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

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 ‖𝑓‖𝑝𝐽𝑁𝑝(𝑄0)∶=sup𝑄𝑗𝑗𝑗||𝑄𝑗||𝑄𝑗|||𝑓−𝑓𝑄𝑗|||d𝑥𝑝<∞,(1.1) 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 𝑓𝑄∶=𝑄1𝑓d𝑥∶=||𝑄||𝑄𝑓d𝑥.(1.2) 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 𝐽𝑁𝑝limğ‘â†’âˆžâ€–ğ‘“â€–ğ½ğ‘ğ‘(𝑄0)=sup𝑄⊆𝑄0𝑄||𝑓−𝑓𝑄||d𝑥=∶‖𝑓‖BMO(𝑄0).(1.3) 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 ||𝑥∈𝑄0∶||𝑓(𝑥)−𝑓𝑄0||||>𝜆≤𝐶‖𝑓‖𝐽𝑁𝑝(𝑄0)𝜆𝑝,(1.4) 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 cube ∏𝑄=𝑛𝑖=1[ğ‘Žğ‘–,ğ‘Žğ‘–+ℎ] and 𝑟>0, we define the “forward in time” 𝑟-translation 𝑄+,𝑟=𝑛−1𝑖=1î€ºğ‘Žğ‘–,ğ‘Žğ‘–î€»Ã—î€ºğ‘Ž+â„Žğ‘›+ğ‘Ÿâ„Ž,ğ‘Žğ‘›î€».+(𝑟+1)ℎ(2.1) For short, we write 𝑄+,1=𝑄+ for the cube of the same size above 𝑄. We write 𝑓∈BMO+(ℝ𝑛), if we have ‖𝑓‖BMO+(ℝ𝑛)∶=sup𝑄⊆ℝ𝑛𝑄𝑓−𝑓𝑄++d𝑥<∞,(2.2) 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|||𝑥∈𝑄∶𝑓(𝑥)−𝑓𝑄+,𝑟+|||>𝜆≤𝐵𝑒−𝑏𝜆/‖𝑓‖BMO+𝑛)(ℝ||𝑄||,(2.3) 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∈𝐽𝑁+𝑝(𝑄0) if ‖𝑓‖𝑝𝐽𝑁+𝑝𝑄0∶=sup𝑄𝑗𝑗𝑗||𝑄𝑗||𝑄𝑗∪𝑄+𝑗𝑓−𝑓𝑄𝑗+,2+d𝑥𝑝<∞,(3.1) 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 𝐽𝑁+𝑝 as ğ‘â†’âˆž. Precisely, limğ‘â†’âˆžâ€–ğ‘“â€–ğ½ğ‘+𝑝(𝑄0)=sup𝑄⊆𝑄0𝑄∪𝑄+𝑓−𝑓𝑄+,2+d𝑥,(3.2) where the quantity on the right-hand side is globally equivalent to the BMO+ norm of 𝑓, that is, sup𝑄⊆ℝ𝑛𝑄∪𝑄+𝑓−𝑓𝑄+,2+d𝑥≈‖𝑓‖BMO+(ℝ𝑛),(3.3) 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. Assume 𝑓∈𝐽𝑁+𝑝(𝑄0). Then, for every 𝜆>0, one has ||||𝑥∈𝑄0∶𝑓(𝑥)−𝑓𝑄0+,2+||||>𝜆≤𝐶‖𝑓‖𝐽𝑁+𝑝(𝑄0)𝜆𝑝,(3.4) 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 ||||𝑥∈𝑄0∶𝑓(𝑥)−𝑓𝑄0+,𝑟+||||>𝜆≤𝐶‖𝑓‖𝐽𝑁+𝑝(𝑄0)𝜆𝑝,(3.5) 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 𝑀𝑄+,𝑑0𝑓(𝑥)∶=sup𝑄∈Δ𝑥∈𝑄𝑄+𝑓d𝑥.(4.1) A standard stopping-time argument shows that we have 𝑥∈𝑄0∶𝑀𝑄+,𝑑0=𝑓(𝑥)>𝜆𝑗𝑄𝑗,(4.2) where 𝑄𝑗's are the maximal dyadic subcubes of 𝑄0 satisfying 𝑄+𝑗𝑓d𝑥>𝜆.(4.3) Maximality implies that the cubes 𝑄𝑗 are pairwise nonoverlapping. Moreover, if 𝜆≥𝑓𝑄+0, 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⊆𝑄+𝑗−, we conclude 𝑄𝑗+,2𝑓d𝑥≤2𝑛𝜆,(4.4) provided 𝜆≥𝑓𝑄+0. Standard arguments imply a weak-type estimate for 𝑀𝑄+,𝑑0. Indeed, we have |||𝑥∈𝑄0∶𝑀𝑄+,𝑑0|||=𝑓(𝑥)>𝜆𝑗||𝑄𝑗||.(4.5) While the cubes 𝑄𝑗 are non-overlapping, the cubes 𝑄+𝑗 may not be. Let us replace {𝑄+𝑗}𝑗 by the maximal non-overlapping subfamily {𝑄+𝑗}𝑗 which we form by collecting those 𝑄+𝑗 which are not properly contained in any other 𝑄+ğ‘—î…ž. Maximality of {𝑄+𝑗}𝑗 enables us to partition the family {𝑄𝑗}𝑗 as follows. Given 𝑄+𝑗, we define 𝐼𝑗∶={𝑖∶𝑄+𝑖⊆𝑄+𝑗}, and we may write {𝑄𝑗}𝑗=⋃𝑗{𝑄𝑖∶𝑖∈𝐼𝑗}. Now, whenever 𝑖∈𝐼𝑗, we have 𝑄𝑖⊆Q𝑗∪𝑄+𝑗 and we get the estimate 𝑗||𝑄𝑗||=𝑗𝑖∈𝐼𝑗||𝑄𝑖||≤2𝑗|||𝑄+𝑗|||≤2𝜆𝑄0∪𝑄+0𝑓d𝑥.(4.6) Combining the previous estimates, we arrive at |||𝑥∈𝑄0∶𝑀𝑄+,𝑑0|||≤2𝑓(𝑥)>𝜆𝜆𝑄0∪𝑄+0𝑓d𝑥.(4.7)

We begin by proving the following good 𝜆 inequality for the forward in time dyadic maximal operator.

Lemma 4.1. Assume 𝑓∈𝐽𝑁+𝑝(𝑄0) and take 0<𝑏<2−𝑛. Then, one has ||||𝑥∈𝑄0∶𝑀𝑄+,𝑑0𝑓−𝑓𝑄0+,2+||||≤(𝑥)>ğœ†ğ‘Žâ€–ğ‘“â€–ğ½ğ‘+𝑝(𝑄0)𝜆||||𝑥∈𝑄0∶𝑀𝑄+,𝑑0𝑓−𝑓𝑄0+,2+||||(𝑥)>𝑏𝜆1/ğ‘ž,(4.8) whenever 𝑏𝜆≥𝑄+0𝑓−𝑓𝑄0+,2+d𝑥.(4.9) Here ğ‘Ž=4(1−2𝑛𝑏)−1 and ğ‘ž is the conjugate exponent of 𝑝.

Proof. Without loss of generality, we may assume ‖𝑓‖𝐽𝑁+𝑝(𝑄0)=1. Setting 𝐸𝑄(𝜆)∶=𝑥∈𝑄∶𝑀𝑄+,𝑑𝑓−𝑓𝑄0+,2+(𝑥)>𝜆,(4.10) we may write the statement as ||𝐸𝑄0||≤4(𝜆)(1−2𝑛𝑏||𝐸)𝜆𝑄0||(𝑏𝜆)1/ğ‘ž.(4.11) 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 𝐸𝑄0(𝑏𝜆)=𝑗𝑄𝑗.(4.12) Since 𝑏𝜆<𝜆, we have 𝐸𝑄0(𝜆)⊆𝐸𝑄0(𝑏𝜆). It now follows that 𝐸𝑄0(𝜆)=𝑗𝐸𝑄𝑗(𝜆).(4.13) We claim that for every 𝑗, |||𝐸𝑄𝑗(|||≤2𝜆)(1−2𝑛𝑏)𝜆𝑄𝑗∪𝑄+𝑗𝑓−𝑓𝑄𝑗+,2+d𝑥.(4.14) Consider the functions 𝑔𝑗∶=(𝑓−𝑓𝑄𝑗+,2)+. To prove (4.14) it suffices to show that 𝐸𝑄𝑗(𝜆)⊆𝑥∈𝑄𝑗∶𝑀𝑄+,𝑑𝑗𝑔𝑗(𝑥)>(1−2𝑛𝑏)𝜆.(4.15) 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 𝑄+𝑓−𝑓𝑄0+,2+>𝜆.(4.16) From (4.4) we have 𝑄𝑗+,2𝑓−𝑓𝑄0+,2+≤2𝑛𝑏𝜆.(4.17) Combining these, we obtain (1−2𝑛𝑏)𝜆<𝑄+𝑓−𝑓𝑄0+,2+d𝑥−𝑄𝑗+,2𝑓−𝑓𝑄0+,2+≤d𝑥𝑄+𝑓−𝑓𝑄0+,2+d𝑥−𝑄𝑗+,2𝑓−𝑓𝑄0+,2d𝑥+=𝑄+𝑓−𝑓𝑄0+,2+−𝑓𝑄𝑗+,2−𝑓𝑄0+,2+≤d𝑥𝑄+𝑓−𝑓𝑄𝑗+,2+d𝑥≤𝑀𝑄+,𝑑𝑗𝑔𝑗(𝑥).(4.18)
Having now seen that (4.14) holds, we use (4.13) and sum over all 𝑗 to obtain ||𝐸𝑄0||=(𝜆)𝑗|||𝐸𝑄𝑗|||≤2(𝜆)(1−2𝑛𝑏)𝜆𝑗𝑄𝑗∪𝑄+𝑗𝑓−𝑓𝑄𝑗+,2+=2d𝑥(1−2𝑛𝑏)𝜆𝑗||𝑄𝑗||1/ğ‘ž||𝑄𝑗||−1/ğ‘žî€œğ‘„ğ‘—âˆªğ‘„+𝑗𝑓−𝑓𝑄𝑗+,2+≤4d𝑥(1−2𝑛𝑏)𝜆𝑗||𝑄𝑗||1/ğ‘ž,(4.19) where the last inequality follows from the Hölder inequality and the assumption ‖𝑓‖𝐽𝑁+𝑝(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‖𝐽𝑁+𝑝(𝑄0)=1, we shall show ||𝐸𝑄0(||≤𝐶𝜆)𝜆𝑝.(4.20) Let us choose 𝜆02∶=𝑏||𝑄0||1/𝑝,(4.21) and assume 𝜆>𝜆0. Then take 𝑁∈ℤ+ such that 𝑏−𝑁𝜆0≤𝜆<𝑏−(𝑁+1)𝜆0=2𝑏−(𝑁+2)||𝑄0||1/𝑝.(4.22) By the assumption ‖𝑓‖𝐽𝑁+𝑃(𝑄0)=1, we have 1||𝑄0||𝑄0∪𝑄+0𝑓−𝑓𝑄0+,2+2d𝑥≤||𝑄0||1/𝑝=𝑏𝜆0.(4.23) In particular, this implies that 1𝑏𝑄+0𝑓−𝑓𝑄0+,2+d𝑥≤𝜆0≤𝑏−1𝜆0≤⋯≤𝑏−𝑁𝜆0,(4.24) allowing us to apply the previous lemma successively 𝑁 times to estimate the left-hand side of (4.20) as follows: ||𝐸𝑄0||≤||𝐸(𝜆)𝑄0𝑏−𝑁𝜆0||â‰¤ğ‘Žğ‘âˆ’ğ‘ğœ†0â‹…îƒ©ğ‘Žğ‘âˆ’ğ‘+1𝜆01/ğ‘žîƒ©ğ‘Žâ‹…â‹¯â‹…ğ‘âˆ’1𝜆01/ğ‘žğ‘âˆ’1||𝐸𝑄0𝜆0||1/ğ‘žğ‘â‰¤ğ‘Žâ‹…î‚µğ‘Žğ‘ğœ†ğ‘2𝜆1/ğ‘žî‚µğ‘Žâ‹…â‹¯â‹…ğ‘ğ‘ğœ†î‚¶1/ğ‘žğ‘âˆ’1⋅2𝜆0𝑄0∪𝑄+0(𝑓−𝑓𝑄0+,2)+d𝑥1/ğ‘žğ‘,(4.25) 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 ||𝐸𝑄0||â‰¤î‚€ğ‘Ž(𝜆)𝜆1+ğ‘žâˆ’1+⋯+ğ‘žâˆ’(𝑁−1)⋅𝑏−(1+2ğ‘žâˆ’1+⋯+ğ‘ğ‘žâˆ’(𝑁−1))⋅||𝑄2𝑏0||1/ğ‘žğ‘=î‚€ğ‘Žğœ†î‚ğ‘âˆ’ğ‘/ğ‘žğ‘â‹…ğ‘âˆ’(1+2ğ‘žâˆ’1+⋯+ğ‘ğ‘žâˆ’(𝑁−1))+ğ‘žâˆ’ğ‘â‹…21/ğ‘žğ‘â‹…||𝑄0||1/ğ‘žğ‘.(4.26) Since both 1+2ğ‘žâˆ’1+⋯+ğ‘ğ‘žâˆ’(𝑁−1) and 𝑝−𝑝/ğ‘žğ‘ remain bounded as ğ‘â†’âˆž, we have ||𝐸𝑄0||||𝑄(𝜆)≤𝐶0||1/ğ‘žğ‘î‚€1𝜆𝑝−𝑝/ğ‘žğ‘.(4.27) Finally, we notice that from the second inequality in (4.22) we get ||𝑄0||1/ğ‘žğ‘î‚€1𝜆−𝑝/ğ‘žğ‘=𝜆𝑝/ğ‘žğ‘||𝑄0||1/ğ‘žğ‘â‰¤2𝑝/ğ‘žğ‘ğ‘âˆ’(𝑁+2)𝑝/ğ‘žğ‘â‰¤ğ¶,(4.28) with 𝐶 independent of 𝑁. Thus we have arrived at the desired estimate.
For 0<𝜆≤𝜆0 we use the trivial estimate ||||𝑥∈𝑄0∶𝑓(𝑥)−𝑓𝑄0+,2+||||≤||𝑄>𝜆0||=2𝑝𝑏𝑝𝜆𝑝0≤𝐶𝜆𝑝.(4.29)


The author was supported by the Finnish Cultural Foundation. The author wishes to thank J. Kinnunen for proposing the problem.