The John–Nirenberg Space: Equality of the Vanishing Subspaces $VJN_p$ and $CJN_p$

Abstract

The John–Nirenberg spaces $JN_p$ are generalizations of the space of bounded mean oscillation BMO with $JN_\infty =BMO$. Their vanishing subspaces $VJN_p$ and $CJN_p$ are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that $VJN_p$ and $CJN_p$ coincide by showing that certain Morrey type integrals of $JN_p$ functions tend to zero for small and large cubes. We also show that $JN_{p,q}(\mathbb {R}^n) =L^p(\mathbb {R}^n) / \mathbb {R}$, if $p = q$.

1 Introduction

In 1961, John and Nirenberg studied the well-known space of bounded mean oscillation BMO and proved the profound John–Nirenberg inequality for BMO functions [10]. The space BMO plays a vital role in harmonic analysis and it has been studied very extensively. For example, a celebrated result of Fefferman and Stein states that BMO can be characterized as the dual space of the real Hardy space $H^1$ [7]. In [10], John and Nirenberg also defined a generalization of BMO, which is now known as the John–Nirenberg space, or $JN_p$, with a parameter $1< p < \infty$, see Definition 2.3 below. In addition, they proved the John–Nirenberg inequality for $JN_p$ functions, see Theorem 2.4 below. From this theorem, it follows that $JN_p(Q_0) \subset L^{p,\infty }(Q_0)$, where $Q_0 \subset \mathbb {R}^n$ is a bounded cube. It is also easy to see that $L^p(Q_0) \subset JN_p(Q_0)$. Both of these inclusions are strict; however, this is far from trivial. An example of a function in $JN_p \setminus L^p$ was discovered in 2018 [4]. Thus, the space $JN_p$ is a nontrivial space between $L^p$ and $L^{p,\infty }$. However, there are still many unanswered questions related to the study of John–Nirenberg spaces.

Various John–Nirenberg type spaces have attracted attention in recent years, including the dyadic John–Nirenberg space [11], the congruent John–Nirenberg space [9, 22], the John–Nirenberg–Campanato space [17, 19], and the sparse John–Nirenberg space [5]. The John–Nirenberg space can also be defined with medians instead of using integral averages [13]. Hurri-Syrjänen et al. established a local-to-global result for the space $JN_p(\Omega )$, where $\Omega \subset \mathbb {R}^n$ is an open set [8] and Marola and Saari found similar results for $JN_p$ in the setting of metric measure spaces [12].

The spaces VMO and CMO are well-known vanishing subspaces of BMO. They were introduced by Sarason [16] and Neri [14], respectively. The aforementioned duality phenomenon of Fefferman and Stein was later complemented by Coifman and Weiss who showed that $H^1$ is the dual space of CMO [3]. Recently, there has been research on the $JN_p$ counterparts of these spaces, which are denoted by $VJN_p$ and $CJN_p$ [20]. Brudnyi and Brudnyi showed that $VJN_p$ is a predual to the Hardy-type space $HK_{p'}$ among other duality results for a family of related spaces $V_{pq}^{k \lambda }$ on a bounded cube [2]. The space $HK_{p'}$ was first introduced by Dafni et al. as a predual to the space $JN_p$ [4]. Here, $p'$ is the conjugate index of p, i.e., $1/p + 1/p' = 1$. It follows directly from the definitions of $VJN_p$ and $CJN_p$ that $L^p \subseteq CJN_p \subseteq VJN_p \subseteq JN_p$. Moreover, examples in [18] demonstrate that $L^p \ne CJN_p$ and $VJN_p \ne JN_p$. However, it has been an open question whether the set $VJN_p\setminus CJN_p$ is nonempty, see [18, 20]. As our main result, we show that $VJN_p$ and $CJN_p$ coincide.

Our method is to study Morrey or weak $L^p$ type integrals

$$\begin{aligned} |Q|^{\frac{1}{p}-1} \int _Q |f|, \end{aligned}$$

(1.1)

where Q is a cube. We prove that if $f \in JN_p$, then these integrals tend to zero both when $|Q| \rightarrow 0$ and when $|Q| \rightarrow \infty$. See Theorems 3.5 and 3.8 below for precise statements of these results. Note that $L^p$ functions have this property, but weak $L^p$ functions do not. From Theorem 3.5, it follows easily that $CJN_p = VJN_p$, see Corollary 3.7.

In Sect. 2 we briefly study the more general version of the John–Nirenberg type spaces $JN_{p,q}(X)$, where the $L^1$-norm of the oscillation term is replaced with the $L^q$-norm where $q \ge 1$. This generalization has been studied in [4, 21, 22]. It has turned out that in case X is a bounded cube, the $JN_{p,q}$ norm is equivalent with the $JN_p$ norm (for $q<p$) or $L^q$ norm (for $q \ge p$). In case $X = \mathbb {R}^n$, the $JN_{p,q}$ norm is equivalent with the $JN_p$ norm (for $q<p$), and if $q>p$, the space contains only functions that are constant almost everywhere. We complete this picture by showing that in the borderline case $p = q$ and $X = \mathbb {R}^n$, this space is equivalent with the space $L^p(\mathbb {R}^n) / \mathbb {R}$, i.e., the space of functions f for which there is a constant b such that $f - b \in L^p(\mathbb {R}^n)$. The result answers a question raised in [22, Remark 2.9].

2 Preliminaries

Throughout this paper by a cube, we mean an open cube with edges parallel to the coordinate axes. We let $X \subseteq \mathbb {R}^n$ be either a bounded cube or the entire space $\mathbb {R}^n$. If Q is a cube, we denote by l(Q) its side length. For any $r > 0$, we denote by rQ the cube with the same center as Q but with side length $r \cdot l(Q)$. For any measurable set $E \subset \mathbb {R}^n$, such that $0< |E| < \infty$, we denote the integral average of a function f over E by

Definition 2.1

(Weak $L^p$-spaces) Let $1 \le p < \infty$. For a measurable function f, we define

$$\begin{aligned} \Vert f \Vert _{L^{p,\infty }(X)}:= \sup _{t> 0} t | \{ x \in X: |f(x)| > t \} |^{1/p}. \end{aligned}$$

We say that f is a weak $L^p$ function, or $f \in L^{p,\infty }(X)$, if $\Vert f \Vert _{L^{p,\infty }(X)}$ is finite. We define

where E is any measurable set. We say that $f \in L^{p,w}(X)$, if $\Vert f \Vert _{L^{p,w}(X)}$ is finite.

Remark 2.2

The expression $\Vert \cdot \Vert _{L^{p,\infty }(X)}$ is not a norm, since the triangle inequality fails to hold. However, $\Vert \cdot \Vert _{L^{p,w}(X)}$ does define a norm. Additionally, if $p > 1$, $\Vert f \Vert _{L^{p,\infty }(X)}$ and $\Vert f \Vert _{L^{p,w}(X)}$ are comparable and therefore $L^{p,w}(X) = L^{p,\infty }(X)$, see Chap. 2.8.3 in [6].

Definition 2.3

($JN_p$) Let $1< p < \infty$. A function f is in $JN_p(X)$ if $f \in L_{loc}^1(X)$ and there is a constant $K < \infty$ such that

for all countable collections of pairwise disjoint cubes $(Q_i)_{i=1}^{\infty }$ in X. We denote the smallest such number K by $\Vert f\Vert _{JN_p(X)}$.

The space $JN_p$ is related to BMO in the sense that the BMO norm of a function is the limit of the function’s $JN_p$ norm when p tends to infinity. It is easy to see that

$$\begin{aligned} \Vert |f| \Vert _{JN_p(X)} \le 2 \Vert f \Vert _{JN_p(X)}. \end{aligned}$$

(2.1)

Likewise, it is clear that $L^p(X) \subset JN_p(X)$, as we get from Hölder’s inequality that $\Vert f \Vert _{JN_p(X)} \le 2 \Vert f \Vert _{L^p(X)}$. If X is a bounded cube, then $JN_p(X) \subset L^{p, \infty } (X)$. This is known as the John–Nirenberg inequality for $JN_p$ functions.

Theorem 2.4

(John–Nirenberg inequality for $JN_p$) Let $1< p < \infty$, $Q_0 \subset \mathbb {R}^n$ a bounded cube and $f \in JN_p(Q_0)$. Then $f \in L^{p,\infty }(Q_0)$ and

$$\begin{aligned} \Vert f - f_{Q_0} \Vert _{L^{p,\infty }(Q_0)} \le c \Vert f \Vert _{JN_p(Q_0)} \end{aligned}$$

with some constant $c = c(n,p)$.

The proof can be found in [1, 10], for example. In [18], this result was extended to the space $JN_p(\mathbb {R}^n)$. Also a more general John–Nirenberg space $JN_{p,q}$ has been studied, for example, in [4, 21, 22].

Definition 2.5

($JN_{p,q}$) Let $1 \le p < \infty$ and $1 \le q < \infty$. A function f is in $JN_{p,q}(X)$ if $f \in L_{loc}^1(X)$, and there is a constant $K < \infty$ such that

for all countable collections of pairwise disjoint cubes $(Q_i)_{i=1}^{\infty }$ in X. We denote the smallest such number K by $\Vert f\Vert _{JN_{p,q}(X)}$.

It was shown in [4, Proposition 5.1] that if X is a bounded cube, then $JN_{p,q}(X) = JN_p(X)$, if $1 \le q < p$ and $JN_{p,q}(X) = L^q(X)$ if $p \le q < \infty$. The same proof also shows us that $JN_{p,q}(\mathbb {R}^n) = JN_p(\mathbb {R}^n)$, if $1 \le q < p$. It was shown in [22, Corollary 2.8] that the space $JN_{p,q}(\mathbb {R}^n)$ contains only functions that are constant almost everywhere, if $p< q < \infty$. However, it was stated in [22, Remark 2.9] that the situation is unclear if $q = p$. We complete the picture by showing that $JN_{p,p}(\mathbb R^n)$ is equal to $L^p(\mathbb R^n)$ up to a constant. This also answers [21, Question 15].

Proposition 2.6

Let $1 \le p < \infty$. Then $JN_{p,p}(\mathbb {R}^n) = L^p(\mathbb {R}^n) / \mathbb {R}$ and there is a constant $c = c(p)$ such that for any function $f \in L_{loc}^1(\mathbb {R}^n)$, we have

$$\begin{aligned} \frac{1}{c} \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)} \le \inf _{b \in \mathbb {R}} \Vert f-b \Vert _{L^p(\mathbb {R}^n)} \le c \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)}. \end{aligned}$$

Proof

First assume that $f \in L^p / \mathbb {R}$, that is there is a constant b such that $f - b \in L^p$. Then for any set of pairwise disjoint cubes $Q_i$,

$$\begin{aligned} \sum _{i=1}^{\infty } \int _{Q_i} | f - f_{Q_i} |^p= & {} \sum _{i=1}^{\infty } \int _{Q_i} | (f-b) - (f-b)_{Q_i} |^p \nonumber \\\le & {} \sum _{i=1}^{\infty } 2^{p} \int _{Q_i} |f-b|^p \le 2^p \int _{\mathbb {R}^n} |f-b|^p \end{aligned}$$

and therefore $f \in JN_{p,p}(\mathbb {R}^n)$.

Now assume that $f \in JN_{p,p}(\mathbb {R}^n)$. Clearly

$$\begin{aligned} \int _Q |f - f_Q|^p \le \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)}^p \end{aligned}$$

for every cube $Q \subset \mathbb {R}^n$. Let $(Q_k)_{k=1}^{\infty }$ be a sequence of cubes such that the center of every cube is the origin and $|Q_k| = 2^k$. Then $Q_1 \subset Q_2 \subset ...$ and $\cup _{k=1}^{\infty } Q_k = \mathbb {R}^n$. We shall prove that the sequence of integral averages $(f_{Q_k})_{k=1}^{\infty }$ is a Cauchy sequence. For any integer i, we have

This means that

Then for any positive integers m and k, we get

$$\begin{aligned} \begin{aligned} |f_{Q_m} - f_{Q_k}|&\le \sum _{i = \min (m,k)}^{\max (m,k) - 1} |f_{Q_{i+1}} - f_{Q_i}| \le \sum _{i = \min (m,k)}^{\infty } 2^{1 - \frac{1}{p}} \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)} 2^{-i/p} \\&= c \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)} 2^{-\min (m,k)/p}, \end{aligned} \end{aligned}$$

(2.2)

where the constant c depends only on p. Therefore, $(f_{Q_k})_{k=1}^{\infty }$ is a Cauchy sequence. Then by using (2.2), we get

$$\begin{aligned} \int _{\mathbb {R}^n} \left| f - \lim _{k \rightarrow \infty } f_{Q_k} \right| ^p&= \lim _{m \rightarrow \infty } \int _{Q_m} \left| f - \lim _{k \rightarrow \infty } f_{Q_k} \right| ^p \\&\le \lim _{m \rightarrow \infty } \int _{Q_m} 2^{p-1} \left( |f - f_{Q_m}|^p + \left| f_{Q_m} - \lim _{k \rightarrow \infty } f_{Q_k} \right| ^p \right) \\&\le \lim _{m \rightarrow \infty } 2^{p-1} \left( \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)}^p + 2^m \lim _{k \rightarrow \infty } \left| f_{Q_m} - f_{Q_k} \right| ^p \right) \\&\le 2^{p-1} \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)}^p \left( 1 + \lim _{m \rightarrow \infty } 2^m \lim _{k \rightarrow \infty } c 2^{-\min (m,k)} \right) \\&= 2^{p-1} \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)}^p \left( 1 + \lim _{m \rightarrow \infty } 2^m c 2^{-m} \right) = c \Vert f \Vert _{JN_{p,p}(\mathbb {R}^n)}^p, \end{aligned}$$

where the constant c depends only on p. This means that $f - \lim _{k \rightarrow \infty } f_{Q_k} \in L^p$ and therefore $f \in L^p / \mathbb {R}$. This completes the proof. $\square$

The spaces $VJN_p$ and $CJN_p$ were studied in [18, 20]. These spaces are $JN_p$ counterparts of the spaces VMO and CMO, which are subspaces of BMO. The spaces $VJN_p$ and $CJN_p$ can also be defined in a bounded cube $Q_0$ instead of $\mathbb {R}^n$ as in the following definitions. However, in that case, it is clear that the spaces coincide [18, 20]. In this paper, we only define the spaces in $\mathbb {R}^n$ and we write $VJN_p = VJN_p(\mathbb {R}^n)$ and $CJN_p = CJN_p(\mathbb {R}^n)$ to simplify the notation.

Definition 2.7

($VJN_p$) Let $1< p < \infty$. Then the vanishing subspace $VJN_p$ of $JN_p$ is defined by setting

$$\begin{aligned} VJN_p:= \overline{D_p(\mathbb {R}^n) \cap JN_p}^{JN_p}, \end{aligned}$$

where

$$\begin{aligned} D_p(\mathbb {R}^n):= \{ f \in C^{\infty }(\mathbb {R}^n): |\nabla f| \in L^p(\mathbb {R}^n) \}. \end{aligned}$$

Definition 2.8

($CJN_p$) Let $1< p < \infty$. Then the subspace $CJN_p$ of $JN_p$ is defined by setting

$$\begin{aligned} CJN_p:= \overline{C_c^{\infty }(\mathbb {R}^n) }^{JN_p}, \end{aligned}$$

where $C_c^{\infty }(\mathbb {R}^n)$ denotes the set of smooth functions with compact support in $\mathbb {R}^n$.

As in the case of vanishing subspaces of BMO, there exist characterizations of $VJN_p$ and $CJN_p$ as $JN_p$ functions for which certain integrals vanish, see [20, Theorems 3.2 and 4.3].

Theorem 2.9

Let $1< p < \infty$. Then $f \in VJN_p$ if and only if $f \in JN_p$ and

where the supremum is taken over all collections of pairwise disjoint cubes $(Q_i)_{i=1}^{\infty }$ in $\mathbb {R}^n$, such that the side length of each $Q_i$ is at most a.

Theorem 2.10

Let $1< p < \infty$. Then $f \in CJN_p$ if and only if $f \in VJN_p$ and

where the supremum is taken over all cubes $Q \subset \mathbb {R}^n$ such that the side length of Q is at least a.

From the definitions, we can see that $L^p / \mathbb {R} \subseteq CJN_p \subseteq VJN_p \subseteq JN_p$. It was shown in [18] that $L^p / \mathbb {R} \ne CJN_p$ and $VJN_p \ne JN_p$. However, the question of whether $CJN_p$ and $VJN_p$ coincide remained open.

3 Equality of $VJN_p$ and $CJN_p$

Let $1< p < \infty$. In this section, we prove the equality of $VJN_p$ and $CJN_p$ by showing that for any $JN_p$ function f, integrals of the type

tend to zero both when $|Q| \rightarrow 0$ and when $|Q| \rightarrow \infty$. This type of integral appears in the Morrey norm, see for example [15]. Compare it also to the weak $L^p$ norm (Definition 2.1), where the supremum is taken over such integrals with the cube Q replaced with an arbitrary measurable set.

The aforementioned results follow from Proposition 3.1.

Proposition 3.1

Let $X \subseteq \mathbb {R}^n$ be either a bounded cube or the entire space $\mathbb {R}^n$. Let $Q \subset X$ be a cube such that $3 Q \subseteq X$. Let $1< p < \infty$, $0< A < \infty$ and $0 < \epsilon \le \epsilon _0(n,p)$. Suppose that $f \in L_{loc}^1(X)$ is a nonnegative function such that

(3.1)

and for any cube $Q' \subset X$ with $l(Q') = \frac{2}{3} l(Q)$ or $l(Q') = \frac{4}{3} l(Q)$ we have

(3.2)

Then there exist two cubes $Q_1 \subset 3 Q$ and $Q_2 \subset 3Q$ such that for $i \in \{ 1, 2\}$, we have

(3.3)

(3.4)

(3.5)

where $c = c(n,p)$ is a positive constant.

To prove this proposition, we first need to prove Lemmas 3.2 and 3.4.

Lemma 3.2

Let $0< \alpha< 1 < \beta$. Let $Q' \subset Q \subset {\tilde{Q}} \subset \mathbb {R}^n$ be cubes such that $l(Q') = \alpha l(Q)$ and $l({\tilde{Q}}) = \beta l(Q)$. Suppose that $f \in L^1({\tilde{Q}})$ is a nonnegative function, $1< p < \infty$, $0< A < \infty$ and $0 < \epsilon \le \epsilon _0(n, p, \alpha , \beta )$. Assume also that (3.1) holds for Q and (3.2) holds for $Q'$ and ${\tilde{Q}}$. Then we have

where $c_1 = c_1(n, p, \alpha , \beta )$ is a positive constant.

Proof

From the assumptions of the lemma, we get directly

$$\begin{aligned} \int _{{\tilde{Q}} \setminus Q'} \left| f - f_{{\tilde{Q}} \setminus Q'} \right|&\ge \left| \int _{{\tilde{Q}} \setminus Q} f - \frac{|{\tilde{Q}} \setminus Q|}{|{\tilde{Q}} \setminus Q'|} \int _{{\tilde{Q}} \setminus Q'} f \right| + \left| \int _{Q \setminus Q'} f - \frac{|Q \setminus Q'|}{|{\tilde{Q}} \setminus Q'|} \int _{{\tilde{Q}} \setminus Q'} f \right| \\&= 2 \left| \int _Q f - \frac{1 - \alpha ^n}{\beta ^n - \alpha ^n} \int _{{\tilde{Q}}} f - \frac{\beta ^n - 1}{\beta ^n - \alpha ^n} \int _{Q'} f \right| \\&\ge 2 \bigg ( A (1 - \epsilon ) |Q|^{1 - \frac{1}{p}} - \frac{1 - \alpha ^n}{\beta ^n - \alpha ^n} A (1 + \epsilon ) |{\tilde{Q}}|^{1 - \frac{1}{p}} \\&\quad - \frac{\beta ^n - 1}{\beta ^n - \alpha ^n} A (1 + \epsilon ) |Q'|^{1 - \frac{1}{p}} \bigg ) \\&= 2 A |Q|^{1 - \frac{1}{p}} \bigg ( 1 - \epsilon - \frac{1 - \alpha ^n}{\beta ^n - \alpha ^n} (1 + \epsilon ) \left( \beta ^n \right) ^{1 - \frac{1}{p}} \\&\quad - \frac{\beta ^n - 1}{\beta ^n - \alpha ^n} (1 + \epsilon ) \left( \alpha ^n \right) ^{1 - \frac{1}{p}} \bigg ). \end{aligned}$$

Thus, we have

Notice that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} C(n , p , \alpha , \beta , \epsilon )&= C(n , p , \alpha , \beta , 0) \\&= 2 \left( \beta ^n - \alpha ^n \right) ^{\frac{1}{p} - 1} \left( 1 - \frac{1 - \alpha ^n}{\beta ^n - \alpha ^n} \beta ^{n - \frac{n}{p}} - \frac{\beta ^n - 1}{\beta ^n - \alpha ^n} \alpha ^{n - \frac{n}{p}} \right) . \end{aligned}$$

This is positive for every n, p, $\alpha$, and $\beta$. Indeed, we have $C(n, p, \alpha , \beta , 0 ) = 2 (\beta ^n - \alpha ^n)^{\frac{1}{p} - 1} h(\frac{1}{p})$ with

$$\begin{aligned} h(x) = 1 - \frac{1 - \alpha ^n}{\beta ^n - \alpha ^n} \beta ^{n - n x} - \frac{\beta ^n - 1}{\beta ^n - \alpha ^n} \alpha ^{n - n x}. \end{aligned}$$

We notice that $h(0) = h(1) = 0$ and the second derivative of h is strictly negative. Thus h is concave and $h(x) > 0$ for every $0< x < 1$. In conclusion, if $\epsilon$ is small enough, we have

$$\begin{aligned} C(n, p, \alpha , \beta , \epsilon ) \ge \frac{1}{2} C(n, p, \alpha , \beta , 0):= c_1 (n, p, \alpha , \beta ) > 0. \end{aligned}$$

This completes the proof. $\square$

For the reader’s convenience, we start by giving a proof of Proposition 3.1 in the special case $n=1$ as it is technically much simpler. The idea of the proof is the same also in the multidimensional case.

Proof of Proposition 3.1 in the case $n = 1$

Let us assume that $Q = [a, a + L]$. Define ${\tilde{Q}} = \left[ a, a + \frac{4}{3} L \right]$ and $Q' = \left[ a, a + \frac{2}{3} L \right]$. We set $Q_1:= {\tilde{Q}} {\setminus } Q' = \left[ a + \frac{2}{3} L, a + \frac{4}{3} L \right]$ and we get from Lemma 3.2 and the assumptions in Proposition 3.1 that

if $\epsilon$ is small enough. Here $c_1 = c_1(n,p,\alpha ,\beta )$ with $n=1$, $\alpha = \frac{2}{3}$ and $\beta = \frac{4}{3}$.

On the other hand, if we set ${\tilde{Q}} = \left[ a - \frac{1}{3} L, a + L \right]$ and $Q' = \big [ a + \frac{1}{3} L, a + L \big ]$ and define $Q_2:= {\tilde{Q}} {\setminus } Q' = \left[ a - \frac{1}{3} L, a + \frac{1}{3} L \right]$, then we get from Lemma 3.2 that

if $\epsilon$ is small enough. Finally we notice that the distance between the cubes $Q_1$ and $Q_2$ is $\frac{1}{3} L$. This completes the proof. $\square$

The case $n \ge 2$ is more complicated as the set ${\tilde{Q}} {\setminus } Q'$ is usually not a cube. Before the actual proof, we fix some notation about directions and projections.

Definition 3.3

Let $\{v_1,\ldots ,v_n\}$ denote the standard orthonormal basis for $\mathbb R^n$. Let $Q_1 \subset \mathbb {R}^n$ and $Q_2 \subset \mathbb {R}^n$ be cubes. The cubes can be presented as Cartesian products of intervals as

$$\begin{aligned} Q_1&= I_1^1 \times I_2^1 \times ... \times I_n^1 \text { and } \\ Q_2&= I_1^2 \times I_2^2 \times ... \times I_n^2. \end{aligned}$$

We say that $P_k(Q_1):= I_k^1$ is the projection of cube $Q_1$ to the subspace spanned by the base vector $v_k$. Fix an index k with $1\le k\le n$. If for every $x_k \in P_k(Q_1)$ and $y_k \in P_k(Q_2)$ we have $x_k \le y_k$, then we say that $Q_2$ is located in direction $\uparrow ^k$ from $Q_1$ and $Q_1$ is located in direction $\downarrow ^k$ from $Q_2$.

Lemma 3.4

Let $Q \subset \mathbb {R}^n$ be a cube. Let $Q' \subset Q \subset {\tilde{Q}}$ be cubes with $l(Q') = \frac{2}{3} l(Q)$ and $l({\tilde{Q}}) = \frac{4}{3} l(Q)$ such that all the cubes share a corner. By symmetry, we may assume that $Q = \left[ 0, L \right] ^n$, $Q' = \left[ 0, \frac{2}{3} L \right] ^n$ and ${\tilde{Q}} = \left[ 0, \frac{4}{3} L \right] ^n$. Let $f \in L^1 \left( [0, 2\,L]^n \right)$ be a nonnegative function, $1< p < \infty$, $0< A < \infty$ and $0 < \epsilon \le \epsilon _0$ from Lemma 3.2 with $\alpha = \frac{2}{3}$ and $\beta = \frac{4}{3}$. Suppose also that (3.1) holds for Q and (3.2) holds for $Q'$ and ${\tilde{Q}}$. Then there exists a cube ${\bar{Q}} \subset [0, 2L]^n \setminus Q'$ such that either

• (a) $l({\bar{Q}}) = \frac{2}{3} L$ and ${\bar{Q}} \subset {\tilde{Q}}$

  or

• (b) $l({\bar{Q}}) = \frac{4}{3} L$ and $P_k({\bar{Q}}) = P_k({\tilde{Q}}) = [0, \frac{4}{3} L]$ for every $1 \le k \le n$ except one

and in addition

where $c_2 = c_2(n,p)$ is a positive constant.

Proof

Let us divide the cube ${\tilde{Q}}$ dyadically into $2^n$ subcubes. Then one of them is $Q'$. Let us define $Q_0 = \left[ \frac{2}{3} L, \frac{4}{3} L \right] ^n$ and let us denote the rest of the subcubes by $(Q_i)_{i=1}^{2^n - 2}$. Notice that $Q'$ does not have an index unlike all the other dyadic subcubes. For any $1 \le j \le n$, we define

$$\begin{aligned} Q_j':= I_1 \times I_2 \times ... \times I_n, \end{aligned}$$

where

$$\begin{aligned} I_k= {\left\{ \begin{array}{ll} {[}\frac{2}{3} L, 2L],&{} k=j,\\ {[}0, \frac{4}{3} L], &{} k\ne j.\\ \end{array}\right. } \end{aligned}$$

See Fig. 1 to see how these cubes are located with respect to each other. It is simple to check that then

$$\begin{aligned} Q_0 \subset Q_j' \subset [0, 2L]^n \setminus Q' \quad \text { and }\quad l(Q_j') = 2 l(Q_0) \end{aligned}$$

for every j. Also for every $Q_i$ with $1\le i\le 2^n-2$, there exists at least one cube $Q_j'$ such that $Q_i \subset Q_j'$. For every $Q_i$, let us denote by $Q_{j_i}'$ one of these cubes $Q_j'$.

Fig. 1

The cubes Q, $Q'$, ${\tilde{Q}}$, $(Q_i)_{i=0}^{2^n-2}$ and $(Q_j')_{j=1}^n$, when $n = 2$. We have ${\tilde{Q}} = Q' \cup Q_0 \cup Q_1 \cup Q_2$, $Q_0 \cup Q_1 \subset Q_1'$ and $Q_0 \cup Q_2 \subset Q_2'$

Let us prove that for at least one of the cubes $Q_i$ or $Q_j'$, we have

(3.6)

where $c_1 = c_1(n, p, \alpha , \beta )$ is the constant from Lemma 3.2 with $\alpha = \frac{2}{3}$ and $\beta = \frac{4}{3}$. We prove this by contradiction. Assume that (3.6) does not hold for any $Q_i$ or $Q_j'$. We get from Lemma 3.2 that

(3.7)

We continue estimating one of the integrals in the sum above

$$\begin{aligned}&\int _{Q_i} \Bigg | f - \frac{1}{(2^n - 1)|Q'|} \sum _{k=0}^{2^n-2} \int _{Q_k} f \Bigg | \\&\quad = \int _{Q_i} \Bigg | f - f_{Q_i} + \frac{1}{(2^n - 1)|Q'|} \sum _{k=0}^{2^n - 2} \left( \int _{Q_i} f - \int _{Q_k} f \right) \Bigg | \\&\quad \le \int _{Q_i} |f - f_{Q_i}| + \frac{1}{2^n - 1} \sum _{k=0}^{2^n - 2} \left| \int _{Q_i} f - \int _{Q_k} f \right| . \end{aligned}$$

Assume that $k \ge 1$. Then because $Q_k \cup Q_0 \subset Q_{j_k}'$ and $Q_k \cap Q_0 = \varnothing$, we have

$$\begin{aligned} \left| \int _{Q_k} f - \int _{Q_0} f \right| \le \left| \int _{Q_k} \left( f - f_{Q_{j_k}'} \right) \right| + \left| \int _{Q_0} \left( f_{Q_{j_k}'} - f \right) \right| \le \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| . \end{aligned}$$

Thus if $i = 0$, we get

$$\begin{aligned} \sum _{k=0}^{2^n - 2} \left| \int _{Q_i} f - \int _{Q_k} f \right| \le \sum _{k=1}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| . \end{aligned}$$

On the other hand, if $i \ge 1$, then

$$\begin{aligned} \sum _{\begin{array}{c} k=0 \\ k \ne i \end{array}}^{2^n - 2} \left| \int _{Q_i} f - \int _{Q_k} f \right|&\le (2^n - 2) \left| \int _{Q_i} f - \int _{Q_0} f \right| + \sum _{\begin{array}{c} k=0 \\ k \ne i \end{array}}^{2^n - 2} \left| \int _{Q_0} f - \int _{Q_k} f \right| \\&\le (2^n - 2) \int _{Q_{j_i}'} \left| f - f_{Q_{j_i}'} \right| + \sum _{\begin{array}{c} k=1 \\ k \ne i \end{array}}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| . \end{aligned}$$

We continue by estimating the sum in (3.7) and we get

$$\begin{aligned}&\sum _{i=0}^{2^n - 2} \int _{Q_i} \left| f - \frac{1}{(2^n - 1)|Q'|} \sum _{k=0}^{2^n-2} \int _{Q_k} f \right| \\&\quad \le \sum _{i=0}^{2^n - 2} \left( \int _{Q_i} |f - f_{Q_i}| + \frac{1}{2^n - 1} \sum _{k=0}^{2^n - 2} \left| \int _{Q_i} f - \int _{Q_k} f \right| \right) \\&\quad \le \sum _{i=0}^{2^n - 2} \int _{Q_i} |f - f_{Q_i}| + \frac{1}{2^n - 1} \sum _{k=1}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| \\&\qquad + \sum _{i=1}^{2^n - 2} \left( \frac{1}{2^n - 1} \left( (2^n - 2) \int _{Q_{j_i}'} \left| f - f_{Q_{j_i}'} \right| + \sum _{\begin{array}{c} k=1 \\ k \ne i \end{array}}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| \right) \right) \\&\quad = \sum _{i=0}^{2^n - 2} \int _{Q_i} |f - f_{Q_i}| + \sum _{k=1}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| + \frac{1}{2^n - 1} \sum _{i=1}^{2^n - 2} \sum _{\begin{array}{c} k=1 \\ k \ne i \end{array}}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| \\&\quad = \sum _{i=0}^{2^n - 2} \int _{Q_i} |f - f_{Q_i}| + \left( 1 + \frac{2^n - 3}{2^n - 1} \right) \sum _{k=1}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| . \end{aligned}$$

Then finally from our assumption, that (3.6) does not hold, we get

$$\begin{aligned}&\sum _{i=0}^{2^n - 2} \int _{Q_i} |f - f_{Q_i}| + \left( 1 + \frac{2^n - 3}{2^n - 1} \right) \sum _{k=1}^{2^n - 2} \int _{Q_{j_k}'} \left| f - f_{Q_{j_k}'} \right| \\&\quad < \sum _{i=0}^{2^n - 2} \left( 2^n - 1 \right) ^{-\frac{1}{p}} \left( 1 + 2^{1 + n - \frac{n}{p}} \left( \frac{2^n - 2}{2^n - 1} \right) ^2 \right) ^{-1} c_1 A |Q_i|^{1 - \frac{1}{p}} \\&\qquad + 2 \cdot \frac{2^n - 2}{2^n - 1} \sum _{k=1}^{2^n - 2} \left( 2^n - 1 \right) ^{-\frac{1}{p}} \left( 1 + 2^{1 + n - \frac{n}{p}} \left( \frac{2^n - 2}{2^n - 1} \right) ^2 \right) ^{-1} c_1 A |Q_{j_k}'|^{1 - \frac{1}{p}} \\&\quad = \left( 2^n - 1 \right) ^{1-\frac{1}{p}} \left( 1 + 2^{1 + n - \frac{n}{p}} \bigl ( \tfrac{2^n - 2}{2^n - 1} \bigr )^2 \right) ^{-1} c_1 A \biggl ( |Q'|^{1 - \frac{1}{p}} + \tfrac{2(2^n - 2)^2}{(2^n - 1)^2} \left( 2^n |Q'| \right) ^{1 - \frac{1}{p}} \biggr ) \\&\quad = (2^n - 1)^{1 - \frac{1}{p}} c_1 A |Q'|^{1 - \frac{1}{p}}. \end{aligned}$$

In conclusion, we have

which is a contradiction. Hence, there is at least one cube ${\bar{Q}} \subset [0, 2\,L]^n {\setminus } Q'$ that satisfies the conditions of the lemma and

where $c_2 = c_2(n,p)$ is a positive constant. This completes the proof. $\square$

Now we are ready to prove Proposition 3.1.

Proof of Proposition 3.1 in the case $n \ge 2$

Let $0 < \epsilon \le \epsilon _0$ from Lemma 3.2 with $\alpha = \frac{2}{3}$ and $\beta = \frac{4}{3}$. We apply Lemma 3.4$2^n$ times to each corner of Q and obtain $2^n$ sets of cubes $Q'$, ${\tilde{Q}}$ and ${\bar{Q}}$. Here, every ${\bar{Q}}$ satisfies (3.3) and (3.5). No matter which corner the cubes $Q'$, Q and ${\tilde{Q}}$ share, we always have $\frac{1}{3} Q \subset Q'$. Since ${\bar{Q}}$ and $Q'$ are disjoint, we get that ${\bar{Q}}$ and $\frac{1}{3} Q$ are also disjoint and thus each ${\bar{Q}}$ is located in at least one direction from $\frac{1}{3} Q$ in the sense of Definition 3.3.

Because $Q'$ is in the corner of ${\tilde{Q}}$, there is one direction for each $k \in \{1, 2,..., n\}$ such that ${\bar{Q}}$ cannot be located in that direction from $\frac{1}{3} Q$. For example, if $Q = [0,L]^n$, $Q' = [0, \frac{2}{3} L]^n$ and ${\tilde{Q}} = [0, \frac{4}{3} L]^n$, then the possible directions where ${\bar{Q}}$ may be located in from $\frac{1}{3} Q$ are $\uparrow ^1$, $\uparrow ^2$,... and $\uparrow ^n$. The cube ${\bar{Q}}$ cannot be located in any of the directions $\downarrow ^1$, $\downarrow ^2$,... and $\downarrow ^n$ from $\frac{1}{3} Q$. Thus, for each cube ${\bar{Q}}$, there are n possible directions and for any two cubes ${\bar{Q}}$, the sets of possible directions do not coincide.

If one cube ${\bar{Q}}$ is located in direction $\uparrow ^k$ from $\frac{1}{3} Q$ and another is located in direction $\downarrow ^k$, then the distance between those two cubes is at least $\frac{1}{3} l(Q)$ – thus the proposition is true. Therefore let us assume by contradiction that no two cubes ${\bar{Q}}$ are located in opposite directions.

Let S be the set of all directions in which all the cubes ${\bar{Q}}$ are located from $\frac{1}{3} Q$. Then we clearly have $|S| \le n$, because by our assumption, there is at most one direction in S for each $k \in \{1, 2,..., n\}$. However, there is always at least one cube ${\bar{Q}}$ for which the possible directions are all exactly opposite to the directions in S. If for example the directions in S are $\uparrow ^1$, $\uparrow ^2$,... and $\uparrow ^m$ for some $m \le n$, then there is no direction in S for the cube ${\bar{Q}}$ for which the possible directions are only $\downarrow ^1$, $\downarrow ^2$,... and $\downarrow ^n$. This contradicts with the assumption that the directions of all ${\bar{Q}}$ are represented in S. Thus, we conclude that there must be two cubes ${\bar{Q}}$ in opposite directions. This completes the proof. $\square$

Now we can show that the Morrey type integral (1.1) vanishes as the measure of the cube tends to infinity.

Theorem 3.5

Let $1< p < \infty$ and suppose that $f \in JN_p(\mathbb {R}^n)$. Then there is a constant b such that

(3.8)

where the supremum is taken over all cubes $Q \subset \mathbb {R}^n$ such that the side length of Q is at least a.

Proof

Assume first that $f \in JN_p(\mathbb {R}^n) \cap L^{p,\infty }(\mathbb {R}^n)$ and f is nonnegative. Let

where $A \ge 0$. The limit exists as this sequence is decreasing, the elements are finite – this follows from Remark 2.2 because $f \in L^{p,\infty }(\mathbb {R}^n)$ – and the sequence is bounded from below by 0.

Let us assume that $A > 0$. Let $0 < \epsilon \le \epsilon _0(n, p)$ as in Proposition 3.1. Then there exists a number $N < \infty$ such that (3.2) holds for every cube Q where $l(Q) \ge N$. Also for any number $M < \infty$, there exists a cube Q such that $l(Q) \ge M$ and (3.1) holds. This means that we can find a sequence of cubes $(Q_i)_{i=1}^{\infty }$ such that $l(Q_1) \ge \frac{3}{2} N$, $l(Q_{i+1}) > l(Q_i)$, $\lim _{i \rightarrow \infty } l(Q_i) = \infty$ and

for every $i \in \mathbb {N}$.

Let $Q_i$ be one of these cubes. According to Proposition 3.1, there exist two cubes $Q_{i,1} \subset 3 Q_i$ and $Q_{i,2} \subset 3 Q_i$ that satisfy (3.3), (3.4) and (3.5). The cubes $(Q_{i,1})_{i=1}^{\infty }$ may not be pairwise disjoint. However, for every cube $Q_i$, we have two cubes to choose from.

Let us construct a new sequence of cubes $(Q_{{i_j}}')_{j=1}^{\infty }$ iteratively. We start with $Q_{i_1}:= Q_1$ and choose $Q_{i_1}':= Q_{1,1}$. Let $Q_{i_2}$, $i_2 > 1$, be the smallest cube in the sequence $(Q_i)_{i=1}^{\infty }$ such that $\frac{1}{3} l(Q_{i_2}) \ge l(Q_{i_1}')$. Then at least one of the cubes $Q_{i_2,1}$ and $Q_{i_2,2}$ is pairwise disjoint with $Q_{i_1}'$. Let’s say that $Q_{i_2,1}$ is the disjoint one and set $Q_{i_2}':= Q_{i_2,1}$.

Let us denote by Q the smallest cube such that $Q_{i_1}' \cup Q_{i_2}' \subset Q$. Let $Q_{i_3}$, $i_3 > i_2$, be the smallest cube in the sequence $(Q_i)_{i=1}^{\infty }$ such that $\frac{1}{3} l(Q_{i_3}) \ge l(Q)$. Then at least one of the cubes $Q_{i_3, 1}$ and $Q_{i_3, 2}$ is pairwise disjoint with both $Q_{i_1}'$ and $Q_{i_2}'$.

By repeating this process and taking a subsequence of $(Q_i)_{i=1}^{\infty }$, if necessary, we get infinitely many pairwise disjoint cubes $(Q_{i_j}')_{j=1}^{\infty }$. Then we get

This contradicts with the assumption that $f \in JN_p(\mathbb {R}^n)$. Thus, we conclude that $A = 0$.

Now assume only that $f \in JN_p(\mathbb {R}^n)$. From [18, Theorem 5.2], we get that there is a constant b such that $f - b \in L^{p,\infty }(\mathbb {R}^n)$. Obviously $f - b \in JN_p(\mathbb {R}^n)$. Then from (2.1), we get that $|f-b| \in JN_p(\mathbb {R}^n) \cap L^{p,\infty }(\mathbb {R}^n)$. Finally by the same reasoning as before, we conclude that

This completes the proof. $\square$

Remark 3.6

Naturally Theorem 3.5 implies that the result holds also for $L^p$ functions with $p>1$, but this can also be seen easily by considering the Hardy-Littlewood maximal function. Indeed assume by contradiction that

Then for any number $a < \infty$, there exists a cube Q such that $l(Q) \ge a$ and

Thus for every $x \in Q$, we have $M f(x) \ge \frac{A}{2} |Q|^{- \frac{1}{p}}$, where Mf is the non-centered Hardy-Littlewood maximal function. Thus $Mf(x) \notin L^p$ and consequently $f \notin L^p$.

Clearly the theorem does not hold for weak $L^p$ functions. For example let $n = 1$ and $f(x) = x^{-1/p}$, when $x > 0$. Then we have $f \in L^{p,\infty }(\mathbb {R})$ but (3.8) does not hold for any b. The same function shows us that Theorem 3.8 does not hold for weak $L^p$ functions.

As a corollary from Theorem 3.5, we get that $VJN_p$ and $CJN_p$ coincide. This answers a question that was posed in [18] and it answers [20, Question 5.6] and [21, Question 17].

Corollary 3.7

Let $1< p< \infty$. Then $CJN_p(\mathbb {R}^n) = VJN_p(\mathbb {R}^n)$.

Proof

It is clear that $CJN_p \subseteq VJN_p$. Let $f \in VJN_p \subset JN_p$. Then we get from Theorem 3.5 that there is a constant b such that

Then by Theorem 2.10, we get that $f \in CJN_p$. This completes the proof. $\square$

From the following result, we can infer that the additional condition in [18, Lemma 5.8] is not necessary, answering the question posed in [18].

Theorem 3.8

Let $X \subseteq \mathbb {R}^n$ be either a bounded cube or the entire space $\mathbb {R}^n$ and $1< p < \infty$. Suppose that $f \in JN_p(X)$. Then

(3.9)

where the supremum is taken over all cubes $Q \subseteq X$, such that the side length of Q is at most a.

Remark 3.9

The theorem implies that $L^p$ functions also satisfy (3.9). However, in that case, the result is well known as it follows simply from Hölder’s inequality and the dominated convergence theorem. Notice that for $L^p$ functions, (3.9) holds also when $p = 1$, whereas in Remark 3.6, it is necessary that $p > 1$.

Remark 3.10

In particular, this result implies that for any $1< p < \infty$ and bounded cube $X \subset \mathbb {R}^n$, the John–Nirenberg space $JN_p(X)$ is a subspace of the vanishing Morrey space $VL^{1,n-\frac{n}{p}}(X)$ as defined in [15].

Proof of Theorem 3.8

The proof is similar to the proof of Theorem 3.5, but we have to take into account the possibility that we might have $3 Q \not \subseteq X$, even though $Q \subseteq X$.

Let $f \in JN_p(X)$. Let us assume first that f is nonnegative and $f \in L^{p,\infty }(X)$. Let

(3.10)

The limit exists as the sequence is decreasing and bounded from below by 0.

Assume that $A > 0$. Let $0 < \epsilon \le \epsilon _0(n,p)$ as in Proposition 3.1. Then there is a number $\delta > 0$ such that (3.2) holds for any cube Q where $l(Q) \le \delta$. Also because of (3.10), we know that for every $a > 0$, there exists a cube $Q \subseteq X$ such that $l(Q) \le a$ and (3.1) holds for Q. Therefore, we can find a sequence of cubes $(Q_i)_{i=1}^{\infty }$ such that $l(Q_1) \le \frac{3}{4} \delta$, $l(Q_{i+1}) < l(Q_i)$, $\lim _{i \rightarrow \infty } l(Q_i) = 0$ and

for every $i \ge 1$.

Case 1: There are infinitely many cubes $Q_i$ in the sequence such that $3 Q_i \subseteq X$.

By taking a subsequence, we may assume that $3 Q_i \subseteq X$ for every i. Let $Q_i$ be one of these cubes. Then according to Proposition 3.1, there exist two cubes $Q_{i,1} \subset 3 Q_i$ and $Q_{i,2} \subset 3 Q_i$ that satisfy (3.3), (3.4) and (3.5). The cubes $(Q_{i,1})_{i=1}^{\infty }$ may not be pairwise disjoint. However, for every cube $Q_i$, we have two cubes to choose from.

Let us construct a new sequence of cubes $(Q_{i_j}')_{j=1}^{\infty }$ iteratively. We start with $Q_{i_1}:= Q_1$. Then there exist infinitely many cubes $Q_i$ in the sequence such that $Q_{1,1} \cap 3 Q_i = \varnothing$ for every i or $Q_{1,2} \cap 3 Q_i = \varnothing$ for every i. This is true because ${\text {dist}}(Q_{1,1}, Q_{1,2}) \ge l(Q_1) / 3$ and $\lim _{i \rightarrow \infty } l(Q_i) = 0$. Without loss of generality, we may assume that $Q_{1,1} \cap 3 Q_i = \varnothing$ for infinitely many $i > 1$. We set $Q_{i_1}':= Q_{1,1}$. Let $Q_{i_2}$, $i_2 > 1$, be the first cube in the sequence $(Q_i)_{i=1}^{\infty }$ such that $Q_{i_1}' \cap 3 Q_{i_2} = \varnothing$. Then the cubes $Q_{i_1}'$, $Q_{i_2, 1}$ and $Q_{i_2, 2}$ are all pairwise disjoint. Thus, we can continue by choosing one of the cubes $Q_{i_2, 1}$ and $Q_{i_2, 2}$ such that there are still infinitely many cubes $3 Q_i$, $i > i_2$, that are pairwise disjoint with both the chosen cube and $Q_{i_1}'$.

By repeating this process and taking a subsequence of $(Q_i)_{i=1}^{\infty }$, if necessary, we get infinitely many pairwise disjoint cubes $(Q_{i_j}')_{j=1}^{\infty }$ in X that all satisfy (3.5).

Case 2: There are only finitely many cubes $Q_i$ in the sequence such that $3 Q_i \subseteq X$. This can only happen if X is a bounded cube. In this case, we shall also construct a new sequence of pairwise disjoint cubes $(Q_{i_j}')_{j=1}^{\infty }$ iteratively. By taking a subsequence of $(Q_i)_{i=1}^{\infty }$, we may assume that $l(Q_1) \le \frac{1}{4} l(X)$ and $l(Q_{i+1}) \le \frac{2}{9} l(Q_i)$. Assume that $3 Q_i \not \subset X$. Then we know that for any $k \in \{1, 2,..., n\}$, the projection $P_k(3 Q_i)$ can only intersect one of the endpoints of $P_k(X)$. Let m be the number of base vectors $v_k$ such that $P_k(3 Q_i) \subset P_k(X)$. Then $0 \le m < n$ and there exist $2^m$ cubes ${\bar{Q}}_i \subset X \cap 3 Q_i {\setminus } \frac{1}{3} Q_i$ as they are defined in Lemma 3.4.

Every such cube ${\bar{Q}}_i$ is located in some direction from the cube $\frac{1}{3} Q_i$, in a similar way as in the proof of Proposition 3.1. Without loss of generality, we may assume that the possible directions for these cubes are $\uparrow ^1$, $\downarrow ^1$, $\uparrow ^2$, $\downarrow ^2$,..., $\uparrow ^m$, $\downarrow ^m$, $\downarrow ^{m+1}$, $\downarrow ^{m+2}$,..., $\downarrow ^n$, but for each ${\bar{Q}}_i$ there is naturally only one possible direction for each $k \in \{1, 2,..., n\}$. If one of the cubes is located in direction $\uparrow ^k$ and another is in direction $\downarrow ^k$, $k \le m$, then the distance between these two cubes is at least $\frac{1}{3} l(Q_i)$. Thus, we can choose one of these cubes into the sequence $(Q_{i_j}')_{j=1}^{\infty }$ the same way as in the first case and there are still infinitely many cubes $Q_l$, $l > i$, such that $3 Q_l \cap {\bar{Q}}_i = \varnothing$.

Let us assume that there are no two cubes ${\bar{Q}}_i$ that are located in opposite directions from $\frac{1}{3} Q_i$. Then by similar reasoning as in the proof of Proposition 3.1, we know that at least one of the cubes ${\bar{Q}}_i$ must be located in direction $\downarrow ^k$ for some $k > m$. Let us denote such a cube by $Q_{i,1}$. If there are infinitely many cubes $Q_l$, $l > i$, such that $3 Q_l \cap Q_{i,1} = \varnothing$, then we may choose $Q_{i,1}$ into the sequence $(Q_{i_j}')_{j=1}^{\infty }$.

Assume that there are only finitely many cubes $Q_l$, $l > i$, such that $3 Q_l \cap Q_{i,1} = \varnothing$. Then there are infinitely many cubes $Q_l$, $l > i$, such that $3 Q_l \cap Q_{i,1} \ne \varnothing$. Let $Q_l$ be one of these cubes. Because $l(Q_l) \le \frac{2}{9} l(Q_i)$, we know that $P_k(3 Q_l) \subset P_k(X)$. This is because the distance from $P_k(Q_{i,1})$ to $\partial P_k(X)$ is at least $\frac{2}{3} l(Q_i)$.

In addition for every $1 \le s \le m$, we have $P_s(3 Q_l) \subset P_s(X)$. This is because of how the cube ${\bar{Q}}$ was chosen in Lemma 3.4. Because the cube $Q_{i,1}$ is located in direction $\downarrow ^k$ from the cube $\frac{1}{3} Q_i$, we have either $Q_{i,1} \subset {\tilde{Q}}_i$ (where ${\tilde{Q}}_i$ is as in Lemma 3.4) or the projection of ${\tilde{Q}}_i$ and the projection of $Q_{i,1}$ coincide for every base vector except $v_k$. Thus $P_s(Q_{i,1}) \subset P_s({\tilde{Q}}_i)$. Because $P_s(3 Q_i) \subset P_s(X)$ for every $1 \le s \le m$, we get that the distance from $P_s({\tilde{Q}}_i)$ to $\partial P_s(X)$ is at least $\frac{2}{3} l(Q_i)$. See Fig. 2 to get a better understanding of the situation.

Fig. 2

An example of how the cubes X, $Q_i$, ${\tilde{Q}}_i$, $Q_{i,1}$ and $Q_l$ may be situated with respect to each other. In the picture, we have the projections of these cubes to the mk-plane. The cube $Q_{i,1}$ is located in direction $\downarrow ^k$ from $\frac{1}{3} Q_i$. Therefore, we have either $Q_{i,1}$ as shown in the picture or $Q_{i,1} \subset {\tilde{Q}}_i$

In conclusion, if $3 Q_i \not \subset X$ and there are m base vectors $v_s$ such that $P_s(3 Q_i) \subset P_s(X)$, then we can find a cube $Q_{i,1} \subset X \cap 3 Q_i$ such that (3.5) holds and there are infinitely many cubes $Q_l$, $l > i$, such that $Q_{i,1} \cap 3 Q_l = \varnothing$, or, if the first option is not possible, there exist infinitely many cubes $Q_l$, $l > i$, such that $P_s(3 Q_l) \subset P_s(X)$ for at least $m+1$ base vectors $v_s$. Because there are only n base vectors in total, this latter option can only happen at most n times. Thus we can always find infinitely many pairwise disjoint cubes $(Q_{i_j}')_{j=1}^{\infty }$ in X that all satisfy (3.5).

In both cases, we get

This contradicts with the fact that $f \in JN_p(X)$. Therefore, we conclude that $A = 0$.

Now assume only that $f \in JN_p(X)$. Then there is a constant b such that $f - b \in L^{p,\infty }(X)$. Also from (2.1), we get that $|f - b| \in JN_p(X) \cap L^{p,\infty }(X)$. Then using the same argument as earlier, we get

This completes the proof. $\square$