Parabolic John-Nirenberg Spaces

Abstract

We introduce a parabolic version of John-Nirenberg space with exponent p and show that it is contained in local weak-𝐿^𝑝 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‖^𝑝𝐽𝑁_𝑝(𝑄0)∶=sup_{𝑄𝑗𝑗}_𝑗||𝑄_𝑗||_𝑄𝑗|||𝑓−𝑓_𝑄𝑗|||d𝑥^𝑝<∞,(1.1) where the supremum is taken over all partitions {𝑄_𝑗}_𝑗 of a given cube 𝑄₀ into pairwise nonoverlapping subcubes. Here we have used standard notation for integral averages 𝑓_𝑄∶=_𝑄𝑓d𝑥∶=1||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 𝐽𝑁_𝑝(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 𝐽𝑁_𝑝lim_𝑝→∞‖𝑓‖_{𝐽𝑁𝑝(𝑄0)}=sup_𝑄⊆𝑄0_𝑄||𝑓−𝑓_𝑄||d𝑥=∶‖𝑓‖_BMO(𝑄0).(1.3) In contrast to the exponential integrability of BMO functions, functions in 𝐽𝑁_𝑝(𝑄₀) belong to the space weak-𝐿^𝑝(𝑄₀). This was already observed by John and Nirenberg. Precisely, they showed that for 𝜆>0, we have ||𝑥∈𝑄₀∶||𝑓(𝑥)−𝑓_𝑄0||>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[𝑎_𝑖,𝑎_𝑖+ℎ] and 𝑟>0, we define the “forward in time” 𝑟-translation 𝑄^+,𝑟=^𝑛−1_𝑖=1𝑎_𝑖,𝑎_𝑖+ℎ×𝑎_𝑛+𝑟ℎ,𝑎_𝑛+(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⁺(ℝ^𝑛), if we have ‖𝑓‖_BMO⁺_(ℝ^𝑛₎∶=sup_{𝑄⊆ℝ𝑛}_𝑄𝑓−𝑓_𝑄⁺⁺d𝑥<∞,(2.2) where the supremum is taken over all cubes in ℝ^𝑛 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⁺(ℝ^𝑛), stating that functions in BMO⁺(ℝ^𝑛) satisfy the following version of John-Nirenberg inequality. Given 𝑟>1, we have for all 𝜆>0|||𝑥∈𝑄∶𝑓(𝑥)−𝑓_𝑄+,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||𝑄_𝑗||_{𝑄𝑗∪𝑄+𝑗}𝑓−𝑓^𝑄𝑗+,2⁺d𝑥^𝑝<∞,(3.1) where the supremum is taken over countable families {𝑄_𝑗} 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 ℝ^𝑛. The definition is reasonable because in a sense BMO⁺ may be seen as the limit case of 𝐽𝑁^+𝑝 as 𝑝→∞. Precisely, lim_𝑝→∞‖𝑓‖_𝐽𝑁^+𝑝_(𝑄0)=sup_𝑄⊆𝑄0_𝑄∪𝑄+𝑓−𝑓_𝑄^+,2⁺d𝑥,(3.2) where the quantity on the right-hand side is globally equivalent to the BMO⁺ norm of 𝑓, that is, sup_{𝑄⊆ℝ𝑛}_𝑄∪𝑄+𝑓−𝑓_𝑄^+,2⁺d𝑥≈‖𝑓‖_BMO⁺_(ℝ^𝑛₎,(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)−𝑓_𝑄^0+,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)−𝑓_𝑄^0+,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 𝑀^𝑄+,𝑑₀𝑓(𝑥)∶=sup_{𝑄∈Δ𝑥∈𝑄}_𝑄+𝑓d𝑥.(4.1) A standard stopping-time argument shows that we have 𝑥∈𝑄⁰∶𝑀^𝑄+,𝑑₀𝑓(𝑥)>𝜆=_𝑗𝑄_𝑗,(4.2) where 𝑄_𝑗's are the maximal dyadic subcubes of 𝑄₀ satisfying _𝑄+𝑗𝑓d𝑥>𝜆.(4.3) Maximality implies that the cubes 𝑄_𝑗 are pairwise nonoverlapping. Moreover, if 𝜆≥𝑓_𝑄⁺⁰, then 𝑄₀ does not satisfy (4.3). Consequently, in this case every 𝑄_𝑗 is contained in a larger dyadic subcube 𝑄_𝑗− of 𝑄₀ which does not satisfy (4.3). Since 𝑄^𝑗+,2⊆𝑄^+𝑗₋, we conclude _𝑄^𝑗+,2𝑓d𝑥≤2^𝑛𝜆,(4.4) provided 𝜆≥𝑓_𝑄⁺⁰. Standard arguments imply a weak-type estimate for 𝑀^𝑄+,𝑑₀. Indeed, we have |||𝑥∈𝑄₀∶𝑀^𝑄+,𝑑₀𝑓(𝑥)>𝜆|||=_𝑗||𝑄_𝑗||.(4.5) While the cubes 𝑄_𝑗 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^+𝑗. Maximality of {𝑄^+𝑗}_𝑗 enables us to partition the family {𝑄_𝑗}_𝑗 as follows. Given 𝑄^+𝑗, we define 𝐼_𝑗∶={𝑖∶𝑄^+𝑖⊆𝑄^+𝑗}, and we may write {𝑄_𝑗}_𝑗=⋃_𝑗{𝑄_𝑖∶𝑖∈𝐼_𝑗}. Now, whenever 𝑖∈𝐼_𝑗, we have 𝑄_𝑖⊆Q_𝑗∪𝑄^+𝑗 and we get the estimate _𝑗||𝑄_𝑗||=_𝑗𝑖_∈𝐼𝑗≤2_𝑗||𝑄_𝑖||≤2|||𝑄^+𝑗|||𝜆_𝑄0∪𝑄⁺⁰𝑓d𝑥.(4.6) Combining the previous estimates, we arrive at |||𝑥∈𝑄₀∶𝑀^𝑄+,𝑑₀𝑓(𝑥)>𝜆|||≤2𝜆_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≤𝑎‖𝑓‖∶𝑀^𝑄+,𝑑_{0𝐽𝑁+𝑝(𝑄0)}𝑓−𝑓^𝑄0+,2⁺(𝑥)>𝜆||||𝜆||||𝑥∈𝑄⁰∶𝑀^𝑄+,𝑑₀𝑓−𝑓^𝑄0+,2⁺(𝑥)>𝑏𝜆||||^1/𝑞,(4.8) whenever 𝑏𝜆≥_𝑄+0𝑓−𝑓^𝑄0+,2⁺d𝑥.(4.9) Here 𝑎=4(1−2^𝑛𝑏)⁻¹ and 𝑞 is the conjugate exponent of 𝑝.

Proof. Without loss of generality, we may assume ‖𝑓‖_𝐽𝑁^+𝑝_(𝑄0)=1. Setting 𝐸_𝑄(𝜆)∶=𝑥∈𝑄∶𝑀^𝑄+,𝑑𝑓−𝑓^𝑄0+,2⁺(𝑥)>𝜆,(4.10) we may write the statement as ||𝐸_𝑄0(𝜆)||≤4(1−2^𝑛𝑏)𝜆||𝐸^𝑄0(𝑏𝜆)||^1/𝑞.(4.11) Consider the function (𝑓−𝑓_𝑄^0+,2)⁺ and form the decomposition as above at level 𝑏𝜆 to obtain a family of pairwise non-overlapping dyadic subcubes with 𝐸_𝑄0(𝑏𝜆)=_𝑗𝑄_𝑗.(4.12) Since 𝑏𝜆<𝜆, we have 𝐸_𝑄0(𝜆)⊆𝐸_𝑄0(𝑏𝜆). It now follows that 𝐸_𝑄0(𝜆)=_𝑗𝐸_𝑄𝑗(𝜆).(4.13) We claim that for every 𝑗, |||𝐸_𝑄𝑗(𝜆)|||≤2(1−2^𝑛𝑏)𝜆_{𝑄𝑗∪𝑄}^+𝑗𝑓−𝑓^𝑄𝑗+,2⁺d𝑥.(4.14) Consider the functions 𝑔_𝑗∶=(𝑓−𝑓_𝑄^𝑗+,2)⁺. To prove (4.14) it suffices to show that 𝐸_𝑄𝑗(𝜆)⊆𝑥∈𝑄_𝑗∶𝑀^𝑄+,𝑑_𝑗𝑔_𝑗(𝑥)>(1−2^𝑛𝑏)𝜆.(4.15) Indeed, (4.14) then follows at once from the weak-type estimate (4.7) applied to the functions 𝑔_𝑗 with 𝜆 replaced by (1−2^𝑛𝑏)𝜆. Let 𝑥∈𝐸_𝑄𝑗(𝜆) for some 𝑗. Then there exists a dyadic subcube 𝑄 of 𝑄_𝑗 containing 𝑥 and satisfying _𝑄+𝑓−𝑓^𝑄0+,2⁺>𝜆.(4.16) From (4.4) we have _𝑄^𝑗+,2𝑓−𝑓^𝑄0+,2⁺≤2^𝑛𝑏𝜆.(4.17) Combining these, we obtain (1−2^𝑛𝑏)𝜆<≤_𝑄+𝑓−𝑓^𝑄0+,2⁺d𝑥−_𝑄𝑗+,2𝑓−𝑓^𝑄0+,2⁺d𝑥_𝑄+𝑓−𝑓^𝑄0+,2⁺d𝑥−_𝑄𝑗+,2𝑓−𝑓_𝑄^0+,2d𝑥⁺=≤_𝑄+𝑓−𝑓^𝑄0+,2⁺−𝑓_𝑄^𝑗+,2−𝑓_𝑄^0+,2⁺d𝑥_𝑄+≤𝑀_{𝑄+,𝑑𝑗}𝑓−𝑓^𝑄𝑗+,2⁺d𝑥𝑔_𝑗(𝑥).(4.18)
Having now seen that (4.14) holds, we use (4.13) and sum over all 𝑗 to obtain ||𝐸_𝑄0(𝜆)||=≤_𝑗|||𝐸2_𝑄𝑗(𝜆)|||(1−2^𝑛𝑏)𝜆_𝑗_{𝑄𝑗∪𝑄+𝑗}𝑓−𝑓^𝑄𝑗+,2⁺=2d𝑥(1−2^𝑛𝑏)𝜆_𝑗||𝑄_𝑗||^1/𝑞||𝑄_𝑗||^−1/𝑞_{𝑄𝑗∪𝑄+𝑗}𝑓−𝑓^𝑄𝑗+,2⁺≤4d𝑥(1−2^𝑛𝑏)𝜆_𝑗||𝑄_𝑗||^1/𝑞,(4.19) where the last inequality follows from the Hölder inequality and the assumption ‖𝑓‖_𝐽𝑁^+𝑝_(𝑄0)=1. Remembering also that 𝐸_𝑄(𝑏𝜆)=⋃_𝑗𝑄_𝑗, 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 ||𝐸_𝑄0(𝜆)||≤𝐶𝜆^𝑝.(4.20) Let us choose 𝜆₀∶=2𝑏||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−𝑓^𝑄0+,2⁺d𝑥≤2||𝑄₀||^1/𝑝=𝑏𝜆₀.(4.23) In particular, this implies that 1𝑏_𝑄⁺⁰𝑓−𝑓^𝑄0+,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: ||𝐸_𝑄0(𝜆)||≤≤||𝐸𝑎_𝑄0𝑏^−𝑁𝜆₀||𝑏^−𝑁𝜆₀⋅𝑎𝑏^−𝑁+1𝜆₀^1/𝑞⋅⋯⋅𝑎𝑏⁻¹𝜆₀^{1/𝑞𝑁−1}||𝐸_𝑄0𝜆₀||^1/𝑞𝑁≤𝑎𝑏𝜆⋅𝑏𝑎²𝜆^1/𝑞⋅⋯⋅𝑎𝑏^𝑁𝜆^{1/𝑞𝑁−1}⋅2𝜆₀_{𝑄0∪𝑄+0}(𝑓−𝑓_𝑄^0+,2)⁺d𝑥^1/𝑞𝑁,(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 ||𝐸_𝑄0(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}⋅2^1/𝑞𝑁⋅||𝑄₀||^1/𝑞𝑁.(4.26) Since both 1+2𝑞⁻¹+⋯+𝑁𝑞^{−(𝑁−1)} and 𝑝−𝑝/𝑞^𝑁 remain bounded as 𝑁→∞, we have ||𝐸_𝑄0(𝜆)||≤𝐶||𝑄₀||^1/𝑞𝑁𝜆1^{𝑝−𝑝/𝑞𝑁}.(4.27) Finally, we notice that from the second inequality in (4.22) we get ||𝑄₀||^1/𝑞𝑁𝜆1^{−𝑝/𝑞𝑁}=𝜆^{𝑝/𝑞𝑁}||𝑄₀||^1/𝑞𝑁≤2^{𝑝/𝑞𝑁}𝑏^{−(𝑁+2)𝑝/𝑞𝑁}≤𝐶,(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)−𝑓_𝑄^0+,2⁺>𝜆||||≤||𝑄₀||=2^𝑝𝑏^𝑝𝜆^𝑝0≤𝐶𝜆^𝑝.(4.29)

Acknowledgments

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