#### Abstract

We introduce a parabolic version of John-Nirenberg space with exponent *p* and show that it is contained in local weak- 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 where the supremum is taken over all partitions of a given cube into pairwise nonoverlapping subcubes. Here we have used standard notation for integral averages The functional defines a seminorm, and the class of functions satisfying , which we denote by for John-Nirenberg space with exponent , can be seen as a generalization of BMO. Indeed, BMO is obtained as the limit case of In contrast to the exponential integrability of BMO functions, functions in belong to the space weak-. This was already observed by John and Nirenberg. Precisely, they showed that for , we have where the constant depends on and . Simpler proofs and generalizations have appeared in [1β11]. In this paper we show that an analogous result holds in the context of parabolic or one-sided spaces, which we hope will increase our understanding of the spaces themselves. Therefore, the main purpose is to provide a step towards a satisfactory multidimensional theory of the spaces described below.

#### 2. BMO^{+} Spaces

We will introduce some notations and terminologies. Given a Euclidean cube and , we define the βforward in timeβ -translation For short, we write for the cube of the same size above . We write , if we have where the supremum is taken over all cubes in 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 class was first introduced by MartΓn-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 , stating that functions in satisfy the following version of John-Nirenberg inequality. Given , we have for all where the constants and 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 . The weaker result (2.3) for 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 and , we write if where the supremum is taken over countable families of pairwise nonoverlapping subcubes of . Observe that even if is contained in , or may not be. Hence, we assume that is a priori defined on all of . The definition is reasonable because in a sense may be seen as the limit case of as . Precisely, where the quantity on the right-hand side is globally equivalent to the norm of , that is, 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 . Then, for every , one has
**
where only depends on and .*

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 for or even further to 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 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 and a cube , denote by the family of all dyadic subcubes obtained from by repeatedly bisecting the sides into two parts of equal length. We shall make use of the βforward in time dyadic maximal functionβ defined by A standard stopping-time argument shows that we have where 's are the maximal dyadic subcubes of satisfying Maximality implies that the cubes are pairwise nonoverlapping. Moreover, if , then does not satisfy (4.3). Consequently, in this case every is contained in a larger dyadic subcube of which does not satisfy (4.3). Since , we conclude provided . Standard arguments imply a weak-type estimate for . Indeed, we have While the cubes are non-overlapping, the cubes may not be. Let us replace by the maximal non-overlapping subfamily which we form by collecting those which are not properly contained in any other . Maximality of enables us to partition the family as follows. Given , we define , and we may write . Now, whenever , we have and we get the estimate Combining the previous estimates, we arrive at

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

Lemma 4.1. * Assume and take . Then, one has
**
whenever
**
Here and is the conjugate exponent of .*

*Proof. *Without loss of generality, we may assume . Setting
we may write the statement as
Consider the function and form the decomposition as above at level to obtain a family of pairwise non-overlapping dyadic subcubes with
Since , we have . It now follows that
We claim that for every ,
Consider the functions . To prove (4.14) it suffices to show that
Indeed, (4.14) then follows at once from the weak-type estimate (4.7) applied to the functions with replaced by . Let for some . Then there exists a dyadic subcube of containing and satisfying
From (4.4) we have
Combining these, we obtain

Having now seen that (4.14) holds, we use (4.13) and sum over all to obtain
where the last inequality follows from the HΓΆlder inequality and the assumption . Remembering also that , 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 , we shall show
Let us choose
and assume . Then take such that
By the assumption , we have
In particular, this implies that
allowing us to apply the previous lemma successively times to estimate the left-hand side of (4.20) as follows:
where the last inequality follows from the weak-type estimate (4.7) and the first inequality in (4.22). By the choice of and (4.23) we further estimate
Since both and remain bounded as , we have
Finally, we notice that from the second inequality in (4.22) we get
with independent of . Thus we have arrived at the desired estimate.

For we use the trivial estimate

#### Acknowledgments

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