Parabolic John-Nirenberg spaces

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


Introduction
In the classical paper of F. John and L. Nirenberg [10], 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 {Q j } j of a given cube Q 0 into pairwise non-overlapping subcubes. The functional f → K f defines a seminorm and the class of functions satisfying K f < ∞, which we denote by JN p (Q 0 ) 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 p in the sense that In contrast to the exponential integrability of BMO functions, K f < ∞ implies that f belongs to the space weak-L p (Q 0 ). This was already observed by John and Nirenberg. Precisely, they showed that for λ > 0, we have where the constant C depends on n and p. Simpler proofs and generalizations have appeared in [1,3,5,6,7,8,9,10,11,12,14]. In this note we show that an analogous result holds in the context of parabolic BMO spaces.

Parabolic John-Nirenberg space
We shall introduce some notation and terminology. Given an Euclidean , we define the forward in time translation [a i , a i + h] × [a n + h, a n + 2h].
Moreover, we use the notation Q +,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 (2.1) is not actually a norm.
The one-dimensional BMO + (R) class was first introduced by F. J. Martín-Reyes and A. de la Torre [13], who showed that this class possesses many properties similar to the standard BMO space. Even though steps towards a multidimensional theory has been taken (see [4]), 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), and a slightly weaker version of this result for BMO + (R n ) was obtained in [4].
In this setting we define John-Nirenberg spaces as follows. We write where the supremum is taken over countable families {Q j } of pairwise non-overlapping cubes satisfying j |Q j | < ∞. The definition is reasonable in the sense that the BMO + (R n ) condition may be seen as the limit case of (2.2) as p → ∞. Precisely, where the quantity on the right-hand side is equivalent (up to a multiplication by a universal constant) to the BMO + norm of f , defined by (2.1).
The following theorem is a parabolic version of the weak distribution inequality of John and Nirenberg. Theorem.
Then, for every cube Q 0 and λ > 0, we have where C only depends on n and p.

Proof of the theorem
We follow the argument used in [1]. Given a non-negative f and a cube Q 0 , denote by ∆ = ∆(Q 0 ) the family of all dyadic subcubes obtained from Q 0 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 A standard stopping-time argument shows that we have where Q j 's are the maximal dyadic subcubes of Q 0 satisfying Maximality implies that the cubes Q j are pairwise non-overlapping. Moreover, if λ ≥ f Q + 0 , then Q 0 doesn't satisfy (3.1). Consequently, in this case every Q j is contained in a larger dyadic subcube Q j − of Q 0 which does not satisfy (3.1). Since Q +,2 provided λ ≥ f Q + 0 . Standard arguments imply a weak type estimate for M +,d Q 0 . Indeed, we have 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 Combining the previous estimates, we arrive at We begin by proving the following good λ inequality for the forward in time dyadic maximal operator.
Lemma. Assume f ∈ JN + p (R n ) and take 0 < b < 2 −n . Then, given a cube Q 0 , we have Here a = 4(1 − 2 n b) −1 and q is the conjugate exponent of p.

Proof. Setting
we may write the statement as Consider the function (f −f Q +,2 0 ) + and form the decomposition as above at level bλ to obtain a family of pairwise non-overlapping dyadic subcubes with We claim that for every j, Consider the functions g j := (f − f Q +,2 j ) + . To prove (3.6) it suffices to show that Indeed, (3.6) then follows at once from the weak type estimate (3.3) applied to the functions g j with λ replaced by (1 − 2 n b)λ. Let x ∈ E Q j (λ) for some j. Then there exists a dyadic subcube Q of Q j containing x and satisfying Combining these, we obtain Having now seen that (3.6) holds, we use (3.5) and sum over all j to obtain where the last inequality follows from the Hölder inequality and the definition of K + f . Remembering also that E Q (bλ) = j Q 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 a just a repetition of the argument used in [1].
Proof of the Theorem. Using the same notation as in the proof of the lemma, we shall show Let us choose λ 0 := 2K + f b|Q 0 | 1/p and assume λ > λ 0 . Then take N ∈ Z + such that By the definition of K + f , we have In particular, this implies allowing us to apply the previous lemma successively N times to estimate the left-hand side of (3.8) as follows: where the last inequality follows from the weak type estimate (3.3) and the first inequality in (3.9). By the choice of λ 0 and (3.10) we further estimate ..+N q −(N−1) )+q −N · 2 1/q N · |Q 0 | 1/q N .
Since both 1 + 2q −1 + . . . + Nq −(N −1) and p − p/q N remain bounded as N → ∞, we have Finally, we notice that from the second inequality in (3.9) we get with C independent of N. Thus we have arrived at the desired estimate.