# Parabolic John-Nirenberg Spaces

Authors: Lauri Berkovits
Journal: Journal of Function Spaces and Applications
 (2012)

---

## Abstract

We introduce a parabolic version of John-Nirenberg space with exponentp and show that it is contained in local weak-Lp spaces.


---

## Body

## 1. Introduction

In the classical paper of John and Nirenberg [1], where functions of bounded mean oscillation (BMO) were introduced, they also studied a class satisfying a weaker BMO-type condition
(1.1)‖f‖JNp(Q0)p:=sup⁡{Qj}j∑j‍|Qj|(⨍Qj|f-fQj|dx)p<∞,
where the supremum is taken over all partitions {Qj}j of a given cube Q0 into pairwise nonoverlapping subcubes. Here we have used standard notation for integral averages
(1.2)fQ:=⨍Qfdx:=1|Q|∫Q‍fdx.
The functional f↦∥f∥JNp(Q0) defines a seminorm, and the class of functions satisfying ∥f∥JNp(Q0)<∞, which we denote by JNp(Q0) for John-Nirenberg space with exponent p, can be seen as a generalization of BMO. Indeed, BMO is obtained as the limit case of JNp(1.3)lim⁡p→∞‖f‖JNp(Q0)=sup⁡Q⊆Q0⨍Q|f-fQ|dx=:‖f‖BMO(Q0).
In contrast to the exponential integrability of BMO functions, functions in JNp(Q0) belong to the space weak-Lp(Q0). This was already observed by John and Nirenberg. Precisely, they showed that for λ>0, we have
(1.4)|{x∈Q0:|f(x)-fQ0|>λ}|≤C(‖f‖JNp(Q0)λ)p,
where the constant C depends on n and p. Simpler proofs and generalizations have appeared in [1–11]. In this paper we show that an analogous result holds in the context of parabolic or one-sided BMO+ spaces, which we hope will increase our understanding of the spaces BMO+ themselves. Therefore, the main purpose is to provide a step towards a satisfactory multidimensional theory of the spaces BMO+ described below.


## 2. BMO+ Spaces

We will introduce some notations and terminologies. Given a Euclidean cubeQ=∏i=1n‍[ai,ai+h] and r>0, we define the “forward in time” r-translation
(2.1)Q+,r=∏i=1n-1‍[ai,ai+h]×[an+rh,an+(r+1)h].
For short, we write Q+,1=Q+ for the cube of the same size above Q. We write f∈BMO+(ℝn), if we have
(2.2)∥f∥BMO+(Rn):=sup⁡Q⊆Rn⨍Q(f-fQ+)+dx<∞,
where the supremum is taken over all cubes in ℝn with sides parallel to the coordinate axes. It should be observed that despite the notation, the quantity defined by (2.2) is not actually a norm. The one-dimensional BMO+(ℝ) class was first introduced by Martín-Reyes and de la Torre [12], who showed that this class possesses many properties similar to the standard BMO space. Even though steps towards a multidimensional theory have been taken (see [13]), a satisfactory theory has only been developed in dimension one. In the classical elliptic setting, one of the cornerstones of theory of BMO functions is the celebrated John-Nirenberg inequality, which shows that logarithmic growth is the maximum possible for a BMO function. A corresponding result holds for the class BMO+(ℝn), stating that functions in BMO+(ℝn) satisfy the following version of John-Nirenberg inequality. Given r>1, we have for all λ>0(2.3)|{x∈Q:(f(x)-fQ+,r)+>λ}|≤Be-bλ/‖f‖BMO+(Rn)|Q|,
where the constants B and b only depend on the dimension. This was first proved in the one-dimensional case in [12], in which case (2.3) actually holds true with r=1. The weaker result (2.3) for n≥2 was obtained in [13].


## 3. Parabolic John-Nirenberg Spaces

In this setting we define (local, in the spirit of [1]) John-Nirenberg spaces as follows. Given a cube Q0 and 1<p<∞, we write f∈JNp+(Q0) if
(3.1)‖f‖JNp+(Q0)p:=sup⁡{Qj}j∑j‍|Qj|(⨍Qj∪Qj+(f-fQj+,2)+dx)p<∞,
where the supremum is taken over countable families {Qj} of pairwise nonoverlapping subcubes of Q0. Observe that even if Q is contained in Q0, Q+ or Q+,2 may not be. Hence, we assume that f is a priori defined on all of ℝn. The definition is reasonable because in a sense BMO+ may be seen as the limit case of  JNp+ as p→∞. Precisely,
(3.2)lim⁡p→∞‖f‖JNp+(Q0)=sup⁡Q⊆Q0⨍Q∪Q+(f-fQ+,2)+dx,
where the quantity on the right-hand side is globally equivalent to the BMO+ norm of f, that is,
(3.3)sup⁡Q⊆Rn⨍Q∪Q+(f-fQ+,2)+dx≈‖f‖BMO+(Rn),
up to a multiplication by a universal constant.The following theorem is a parabolic version of the weak distribution inequality of John and Nirenberg.Theorem 3.1.
Assumef∈JNp+(Q0). Then, for every λ>0, one has
(3.4)|{x∈Q0:(f(x)-fQ0+,2)+>λ}|≤C(‖f‖JNp+(Q0)λ)p,
where C only depends on n and p.Comparing this to the one-sided John-Nirenberg inequality (2.3), it is natural to ask whether it is possible to improve (3.4) to
(3.5)|{x∈Q0:(f(x)-fQ0+,r)+>λ}|≤C(‖f‖JNp+(Q0)λ)p,
for r>1 or even further to r=1 in the one-dimensional case. However, the author does not know the answer. In particular, the one-dimensional improvement seems challenging since in the case of BMO+(ℝ) it makes use of the knowledge of one-sided Muckenhoupt's weights, and we do not have such tools at our disposal in the context of John-Nirenberg spaces.


## 4. Proof of the Theorem

We follow the argument used in [2]. Given a nonnegative f and a cube Q0, denote by Δ=Δ(Q0) the family of all dyadic subcubes obtained from Q0 by repeatedly bisecting the sides into two parts of equal length. We shall make use of the “forward in time dyadic maximal function” defined by
(4.1)MQ0+,df(x):=sup⁡Q∈Δx∈Q⨍Q+fdx.
A standard stopping-time argument shows that we have
(4.2){x∈Q0:MQ0+,df(x)>λ}=⋃j‍Qj,
where Qj's are the maximal dyadic subcubes of Q0 satisfying
(4.3)⨍Qj+fdx>λ.
Maximality implies that the cubes Qj are pairwise nonoverlapping. Moreover, if λ≥fQ0+, then Q0 does not satisfy (4.3). Consequently, in this case every Qj is contained in a larger dyadic subcube Qj- of Q0 which does not satisfy (4.3). Since Qj+,2⊆Qj-+, we conclude
(4.4)⨍Qj+,2fdx≤2nλ,
provided λ≥fQ0+. Standard arguments imply a weak-type estimate for MQ0+,d. Indeed, we have
(4.5)|{x∈Q0:MQ0+,df(x)>λ}|=∑j‍|Qj|.
While the cubes Qj are non-overlapping, the cubes Qj+ may not be. Let us replace {Qj+}j by the maximal non-overlapping subfamily {Q~j+}j which we form by collecting those Qj+ which are not properly contained in any other Qj′+. Maximality of {Q~j+}j enables us to partition the family {Qj}j as follows. Given Q~j+, we define Ij:={i:Qi+⊆Q~j+}, and we may write {Qj}j=⋃j‍{Qi:i∈Ij}. Now, whenever i∈Ij, we have Qi⊆Q~j∪Q~j+ and we get the estimate
(4.6)∑j‍|Qj|=∑j‍∑i∈Ij‍|Qi|≤2∑j‍|Q~j+|≤2λ∫Q0∪Q0+‍fdx.
Combining the previous estimates, we arrive at
(4.7)|{x∈Q0:MQ0+,df(x)>λ}|≤2λ∫Q0∪Q0+‍fdx.We begin by proving the following goodλ inequality for the forward in time dyadic maximal operator.Lemma 4.1.
Assumef∈JNp+(Q0) and take 0<b<2-n. Then, one has
(4.8)|{x∈Q0:MQ0+,d(f-fQ0+,2)+(x)>λ}|≤a‖f‖JNp+(Q0)λ|{x∈Q0:MQ0+,d(f-fQ0+,2)+(x)>bλ}|1/q,
whenever
(4.9)bλ≥⨍Q0+(f-fQ0+,2)+dx.
Here a=4(1-2nb)-1 and q is the conjugate exponent of p.Proof.
Without loss of generality, we may assume∥f∥JNp+(Q0)=1. Setting
(4.10)EQ(λ):={x∈Q:MQ+,d(f-fQ0+,2)+(x)>λ},
we may write the statement as
(4.11)|EQ0(λ)|≤4(1-2nb)λ|EQ0(bλ)|1/q.
Consider the function (f-fQ0+,2)+ and form the decomposition as above at level bλ to obtain a family of pairwise non-overlapping dyadic subcubes with
(4.12)EQ0(bλ)=⋃j‍Qj.
Since bλ<λ, we have EQ0(λ)⊆EQ0(bλ). It now follows that
(4.13)EQ0(λ)=⋃j‍EQj(λ).
We claim that for every j,
(4.14)|EQj(λ)|≤2(1-2nb)λ∫Qj∪Qj+‍(f-fQj+,2)+dx.
Consider the functions gj:=(f-fQj+,2)+. To prove (4.14) it suffices to show that
(4.15)EQj(λ)⊆{x∈Qj:MQj+,dgj(x)>(1-2nb)λ}.
Indeed, (4.14) then follows at once from the weak-type estimate (4.7) applied to the functions gj with λ replaced by (1-2nb)λ. Let x∈EQj(λ) for some j. Then there exists a dyadic subcube Q of Qj containing x and satisfying
(4.16)⨍Q+(f-fQ0+,2)+>λ.
From (4.4) we have
(4.17)⨍Qj+,2(f-fQ0+,2)+≤2nbλ.
Combining these, we obtain
(4.18)(1-2nb)λ<⨍Q+(f-fQ0+,2)+dx-⨍Qj+,2(f-fQ0+,2)+dx≤⨍Q+(f-fQ0+,2)+dx-(⨍Qj+,2f-fQ0+,2dx)+=⨍Q+(f-fQ0+,2)+-(fQj+,2-fQ0+,2)+dx≤⨍Q+(f-fQj+,2)+dx≤MQj+,dgj(x).
Having now seen that (4.14) holds, we use (4.13) and sum over all j to obtain
(4.19)|EQ0(λ)|=∑j‍|EQj(λ)|≤2(1-2nb)λ∑j‍∫Qj∪Qj+‍(f-fQj+,2)+dx=2(1-2nb)λ∑j‍|Qj|1/q|Qj|-1/q∫Qj∪Qj+‍(f-fQj+,2)+dx≤4(1-2nb)λ(∑j‍|Qj|)1/q,
where the last inequality follows from the Hölder inequality and the assumption ∥f∥JNp+(Q0)=1. Remembering also that EQ(bλ)=⋃jQj, we obtain the desired estimate.We now complete the proof of the theorem by iterating the previous lemma. Except for a few details, this is just a repetition of the argument used in [2].Proof of the Theorem.
Using the same notation as in the proof of the lemma and still assuming∥f∥JNp+(Q0)=1, we shall show
(4.20)|EQ0(λ)|≤Cλp.
Let us choose
(4.21)λ0:=2b|Q0|1/p,
and assume λ>λ0. Then take N∈ℤ+ such that
(4.22)b-Nλ0≤λ<b-(N+1)λ0=2b-(N+2)|Q0|1/p.
By the assumption ∥f∥JNP+(Q0)=1, we have
(4.23)1|Q0|∫Q0∪Q0+‍(f-fQ0+,2)+dx≤2|Q0|1/p=bλ0.
In particular, this implies that
(4.24)1b⨍Q0+(f-fQ0+,2)+dx≤λ0≤b-1λ0≤⋯≤b-Nλ0,
allowing us to apply the previous lemma successively N times to estimate the left-hand side of (4.20) as follows:
(4.25)|EQ0(λ)|≤|EQ0(b-Nλ0)|≤ab-Nλ0⋅(ab-N+1λ0)1/q⋅⋯⋅(ab-1λ0)1/qN-1|EQ0(λ0)|1/qN≤abλ⋅(ab2λ)1/q⋅⋯⋅(abNλ)1/qN-1⋅(2λ0∫Q0∪Q0+‍(f-fQ0+,2)+dx)1/qN,
where the last inequality follows from the weak-type estimate (4.7) and the first inequality in (4.22). By the choice of λ0 and (4.23) we further estimate
(4.26)|EQ0(λ)|≤(aλ)1+q-1+⋯+q-(N-1)⋅b-(1+2q-1+⋯+Nq-(N-1))⋅(2b|Q0|)1/qN=(aλ)p-p/qN⋅b-(1+2q-1+⋯+Nq-(N-1))+q-N⋅21/qN⋅|Q0|1/qN.
Since both 1+2q-1+⋯+Nq-(N-1) and p-p/qN remain bounded as N→∞, we have
(4.27)|EQ0(λ)|≤C|Q0|1/qN(1λ)p-p/qN.
Finally, we notice that from the second inequality in (4.22) we get
(4.28)|Q0|1/qN(1λ)-p/qN=λp/qN|Q0|1/qN≤2p/qNb-(N+2)p/qN≤C,
with C independent of N. Thus we have arrived at the desired estimate.
For0<λ≤λ0 we use the trivial estimate
(4.29)|{x∈Q0:(f(x)-fQ0+,2)+>λ}|≤|Q0|=2pbpλ0p≤Cλp.



---

*Source: 901917-2012-07-18.xml*