̃T, x≠x′, then the special extension (3.42) and (3.43) of the function f guarantees the existence of
(x¯,t¯),(x¯′,t¯)∈ΩT,
such that:
f̃(x,t)=f(x¯,t¯),f̃(x′,t)=f(x¯′,t¯).
Two cases can be considered. Either x¯=x¯′ (see Figure 1), then we forcedly have
f̃(x,t)=f̃(x′,t),
and therefore
|(Ψf̃)(x,t)-(Ψf̃)(x′,t)||x-x′|γ≤〈Ψ〉x,Ω̃T(γ)‖f̃‖L∞(Ω̃T)≤C‖f‖L∞(ΩT)≤C‖f‖Cγ,γ/2(ΩT),
or x¯≠x¯′, then we forcedly have (see Figure 2)
|x-x′|γ≥|x¯-x¯′|γ.
In this case, we compute
|(Ψf̃)(x,t)-(Ψf̃)(x′,t)||x-x′|γ≤|f̃(x,t)||Ψ(x,t)-Ψ(x′,t)||x-x′|γ+|Ψ(x′,t)||f̃(x,t)-f̃(x′,t)||x-x′|γ≤‖f̃‖L∞(Ω̃T)〈Ψ〉x,Ω̃T(γ)+|f̃(x,t)-f̃(x′,t)||x-x′|γ.
Using (3.50) and (3.53), we deduce that:
|f̃(x,t)-f̃(x′,t)||x-x′|γ=|f(x¯,t¯)-f(x¯′,t¯)||x-x′|γ≤|f(x¯,t¯)-f(x¯′,t¯)||x¯-x¯′|γ≤〈f〉x,ΩT(γ),
therefore, by (3.54), we obtain
|(Ψf̃)(x,t)-(Ψf̃)(x′,t)||x-x′|γ≤‖f̃‖L∞(Ω̃T)〈Ψ〉x,Ω̃T(γ)+〈f〉x,ΩT(γ)≤C‖f‖Cγ,γ/2(ΩT).
The remaining case is when (x,t)∈𝒵2 and (x′,t)∈ℝ2∖Ω̃T (see Figure 3). In this case, we have (Ψf̃)(x′,t)=0 and
|x-x′|γ≥(14)γ,
hence
|(Ψf̃)(x,t)-(Ψf̃)(x′,t)||x-x′|γ≤4γ‖f̃‖L∞(Z2)≤C‖f‖Cγ,γ/2(ΩT).
From (3.48), (3.52), (3.56), and (3.58), we finally deduce that
〈Ψf̃〉x,R2(γ)≤C‖f‖Cγ,γ/2(ΩT).
Arguing in exactly the same way as above, we also find that:
〈Ψf̃〉t,R2(γ/2)≤C‖f‖Cγ,γ/2(ΩT),
with a possibly different constant C that depends on T. Indeed, the term T enters in estimating 〈Ψf̃〉t,ℝ2(γ/2) since (3.57) is now replaced (see again Figure 3) by
|t-t′|γ≥(T4)γ.
This shows the claim.

Case (x,t),(x′,t)∈Ω̃T with x¯=x¯′.

Case (x,t),(x′,t)∈Ω̃T with x¯≠x¯′. (a): x′≠x¯′. (b):x′=x¯′.

Case (x,t)∈𝒵2 and (x′,t)∈ℝ2∖Ω̃T.

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).

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.

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