Nontrivial examples of $JN_p$ and $VJN_p$ functions

We study the John-Nirenberg space $JN_p$, which is a generalization of the space of bounded mean oscillation. In this paper we construct new $JN_p$ functions, that increase the understanding of this function space. It is already known that $L^p(Q_0) \subsetneq JN_p(Q_0) \subsetneq L^{p,\infty}(Q_0)$. We show that if $|f|^{1/p} \in JN_p(Q_0)$, then $|f|^{1/q} \in JN_q(Q_0)$, where $q \geq p$, but there exists a nonnegative function $f$ such that $f^{1/p} \notin JN_p(Q_0)$ even though $f^{1/q} \in JN_q(Q_0)$, for every $q \in (p,\infty)$. We present functions in $JN_p(Q_0) \setminus VJN_p(Q_0)$ and in $VJN_p(Q_0) \setminus L^p(Q_0)$, proving the nontriviality of the vanishing subspace $VJN_p$, which is a $JN_p$ space version of $VMO$. We prove the embedding $JN_p(\mathbb{R}^n) \subset L^{p,\infty}(\mathbb{R}^n)/\mathbb{R}$. Finally we show that we can extend the constructed functions into $\mathbb{R}^n$, such that we get a function in $JN_p(\mathbb{R}^n) \setminus VJN_p(\mathbb{R}^n)$ and another in $CJN_p(\mathbb{R}^n) \setminus L^p(\mathbb{R}^n)/\mathbb{R}$. Here $CJN_p$ is a subspace of $JN_p$ that is inspired by the space $CMO$.


Introduction
In their seminal paper [7] John and Nirenberg studied the famous space of bounded mean oscillation BM O and proved the profound John-Nirenberg inequality for BM O functions. In the same paper they also defined a generalization of BM O which has since become known as the John-Nirenberg space, or JN p , with parameter p ∈ (1, ∞). The space JN p is a generalization of BM O in the sense that the BM O norm of a function is the limit of its JN p norm when p tends to infinity.
In this paper we define JN p as in Definition 2.1 below: for a bounded cube Q 0 ⊂ R n and a number p ∈ (1, ∞), a function f is in JN p (Q 0 ) if f ∈ L 1 (Q 0 ) and where the supremum is taken over all countable collections of pairwise disjoint cubes (Q i ) ∞ i=1 that are contained in Q 0 . Here f Qi denotes the integral average of f over Q i : Many other definitions have been used for JN p , some of which are not equivalent with this definition. This is because the space depends on how much we let the sets Q i overlap. For example many of the definitions in the more general metric measure space, such as in [1,6,9,11], use balls B i instead of cubes, and these balls may overlap with each other in some definitions. This results in different spaces.
Other related function spaces include the dyadic JN p [8], the John-Nirenberg-Campanato spaces [15], their localized versions [14] and the sparse JN p [5]. For an extensive survey of John-Nirenberg type spaces see [17].
It is well-known that L p (Q 0 ) ⊂ JN p (Q 0 ). The space JN p is also embedded in the weak L p space, JN p (Q 0 ) ⊂ L p,∞ (Q 0 ). Further both of these inclusions are strict. An example of a function in JN p \ L p was constructed in a recent paper [4]. Thus the space JN p is a nontrivial space between L p and L p,∞ . Other than that, the behaviour of JN p functions is still very much unknown.
One of the properties with L p spaces is that |f | 1/q ∈ L q if and only if |f | 1/p ∈ L p whenever p, q ∈ [1, ∞). A similar equivalence holds for weak L p spaces. Since JN p is a space between these spaces, this raises the question: does the equivalence hold for JN p spaces? In Section 3 we give a negative answer to this question by constructing an example of a nonnegative function f such that f 1/p / ∈ JN p (Q 0 ) even though f 1/q ∈ JN q (Q 0 ) for every q ∈ (p, ∞). The previous existing example of a JN p function does not have this property of distinguishing JN p spaces with different values of p. We also show that if |f | 1/p ∈ JN p , then |f | 1/q ∈ JN q for every q ≥ p.
Additionally we study the vanishing subspace of JN p , which is denoted by V JN p . This subspace has been studied by Brudnyi and Brudnyi [3] and by Tao et al. [16]. The space V JN p is defined as a John-Nirenberg space counterpart to the famous space of vanishing mean oscillation, V M O, which is a subspace of BM O, and was first studied by Sarason [13]. It is well-known that L p (Q 0 ) ⊂ V JN p (Q 0 ) ⊂ JN p (Q 0 ). Tao et al. proved that L p (Q 0 ) = V JN p (Q 0 ) by showing that the double dual space of V JN p is JN p [16]. In Section 4 we present examples of functions in JN p \ V JN p and in V JN p \ L p , thereby proving that V JN p (Q 0 ) = JN p (Q 0 ). Both of these functions are based on the same type of fractal construction.
Finally in Section 5 we study JN p and V JN p in R n instead of a bounded cube. The embedding JN p ⊂ L p,∞ has been proved in many different ways for a bounded domain -originally in [7] and further discussion can be found in [1,2,6,9,10,15]. However, as far as we know, the question of whether this holds for unbounded domains has not been addressed in the literature. For the sake of completeness we prove that this embedding indeed holds in the case of the whole space R n , if we replace L p,∞ with L p,∞ /R -the space of functions in L p,∞ modulo constant.
We also consider the space CJN p : a subspace of JN p , that has been studied by Tao et al. [16]. The space CJN p is defined analogously to the space of continuous mean oscillation, CM O, which is a subspace of BM O, and was first announced by Neri [12]. We study CJN p only on R n as on bounded domains it coincides with V JN p . It is clear from the definitions that L p /R ⊂ CJN p ⊂ V JN p ⊂ JN p . By extending the functions in Section 4 into the whole space R n we get functions that are in JN p (R n ) \ V JN p (R n ) and in CJN p (R n ) \ L p (R n )/R, proving that the respective inclusions are strict. This answers [16,Question 5.8] and it partially answers [16,Question 5.6] and [17,Question 17].
Throughout this paper we assume that a function has its domain in the Euclidian space R n or in a bounded cube Q 0 ⊂ R n . Every cube in R n is assumed to have edges parallel to the coordinate axes.

Preliminaries
The infimum approach makes calculations sometimes much more simple.
One can also define JN p by using medians which allows us to not use integrals at all. This approach is studied by Myyryläinen [11]. We do not consider JN p with p = 1, because clearly JN 1 (Q 0 ) = L 1 (Q 0 ), see equation (2.3) below. Thus we assume from now on that p > 1.
It is clear that if a function f is in L p (Q 0 ), then it is also in JN p (Q 0 ). This follows from Hölder's inequality: for every partition (Q i ) ∞ i=1 of Q 0 into disjoint cubes. The inclusion L p ⊂ JN p is strict, but functions in JN p \ L p are very complicated. An example of a function in JN p \ L p was constructed in a recent paper [4].
If a function is in JN p (Q 0 ), then it is also in the weak L p -space L p,∞ (Q 0 ). This result is a JN p space counterpart of the famous John-Nirenberg lemma for BM O functions.
Conveniently the constant c does not depend on Q 0 even though the proof does rely on the boundedness of Q 0 . This embedding was originally proved in [7,Lemma 3]. In [6] this was generalized from cubes into John domains. In [1] and [9] the result was proven in a more general metric space with a doubling measure. In [15] a similar result was proven for John-Nirenberg-Campanato spaces, which are a generalization of John-Nirenberg spaces. Milman gave a new characterization of the weak L p space as the Garsia-Rodemich space and proved the embedding of JN p (Q 0 ) in this space [10]. Berkovits et al. proved a good-λ inequality and used this to prove multiple embedding theorems including the embedding of JN p to the weak L p -space [2].
The inclusion JN p (Q 0 ) ⊂ L p,∞ (Q 0 ) is strict as well. Consider the counterexample f (x) = x −1/p in the one-dimensional case n = 1 with Q 0 = (0, 1). In this case f is in the weak L p space. If we divide Q 0 into subintervals Q i = 2 −i , 2 −i+1 , it turns out that for all i ≥ 1. Thus it is easy to see that f / ∈ JN p (Q 0 ). This counterexample can also be easily extended into the multidimensional case Q 0 ⊂ R n by using the following proposition.
For the proof we refer to [4,Proposition 4.1]. In conclusion we know that This raises many questions about the properties of JN p functions. For example it is clear that for nonnegative L p functions the equivalence holds for all p, q ∈ [1, ∞). This also holds for weak L p spaces.
Our example in Section 3 proves that JN p spaces do not have the same property. However there is a weaker result.

2.7.
Proposition. If f : Q 0 → R is a nonnegative function, then Proof. Firstly we notice that f 1/q ∈ L 1 (Q 0 ), because f 1/q ≤ f 1/p whenever f ≥ 1, f 1/p ∈ L 1 (Q 0 ) by assumption, and Q 0 is a bounded cube with finite measure. We notice that for any cube Q i ⊂ Q 0 and for any Here we first used Hölder's inequality and then the inequality a r + b r ≤ (a + b) r , whenever a, b ≥ 0 and r ≥ 1. Also |f 1/p − |c i || ≤ |f 1/p − c i |. By taking the infimum over c i , this implies that .
Here we used Remark 2.2. This completes the proof.
The implication in Proposition 2.7 does not hold in the other direction. In the next section we construct a function for which 3. An example that distinguishes JN p spaces with different parameters p In this section we construct a function f that distinguishes JN p spaces with different p in the sense that f 1/p / ∈ JN p but f 1/q ∈ JN q , where q > p. Clearly we need to have f 1/q ∈ JN q \ L q , because if f 1/q ∈ L q , then it follows from (2.6) that f 1/p ∈ L p ⊂ JN p . This is why the example is influenced by the function g ∈ JN p \ L p given in [4]. Since the spaces L p and JN p coincide for monotone functions [4,Theorem 2.1], all examples in JN p \ L p must have a highly oscillatory structure. It is enough to construct our function in the interval [0, 1]. Then we can make a change of variable to get a similar function in an arbitrary interval.
Let us denote the function f 1/p in Theorem 3.1 by u and the function in [4] by g. Both u and g consist of "towers" of width l i where i ∈ Z + . Each tower has an inclined roof with its left side set at height a i and the right side at height b i . The roof of the tower is linear. On both sides of the tower, at distance d i , there are two similar narrower towers of width l i+1 and height ranging from a i+1 to b i+1 . This construction continues indefinitely, with every tower at a given level i having two towers of the next level i + 1 on both side at distance d i . So there is one tower at level 1, two towers at level 2, 4 towers at level 3 etc. See Figure 1 to get a better qualitative understanding of the function.
To give the exact definition let us use the following dyadic notation. For any dyadic interval I ⊂ 0, 1 2 of length 2 −i , there is a corresponding intervalÎ ⊂ Q 0 of length l i . In [4] the indexing is started from [0, 1), but here we start it from 0, 1 2 for technical reasons. It doesn't change the fact that g ∈ JN p \ L p . The locations ofÎ in Q 0 are defined recursively. For I 1 := 0, 1 2 , letÎ 1 be the interval of length l 1 located in the center of Q 0 . IfÎ is already defined and I ′ is the left (right) half of I, letÎ ′ be the interval of length l i+1 positioned on the left (right) side ofÎ in such a way that dist(Î ′ ,Î) = d i .
We denote the intervals byÎ = [t I , t I + l i ]. For every intervalÎ we define the function for every x ∈Î. The function u I is identically zero elsewhere. The function g I is defined the same way but with different parameters. Note that u I and g I are supported inÎ, not in I. Finally we define the function u (g) as the sum of all these functions u I (g I ): u(x) := Figure 1. First three generations (i = 1, 2, 3) in the construction of u (with p = 2). δ 1 (resp. D 1 ) is the distance from 0, 1 2 to the nearest (resp. farthest)Î after infinitely many iterations of the construction. Notice that a 1 = 0 by definition.
We can recover the function g from [4] by choosing a i = b i . Thus g is constant in every intervalÎ. In this case we denote the height by h i . For g the heights, widths and distances are defined as for every i ≥ 1. It was already shown that g ∈ JN p . Then it is easy to see that also g p/q ∈ JN q for any q > 1, because g p does not depend on p. Therefore it is impossible for that function to separate JN p spaces with different parameters p. The function g had to be modified to get the function u which does prove the result (2.8).
For the function u the widths, distances and heights are defined as for every i ≥ 1. We will use these definitions from now on. Note that the definitions for l i and d i are the same as the respective definitions for g. The key difference is that a i and b i are different. It was shown in [4], that the intervalsÎ are disjoint. Indeed given an interval of length l i , its distance to any other intervalÎ ′ is at least This means that the functions u I have disjoint supports. Similarly we have a bound for D i , the largest distance from an intervalÎ at level i to any of its descendants: For the proof we refer to [4,Lemma 3.3]. From this it follows that the support of u is contained within an interval of length l 1 + 2D 1 ≤ 1 4 + 6 · 1 16 ≤ 1. This means that the function is indeed supported in [0, 1]. It is also useful to notice that b i = (1 + i −1/p )2 i 2 /p ≤ 2 · 2 i 2 /p for every i ≥ 1. We summarize the results so far into the following lemma.
Now that we have defined the function u = f 1/p , we show that u / ∈ JN p (Q 0 ), which together with Proposition 2.7 implies that Proof. Because u is linear in the intervalsÎ, it has a constant slope withinÎ, which we denote by k I . Then it is easy to see thatÎ By plugging in the values of a i and b i we get This finally implies that since the harmonic series diverges.
From now on we fix a number q ∈ (p, ∞). Our goal is to prove that u p/q = f 1/q ∈ JN q (Q 0 ). To simplify the notation we denote u p/q by v. First we need to prove the following lemmas.
In particular v ∈ L 1 (Q 0 ). Here the constants c depend only on p and q. Additionally The fact that v ∈ L 1 follows from the previous result by having The following lemma is easy to prove by finding the local minima of the function , or by using the Taylor approximation (1 + x) r ≈ 1 + rx, when |x| is small.
3.6. Lemma. Let 1 < p < q < ∞ and i ∈ Z + . Then It is clear that if J does not intersect any of the intervalsÎ, then F (J) = 0. Also if J does intersect some of the intervals, then there is a unique widest interval I J , such that J intersectsÎ J . We consider four separate cases how J relates to I J . Three of these cases are the same as in [4]. We added the fourth "contained" one, because this time the function is not constant within the intervalsÎ.
3.8. Lemma. Assume that J intersects some of the intervalsÎ and I = I J . Then where c = c(q) and |I| = 2 −i .
Proof. It is clear that and so we get from the triangle inequality On the other hand if α ≥ inf{v(x) : x ∈ J ∩Î}, then we know that a p/q i Here we used Lemma 3.6 to get the final inequality. Then we get Hence in all cases the lemma is true.
3.9. Proposition. If J is contained, then Proof. In this case the first two terms in Lemma 3.8 are just zero, and so we get the bound for F (J). Then it follows that This is an over-harmonic series, so it converges.
In Proposition 3.9 above we had a convergence specifically because q > p. If we had p = q, the corresponding series would diverge as was seen in Proposition 3.4.
We move on to the other cases where |J \Î J | > 0. In these cases J must intersect the boundary of the intervalÎ J and so for each I there are at most two intervals J such that I = I J and |J \Î| > 0.
3.10. Proposition. If J is short, then Proof. In this case v is zero in J \Î and so the first term in Lemma 3.8 is just zero. Then we get Using this bound we get 3.11. Proposition. If J is medium, then Proof. In this case the bounds for the second and third term in Lemma 3.8 are essentially the same as in Proposition 3.10 because |J \Î| ≤ 2D I ≤ 6d I . For the first term we use Lemma 3.5 to get Then we use the fact that |J| ≥ |J \Î| > δ I to get and so we have the bound for F (J). This implies that To get the bound for F (J) we use Lemma 3.8. First we notice that the bound for the first term is essentially the same as in Proposition 3.11, because |J| ≥ |J \Î| > 2D I ≥ 2d I . For the second and third term we use the fact that |J ∩Î| ≤ |Î| = l I . Then we get Let us say that a dyadic interval I is long, if it is of the form I J for some long J. Then we conclude that Now we are ready to prove the following proposition, which also concludes the proof of Theorem 3.1.
Proof. This follows from Propositions 3.9, 3.10, 3.11 and 3.12. If we take any partition J of Q 0 into disjoint subintervals J ∈ J , then As the estimate does not depend on J , we can conclude that v q JNq(Q0) < ∞ and therefore v ∈ JN q (Q 0 ).
The function f can be extended from the one-dimensional case Q 0 ⊂ R into the multidimensional case Q 0 ⊂ R n with an arbitrary n, by repeatedly using Proposition 2.5.
3.14. Corollary. Let Q 0 ⊂ R n be a bounded cube and let 1 < p < ∞. Then there exists a nonnegative function f such that f 1/q / ∈ JN q (Q 0 ) whenever 1 < q ≤ p, and f 1/q ∈ JN q (Q 0 ) for every q > p.
where the supremum is taken over all collections of pairwise disjoint subcubes Q i ⊂ Q 0 such that the side length of each Q i is at most a.
The following theorem is a characterization of V JN p (Q 0 ).
where C ∞ (Q 0 ) is the set of smooth functions in R n that have been restricted to Q 0 .
For the proof we refer to [16,Theorem 5.3]. To be precise, that theorem applies to the so-called John-Nirenberg-Campanato spaces JN (p,q,s)α (X) that were studied by Tao et al. [15]. Here p ∈ (1, ∞), q ∈ [1, ∞), α ∈ [0, ∞), s is a nonnegative integer and X is either R n or a bounded cube Q 0 ⊂ R n . Brudnyi and Brudnyi studied the vanishing subspace of this John-Nirenberg-Campanato space in [3, Theorem 2.6], though they used some different notation. For consistency of this article, we denote the space by V JN (p,q,s)α (X) .
By choosing q = 1, s = α = 0 and X = Q 0 , the John-Nirenberg-Campanato space JN (p,q,s)α (X) becomes the John-Nirenberg space, JN It follows directly from Theorem 4.2 that L p (Q 0 ) ⊂ V JN p (Q 0 ), by density of C ∞ in L p and by Equation (2.

3). Thus it is immediate that
The obvious question is now if these inclusions are strict. We answer this question by examining two functions. The counterexample presented in [4] is in JN p \V JN p . By modifying the function, we get another function which is in V JN p \ L p .

4.3.
Proposition. Let 1 < p < ∞ and Q 0 ⊂ R a bounded interval. Then there is a function g : Proof. Let g be the same function as in Section 3. It is already known that g ∈ JN p (Q 0 ). To show that g / ∈ V JN p (Q 0 ), let a > 0 be arbitrarily small. Then there is a positive integer m such that 2l m ≤ a. For any integer k > m we notice that d k−1 ≥ l k ≥ D k . This means that for each intervalÎ of length l k we can choose an interval J of length 2l k such that it coversÎ and all of its descendants on one side and the interval J doesn't intersect any other intervalÎ ′ . Then we get On the other hand we know that at least when k is large enough. Thus for k > m large enough, we have Since there are 2 k−1 many intervalsÎ of length l k , this means that and this is true for all a > 0, so especially the limit as a → 0 is positive. Thus by definition g / ∈ V JN p (Q 0 ).

4.4.
Proposition. Let 1 < p < ∞ and Q 0 ⊂ R a bounded interval. Then there is a function g 0 : Proof. Let us construct g 0 by modifying the function g from Section 3. For g 0 we set the height of every tower as h i = 2 i 2 i 1/p . The other parameters l i and d i are the same as in (3.2). Recall that the heights for g were defined as h i = 2 i 2 /p . The proof of g 0 ∈ JN p (Q 0 ) follows the same steps as the proof of g ∈ JN p (Q 0 ) in [4]. Also we notice that thus g 0 / ∈ L p (Q 0 ). To prove that g 0 ∈ V JN p , let a > 0 be a small enough number. Then there is some positive integer k such that δ k ≥ a > δ k+1 . Let J be a countable collection of disjoint intervals such that |J| ≤ a for every J ∈ J . Define Similar to the function u in Section 3, it is clear that if J does not intersect any of the intervalsÎ, then F (J) = 0. If J does intersect some intervals, then there is a unique widest interval I J , such that J intersectsÎ J . Also if J does intersect an intervalÎ, if it does not intersect the boundary of the interval, then F (J) = 0. Therefore we assume that J intersects the boundary ofÎ J . Then there are at most 2 intervals J for every I such that I = I J . Also we can categorize the intervals J as short, medium or long, depending on the length of J \Î J , the same way as in Definition 3.7 and in [4].
We have the following bound for F (J): where c = c(p) and I = I J . The proof is the same as for [4, Lemma 3.6]. If I = I J and |I| = 2 −i with i ≤ k, then the fact that |J \Î| ≤ |J| ≤ a ≤ δ k ≤ δ i implies that J is short. Hence in this case F (J) ≤ c min |J \Î|, |J ∩Î| h p I ≤ cd k h p k . The total number of dyadic intervals I that are not shorter than 2 −k is 1 + 2 + 4 + ...
Since k → ∞ when a → 0, we notice that the limit of this sum is 0. Assume from now on that I = I J and i > k. If J is short, then which tends to zero when a → 0.
If J is medium, then for the second term in (4.5) we have essentially the same bound as in the short case. For the first term we have the bound For the proof we refer to [4, Lemma 3.8], because g 0 ≤ g pointwise. This implies that This also tends to zero when a → 0.
Finally if J is long, we have the same Carleson property as in Proposition 3.12 and in [4, Lemma 3.9]: The bound for the first term in (4.5) is the same as in the medium case: For the second term we have the bound

This implies that
This also tends to zero. In conclusion and this holds for any admissible collection J . This implies that and so by definition g 0 ∈ V JN p (Q 0 ).
It is worth noting that the choice of h i in the previous proof does not need to be exactly what it is. It is sufficient that where s i is a sequence of nonnegative real numbers such that Now we would like to extend this result to the multidimensional case. The following proposition is a counterpart of Proposition 2.5.
Proof. The proof is very similar to the proof of [4,Proposition 4.1]. The only difference is that in the partition of Q 0 into subcubes we have the additional criterion that l(Q i ) ≤ a for every i and for some a > 0. Ultimately we get the result by taking the limit a → 0.
By using Propositions 2.5 and 4.6 multiple times, we can extend both g and g 0 into the multidimensional cube Q 0 ⊂ R n for any n, such that g ∈ JN p (Q 0 ) \ V JN p (Q 0 ) and g 0 ∈ V JN p (Q 0 ) \ L p (Q 0 ). In conclusion we have shown that for any bounded cube Q 0 ⊂ R n

5.
The spaces JN p , V JN p and CJN p defined on R n The definition of the space JN p can be extended from the bounded cube Q 0 into the whole Euclidian space R n . In this section we deal with the entire space R n and always assume that JN p = JN p (R n ) and V JN p = V JN p (R n ) etc. unless otherwise specified.
We denote the smallest such number K by f JNp .
The embedding JN p (Q 0 ) ⊂ L p,∞ (Q 0 ) in Theorem 2.4 has been proven in a number of ways. However all the proofs, that we know of, only consider JN p on bounded sets, not on R n . Clearly any constant function f = c is in the space JN p even though it is in L p,∞ only if c = 0. So let us define for p > 1 the space Then for the sake of completeness we prove the following theorem.

5.2.
Theorem. Let 1 < p < ∞ and f ∈ JN p . Then f ∈ L p,∞ /R and there is a Proof. First let Q ⊂ R n be any bounded cube. It is clear that if we restrict f to this cube, then we have f ∈ JN p (Q) and therefore from Theorem 2.4 we get that . Further from Definition 5.1 it immediately follows that Now let us define a sequence of cubes (Q m ) ∞ m=1 such that |Q m | = 2 m and the center of every cube is the origin. Then clearly we have Q 1 ⊂ Q 2 ⊂ ... and ∞ m=1 Q m = R n . Let us prove that the sequence of integral averages f Qm is a Cauchy sequence. First we notice that for the difference of two consecutive elements we have the following estimate: Now let ǫ > 0 and N = N (ǫ) an integer that will depend on ǫ. Let m, k ≥ N and let us assume that k ≥ m. Then when N is large enough. Thus f Qm is a Cauchy sequence and we can set b := lim m→∞ f Qm . Then we have for any positive integer m for all t > 0. The second term is either equal to |Q m | or 0, but when we take the limit m → ∞ it tends to 0. Therefore we have This completes the proof.
Notice also that the zero-extension of the function we had in Section 2 in L p,∞ (Q 0 ) \ JN p (Q 0 ) is in L p,∞ /R \ JN p . This shows that JN p L p,∞ /R. It is easy to see, by using a similar method as in the previous proof, that JN 1 = L 1 /R. Therefore we assume that p > 1 from now on.
The vanishing subspace V JN p can be defined in the whole space as well.
5.3. Definition (V JN p ). Let 1 < p < ∞. Then the vanishing subspace V JN p of JN p is defined by setting The sup where the supremum is taken over all collections of pairwise disjoint cubes Q i such that the side length of each Q i is at most a.
We also study CJN p , which is another subspace of JN p . This space is defined analogously to the space of continuous mean oscillation CM O, which is a subspace of BM O. In the case of the bounded cube, CJN p (Q 0 ) = V JN p (Q 0 ). Hence we only define CJN p in the whole space. 5.5. Definition (CJN p ). Let 1 < p < ∞. Then the subspace CJN p of JN p is defined by setting where C ∞ c (R n ) denotes the set of smooth functions with compact support in R n . The following characterization of CJN p is proven in [ where the supremum is taken over all cubes Q ⊂ R n such that the side length of Q is at least a.
It follows directly from the definition of CJN p that L p ⊂ CJN p , by density of C ∞ c in L p and by Equation (2.3). Furthermore we notice from Theorems 5.4 and 5.6 and from Definition 5.1 that adding a constant to a function doesn't affect whether the function is in JN p or in the subspaces V JN p or CJN p . Thus if f ∈ L p , then f + c ∈ CJN p for any constant c. If we define L p /R the same way as L p,∞ /R, then it is immediate that Again we can ask whether these inclusions are strict. We examine this by extending the previous functions into multidimensional versions in the whole space. 5.7. Lemma. Let I ⊂ R be a bounded interval such that |I| = L. Without loss of Let us look at Q ∩ Q 0 = J 1 × ... × J n . Clearly we have |J i | ≤ l(Q) for every i. Further we know that |J j | < l(Q) for at least one j -otherwise we would have |Q ∩ Q 0 | = |Q|.
If |J 1 | < l(Q), then the cube Q intersects with the boundary of I. This implies that K, the n − 1-dimensional side of Q ∩ Q 0 , is located at the boundary of Q 0 . Therefore we can estimate Here the first inequality is an equality if Q intersects only one of the boundary points of I. If it intersects the other boundary point, we have |{x ∈ Q ∩ Q 0 : If |J 1 | = l(Q), then |J i | < l(Q) for some i ≥ 2. By the same reasoning as before we know that Q intersects the boundary of the i:th dimensional interval [0, L] and therefore Thus in any case we have |K| ≤ |Q ∩ ∂Q 0 |. Now let Q be a partition of R n into disjoint cubes and define The moral is that if we have any function in JN p (I) for a bounded interval I, we can extend the function into multiple dimensions and into the whole space so that we get a function in JN p (R n ). It is easy to see thatf ∈ L p (R n )/R if and only if f ∈ L p (I) andf ∈ L p,∞ (R n )/R if and only if f ∈ L p,∞ (I). The same results hold also if n = 1. where the supremum is taken over all intervals J such that |J| ≤ a, thenf ∈ CJN p .
Proof. Iff ∈ V JN p and we restrictf to Q 0 , we clearly havef ∈ V JN p (Q 0 ). Then Proposition 4.6 implies that f ∈ V JN p (I).
Now assume that f ∈ V JN p (I) and assumption (5.9) holds. Without loss of generality we may assume that f I = 0. First we notice that then f ∈ JN p (I), which means thatf ∈ JN p , by Lemma 5.7. Now let a > 0. Let Q ⊂ R n be a cube such that Q ∩ Q 0 = ∅, Q \ Q 0 = ∅ and l(Q) ≤ a. Similar to the previous proof, we can write Q ∩ Q 0 = J 1 × J 2 × ... × J n = J 1 × K. Then As in the previous proof, we also have |K| ≤ |Q ∩ ∂Q 0 |. Now let Q be a partition of R n into disjoint cubes such that the side length of each cube is at most a. Define   where the supremum is taken over all intervals J such that |J| ≤ a.
Proof. It is enough that we prove this result for the function g from Proposition 4.3, because clearly g 0 ≤ g. Let a be a small positive number such that δ N < a ≤ δ N −1 for some integer N ≥ 2. Let J ⊂ Q 0 be an interval such that |J| ≤ a. Then there is some integer m ≥ N such that δ m < |J| ≤ δ m−1 . Like in earlier cases, if J does not intersect any intervalÎ, then In conclusion we know that L p /R CJN p ⊆ V JN p JN p L p,∞ /R.
So far an example function that would be in V JN p \ CJN p has not been constructed. It remains an open question whether these two spaces coincide or not. It is also not clear whether the condition (5.9) is necessary for Lemma 5.8. It would be interesting to find a JN p function that doesn't satisfy (5.9).