On the John–Nirenberg inequality

Abstract

We present a version of the John–Nirenberg inequality for a sub-class of BMO by estimating the corresponding mean oscillating distribution function via dyadic decomposition. The dominating functions are of the form of decreasing step functions which are finer than the classical exponential functions and might be much more efficient for some sophisticated analysis. We also prove that the modified BMO-norm is equivalent to the classical BMO-norm under the convexity assumption.

1 Introduction

The space of functions of bounded mean oscillation(BMO) first appeared in the work of John and Nirenberg [1] in the context of nonlinear partial differential equations that arise in the study of minimal surfaces. The space BMO can be characterized by the form of the John–Nirenberg inequality: it says that for $f \in \mathrm{BMO}$, one has

$$\begin{aligned} \frac{1}{ \vert Q \vert } \bigl\vert \bigl\{ x \in Q: \bigl\vert f(x) - f_{Q} \bigr\vert > t \bigr\} \bigr\vert \leq e \cdot \exp \biggl( - \frac{1}{2^{d} e} \frac{t}{ \Vert f \Vert _{\mathrm{BMO}}} \biggr) \end{aligned}$$

(1.1)

for every cube $Q \subset \mathbb{R}^{d}$ and every $t>0$. Here $f_{Q}:= \frac{1}{|Q|} \int _{Q} f(x)\,dx$. One of the important applications of the John–Nirenberg inequality (1.1) is the $L^{p}$ characterization of BMO-norm; that is, for $1< p < \infty$,

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO}_{p}}:= \biggl( \sup_{Q} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert f(x)- f_{Q} \bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}} \end{aligned}$$

(1.2)

is an equivalent to the classical BMO-norm.

This paper refines the John–Nirenberg inequality in two ways. We present a dominating function F that is finer than the exponential functions at the right-hand side of estimate (1.1) (Theorem 4). The function F has a form of decreasing step functions which possesses more merits than the exponential function (see an application of Theorem 4 at the end of Sect. 3). Secondly, we introduce an appropriate base function α such that the functional

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO}_{\alpha }}:= \alpha ^{-1} \biggl( \sup _{Q} \frac{1}{ \vert Q \vert } \int _{Q} \alpha \bigl( \bigl\vert f(x)- f_{Q} \bigr\vert \bigr)\,dx \biggr) \end{aligned}$$

(1.3)

sets the levels for those functions located in the space BMO and forms a norm of the space BMO, which is equivalent to the original norm under the convexity assumption.

The base functions α that we have developed include the base functions of the form $\alpha (x) = x^{p}$ that guarantee the classical result (1.2). We, in fact, designate the base functions α to achieve the Hölder inequality which always incubates the Minkowski type triangle inequality.

In this paper, $(X, \mathfrak{M}, \mu )$ represents an abstract measure space and $\bar{\mathbb{R}}_{+} = \{ x \in \mathbb{R}: x \geq 0 \}$.

2 Hölder functions

Notions of Hölder functions have been developed to find appropriate base functions that permit the Hölder inequality. In this section, we briefly introduce the fundamentals of the Hölder functions; the details can be found in [2–4]. The notions presented here are modified versions without essential differences.

• 1. A pre-Hölder function${\alpha }: \bar{\mathbb{R}}_{+} \rightarrow \bar{\mathbb{R}}_{+}$ is a strictly increasing absolutely continuous function. For example, the exponential function $\alpha (x) = e^{x}$ and $\alpha (x) = x^{p}$ ($0< p <\infty$) are in this category. If there exists a pre-Hölder function β satisfying

  $$ {\alpha }^{-1}(x) {{\beta }}^{-1}(x) = x $$

  (2.1)

  for all $x \in \bar{\mathbb{R}}_{+}$, then β is called the conjugate (pre-Hölder) function of α. In relation (2.1), the notations ${{\alpha }}^{-1}$, ${{\beta }}^{-1}$ are the inverse functions of α, β, respectively. Some examples of pre-Hölder pairs are: $({\alpha }(x), {\beta }(x)) = (x^{p}, x^{q})$ for $p > 1$ with $\frac{1}{p} + \frac{1}{q} = 1$, and

  $$ (\alpha, \beta ):= (\lambda \circ A, \lambda \circ \tilde{A}), $$

  (2.2)

  where we set ${\lambda }(x) = A^{-1}(x) \tilde{A}^{-1}(x)$ for any Orlicz N-function A together with its complementary N-function Ã.

• 2. The spectrum (or exponent function) $p_{\alpha }$ of a pre-Hölder function α is defined as

  $$ p_{\alpha }(x):= x \frac{ \alpha '(x)}{\alpha (x)} $$

  (2.3)

  for almost every $x > 0$. For example, the Lebesgue base function $\alpha (x)= x^{p}$ has a point spectrum: $p_{\alpha }(x) = p$ with $1 \leq p < \infty$.

  A pre-Hölder function α permits a conjugate function if and only if the bijection α satisfies the limit conditions

  $$ \lim_{x \to 0+} \frac{\alpha (x)}{x} =0,\qquad \lim_{x \to \infty } \frac{{\alpha }(x)}{x} = \infty $$

  (2.4)

  together with the spectrum condition

  $$ p_{\alpha }(x) >1 $$

  (2.5)

  for almost every $x>0$. Also, for a pre-Hölder pair $({\alpha }, {\beta })$, we have

  $$ \frac{1}{p_{\alpha }}(s) + \frac{1}{p_{\beta }}(t) = 1 \quad\text{when } \alpha (s) = \beta (t). $$

  For details, we refer to [4].

• 3. Let Φ be a two-variable function on $\bar{\mathbb{R}}_{+} \times \bar{\mathbb{R}}_{+}$ defined by

  $$ \varPhi (x, y):= \alpha ^{-1}(x) \beta ^{-1}(y). $$

  Then we observe that the equation of the tangent plane T of Φ at a point $({\alpha }(a), {\beta }(b))$ is represented by

  $$\begin{aligned} T(x, y) = \frac{1}{p_{\alpha }} \frac{ab }{{\alpha (a)}} x + \frac{1}{p_{\beta }} \frac{ab }{{ \beta (b)}} y + ab \theta _{f} \end{aligned}$$

  (2.6)

  with $\theta _{f} = 1 - \frac{1}{p_{\alpha }} - \frac{1}{p_{\beta }}$. This motivates us to define the Hölder functions as follows.

Definition 1

Let $\hbar > 0$ be given. A pre-Hölder function α with the conjugate function β is said to be a Hölder function if, for any positive constants $a, b >0$, there exist constants $\theta _{1}$, $\theta _{2}$, and $\theta _{f}$ (depending on a and b) such that

$$ \theta _{1} + \theta _{2} + \theta _{f} \leq \hbar $$

and that a dominating condition

$$\begin{aligned} \varPhi (x, y) = \alpha ^{-1}(x) \beta ^{-1}(y) \leq \theta _{1} \frac{ab}{\alpha (a)} x + \theta _{2} \frac{ab}{\beta (b)} y + ab \theta _{f} \end{aligned}$$

(2.7)

holds for all $(x, y) \in \bar{\mathbb{R}}_{+} \times \bar{\mathbb{R}}_{+}$. A Hölder function α is called an s-Hölder function if we can choose $\theta _{f} = 0$ in (2.7).

For example, any (convex) function satisfying

$$\begin{aligned} \alpha (x):= \textstyle\begin{cases} x^{p} & \text{for } 0 \leq x \leq 1, \\ x^{q} & \text{for sufficiently large } x \end{cases}\displaystyle \end{aligned}$$

(2.8)

($1 < p, q < \infty$) is a Hölder function, and so are many variants of (2.8).

• 4. Let α be a Hölder function and β be the corresponding Hölder conjugate function. For the case when $(X, \mathfrak{M}, \mu )$ is an infinite measure space, we define

  $$\begin{aligned} \Vert f \Vert _{\alpha }:= \alpha ^{-1} \biggl( \sup _{\mu (K)< \infty } \frac{1}{\mu (K)} \int _{K} \alpha \bigl( \bigl\vert f(x) \bigr\vert \bigr) \,d \mu \biggr), \end{aligned}$$

  (2.9)

  where the supremum is taken over all measurable subsets K (of X) of finite measure. Hereafter, ${ -\!\!\!\!\!\!\int _{K} f(x) \,d\mu:= \frac{1}{\mu (K)} \int _{K} f(x) \,d\mu }$. Even though we define $\| \cdot \|_{\alpha }$ on an abstract measure space $(X, \mathfrak{M}, \mu )$, we restrict our attention to the Euclidean space $\mathbb{R}^{d}$ equipped with Lebesgue measure, and the measurable sets K at (2.9) are restricted to cubes Q of $\mathbb{R}^{d}$ in the next section.

  For any measurable set K of finite measure, we have a Hölder type inequality

  $$\begin{aligned} \biggl\vert -\!\!\!\!\!\!\int _{K} f(x)g(x) \,d\mu \biggr\vert \leq \hbar \Vert f \Vert _{{\alpha }} \Vert g \Vert _{{\beta }}, \end{aligned}$$

  (2.10)

  if the right-hand side is finite. We name the Hölder functions after Hölder inequality (2.10). For this reason, we briefly sketch the idea of the proof that justifies Definition 1. Let $a:= \| f \|_{\alpha }\neq 0$ and $b:= \| g \|_{\beta } \neq 0$. Then there exist $\theta _{1}$, $\theta _{2}$, and $\theta _{f}$ such that $\theta _{1} + \theta _{2} + \theta _{f} \leq \hbar$ and

  $$\begin{aligned} \bigl\vert f(x) g(x) \bigr\vert &= {\alpha }^{-1}\bigl({\alpha } \bigl( \bigl\vert f(x) \bigr\vert \bigr)\bigr) {\beta }^{-1}\bigl({ \beta }\bigl( \bigl\vert g(x) \bigr\vert \bigr)\bigr) \\ &\leq \theta _{1} \frac{ab}{\alpha (a)} {\alpha }\bigl( \bigl\vert f(x) \bigr\vert \bigr) + \theta _{2} \frac{ab}{\beta (b)} {\beta } \bigl( \bigl\vert g(x) \bigr\vert \bigr) + ab \theta _{f}. \end{aligned}$$

  (2.11)

  Integrating over K and dividing both sides by $\mu (K)$ yield

  $$\begin{aligned} -\!\!\!\!\!\!\int _{K} \bigl\vert f(x)g(x) \bigr\vert \,d \mu \leq{}& \theta _{1} \frac{ab}{\alpha (a)} -\!\!\!\!\!\!\int _{K} {\alpha }\bigl( \bigl\vert f(x) \bigr\vert \bigr) \,d \mu + \theta _{2} \frac{ab}{\beta (b)} -\!\!\!\!\!\!\int _{K} { \beta }\bigl( \bigl\vert g(x) \bigr\vert \bigr) \,d \mu \\ & {}+ \theta _{f} ab -\!\!\!\!\!\!\int _{K} \,d\mu \\ \leq {}&\hbar \Vert f \Vert _{\alpha } \Vert g \Vert _{\beta }. \end{aligned}$$

  When $(X, \mathfrak{M}, \mu )$ is a finite measure space, we define

  $$\begin{aligned} \Vert f \Vert _{\alpha }:= \alpha ^{-1} \biggl( \int _{X} \alpha \bigl( \bigl\vert f(x) \bigr\vert \bigr) \,d \mu \biggr) \end{aligned}$$

  to get a version of Hölder type inequality

  $$\begin{aligned} \biggl\vert \int _{X} f(x)g(x) \,d\mu \biggr\vert \leq \hbar \Vert f \Vert _{{ \alpha }} \Vert g \Vert _{{\beta }}. \end{aligned}$$

  The Hölder inequality always incubates the Minkowski type inequality

  $$\begin{aligned} \Vert f_{1} + f_{2} \Vert _{{\alpha }} \leq \hbar \bigl\{ {{ \Vert f_{1} \Vert _{{ \alpha }}} + { \Vert f_{2} \Vert _{{\alpha }}}} \bigr\} . \end{aligned}$$

  (2.12)

  Also, for any constant $k \geq 0$ and $\| f \|_{\alpha } < \infty$, we have

  $$ \frac{k}{\hbar } \Vert f \Vert _{{\alpha }} \leq \Vert k f \Vert _{{\alpha }} \leq k \hbar \Vert f \Vert _{{\alpha }}. $$

  In particular, when $\hbar =1$, we have the homogeneity

  $$ \Vert k f \Vert _{{\alpha }} = k \Vert f \Vert _{{\alpha }}. $$

  Quasi-homogeneity of the norm is good enough to exploit estimates for the existence theory of nonlinear partial differential equations[2, 3] and to study singular integrals on these genealogical function spaces.

• 5. We close this section by introducing the boundedness of the spectra of the convex base functions.

Proposition 2

(Boundedness of the spectrum)

Let$(\alpha, \beta )$be a convex Hölder pair, and let$p_{\alpha }$be the spectrum ofα. Then there exist constants$c_{1}$, $c_{2}$satisfying

$$\begin{aligned} 1< c_{1} \leq p_{\alpha }(t) \leq c_{2} < \infty \end{aligned}$$

(2.13)

for almost every$t >0$. The same result holds for the spectrum$p_{\beta }$ofβ.

Proof

From the convexity of α, we get

$$ t \alpha '(t) \leq \int _{t}^{2 t} \alpha ' (s) \,ds \leq \int _{0}^{2 t} \alpha ' (s) \,ds =\alpha (2t) \quad\text{for } t>0. $$

Hence in order to prove the second inequality $p_{\alpha }(t) \leq c_{2}$, it is enough to demonstrate that $\alpha (2t) \leq c \alpha (t)$ for some constant $c > 1$.

Suppose that there exists a sequence $\{ t_{j} \}_{j=1}^{\infty }$ of positive numbers such that

$$\begin{aligned} \alpha (2 t_{j}) \geq 2^{j} \alpha (t_{j}),\quad j = 1, 2, \ldots. \end{aligned}$$

(2.14)

We choose a sequence of mutually disjoint measurable subsets $\{ X_{j} \}_{j=1}^{\infty }$ of a finite measurable subset X of $\mathbb{R}^{d}$ such that

$$\begin{aligned} \mu (X_{j}) = \frac{\alpha (1) \mu (X) }{2^{j} \alpha (t_{j}) }. \end{aligned}$$

Define $f(x) = \sum_{j=1}^{\infty } t_{j} \chi _{{X_{j}}}(x)$. Then we have

$$ \int _{X} \alpha \bigl( \bigl\vert f(x) \bigr\vert \bigr) \,d\mu = \sum_{j=1}^{\infty } \alpha (t_{j}) \mu (X_{j}) = \alpha (1) \mu (X). $$

However, from (2.14) one has

$$ \int _{X} \alpha \bigl(2 \bigl\vert f(x) \bigr\vert \bigr) \,d\mu \geq \sum_{j=1}^{\infty } 2^{j} \alpha (t_{j}) \mu (X_{j}) = \infty, $$

which violates the Minkowski type inequality (2.12). In all, we have shown that there is a positive real number $c_{2} > 1$ with

$$ p_{\alpha }(t) \leq c_{2} $$

for almost every $t > 0$.

The same argument on the conjugate function β of α delivers the existence of a positive constant $c_{0} >1$ for which

$$\begin{aligned} p_{\beta }(t) \leq c_{0} \quad\text{for almost every $t > 0$,} \end{aligned}$$

(2.15)

that is,

$$\begin{aligned} \beta '(t) \leq c_{0} \frac{\beta (t)}{t}. \end{aligned}$$

(2.16)

From identity (2.1), we notice that

$$ x = {\beta } \biggl( \frac{x}{\alpha ^{-1}(x)} \biggr) \quad\text{or}\quad {\alpha }(x) = {\beta } \biggl( \frac{\alpha (x)}{x} \biggr). $$

(2.17)

Hence the spectrum condition (2.16) is equivalent to saying

$$\begin{aligned} {{\beta }}' \biggl( \frac{{\alpha }(t)}{t} \biggr) \leq c_{0} t. \end{aligned}$$

(2.18)

On the other hand, differentiate both sides of identity (2.1) to have

$$ \frac{{{\beta }}^{-1}(x)}{{ \alpha '({{\alpha }}^{-1}(x))}} + \frac{{{ \alpha }}^{-1}(x)}{{{{\beta }}'({{\beta }}^{-1}(x))}} =1, $$

which is equivalent to

$$ \frac{{{\beta }}^{-1}({\alpha }(t) )}{{ \alpha '(t)}} + \frac{t }{{{{ \beta }}'({\alpha }(t))}} =1. $$

(2.19)

Then from the second identity of (2.17), we obtain $\beta ^{-1} \circ \alpha (x) = \frac{\alpha (x) }{x}$. Therefore we find that identity (2.19) is equivalent to

$$ \frac{y}{{{\alpha }}'(x)} + \frac{x}{{{\beta }}'(y)} = 1 \quad\text{for } y:= \frac{\alpha (x) }{x}, $$

(2.20)

or

$$\begin{aligned} \alpha '(t) = s+ t \frac{{\alpha }'(t)}{{\beta }'(s)},\quad s = \frac{\alpha (t)}{t}. \end{aligned}$$

(2.21)

Reflecting (2.18) to identity (2.21), we have

$$ \alpha '(t) \geq \frac{c_{0}}{c_{0} - 1} \frac{\alpha (t)}{t}, $$

which implies that

$$\begin{aligned} p_{\alpha }(t) \geq c_{1} \end{aligned}$$

with $c_{1} = \frac{c_{0}}{c_{0} - 1}$. □

As a direct application of Proposition 2, we have the following.

Corollary 3

Let$(\alpha, \beta )$be a convex Hölder pair. Then there exist some constants$c_{1}$, $c_{2}$satisfying

$$\begin{aligned} t^{c_{1}} \leq \alpha (t) \leq t^{c_{2}} \end{aligned}$$

(2.22)

for almost every$t >0$.

Proof

The result follows from solving the two separable ordinary differential inequalities

$$ c_{1} \leq x \frac{\alpha '(x)}{\alpha (x)} \leq c_{2}. $$

 □

3 Refined John–Nirenberg inequality

In this section Q stands for a cube whose sides are parallel to the axes, and $|A|$ is the Lebesgue measure of the set A in $\mathbb{R}^{d}$, $d \geq 1$. For a locally integrable function f on $\mathbb{R}^{d}$, we let

$$\begin{aligned} f_{Q} = -\!\!\!\!\!\!\int _{Q} f(x)\,dx \end{aligned}$$

and define distribution function $\mu _{Q}$ with respect to $f - f_{Q}$ by

$$\begin{aligned} \mu _{Q} (t):= \bigl\vert \bigl\{ x \in Q: \bigl\vert f(x)-f_{Q} \bigr\vert > t \bigr\} \bigr\vert . \end{aligned}$$

(3.1)

Let α be a pre-Hölder function. We denote a class of functions by

$$\begin{aligned} \mathrm{BMO}_{\alpha }\bigl(\mathbb{R}^{d}\bigr)= \bigl\{ f \in L_{loc}^{1}\bigl(\mathbb{R}^{d}\bigr) : \Vert f \Vert _{\mathrm{BMO}_{\alpha }} < \infty \bigr\} , \end{aligned}$$

where we set

$$ \Vert f \Vert _{\mathrm{BMO}_{\alpha }} = \alpha ^{-1} \biggl( \sup _{Q} -\!\!\!\!\!\!\int _{Q} \alpha \bigl( \bigl\vert f(x)- f_{Q} \bigr\vert \bigr)\,dx \biggr). $$

When α is the identity function, we write $\mathrm{BMO}_{\alpha }(\mathbb{R}^{d}):= \mathrm{BMO}(\mathbb{R}^{d})$.

We now state a refined version of the John–Nirenberg inequality on the sub-classes of BMO equipped with the huge class of (convex pre-)Hölder base functions α.

Theorem 4

(Refinement of the John–Nirenberg inequality)

Letαbe a convex pre-Hölder function or Hölder function and$f \in \mathrm{BMO}_{\alpha }$. For every$\lambda >1$, define step functions$F_{\lambda }$by

$$ F_{\lambda }(t) = \sum_{k=0}^{\infty } \biggl( \frac{1}{ \lambda } \biggr)^{k} \chi _{ (0, T]}(t- kT)\quad (t>0), $$

where$T:= 2^{d} \alpha ^{-1}( \lambda \alpha ( \| f \|_{\mathrm{BMO}_{\alpha }}))$. Then one has

$$\begin{aligned} \frac{1}{ \vert Q \vert }\mu _{Q} (t) \leq F_{\lambda }(t)\quad (t>0) \end{aligned}$$

(3.2)

for every cube$Q \subset \mathbb{R}^{d}$and every$\lambda >1$. The upper bound function$F_{\lambda }$is finer than the exponential function, that is to say, for every$\lambda >1$, we have

$$\begin{aligned} \frac{1}{ \vert Q \vert }\mu _{Q} (t) \leq F_{\lambda }(t) \leq \lambda \exp \biggl( - \frac{t \ln \lambda }{2^{d} \alpha ^{-1} ( \lambda \alpha ( \Vert f \Vert _{\mathrm{BMO}_{\alpha }}))} \biggr). \end{aligned}$$

(3.3)

Proof

We define

$$\begin{aligned} &E_{Q}(t) := \bigl\{ x \in Q: \bigl\vert f(x) - f_{Q} \bigr\vert > t \bigr\} , \\ &\varTheta (t) := \sup_{Q}\frac{\mu _{Q} (t)}{ \vert Q \vert }. \end{aligned}$$

We want to show that, for any $\lambda >1$,

$$ \varTheta (t) \leq \sum_{k=0}^{\infty } \frac{1}{ \lambda ^{k}} \chi _{ (0, T]}(t- kT) \lesssim e^{b -\frac{t}{c}} $$

for some positive constants b and c. First, we observe that $\varTheta (t) \leq 1$ for all $t > 0$. So, obviously, we have

$$\begin{aligned} \varTheta (t) \leq \chi _{(0, T]}(t) \leq e^{b-b\frac{t}{T}} \quad\text{for } 0< t \leq T, \end{aligned}$$

(3.4)

where $b>0$ and $T>0$ are constants to be chosen later. Now take a number s so large that $s> 1$ and

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO}_{\alpha }} < s. \end{aligned}$$

Then we have

$$\begin{aligned} -\!\!\!\!\!\!\int _{Q} \alpha \bigl( \bigl\vert f(x) - f_{Q} \bigr\vert \bigr)\,dx < \alpha (s) \end{aligned}$$

(3.5)

for any cube Q.

For a cube Q, we employ a dyadic decomposition of Q as follows: Sub-divide Q into $2^{d}$ equal closed sub-cubes of side length equal to half of the side length of Q by bisecting the sides, and select such sub-cubes $Q_{1}^{1}, Q_{2}^{1}, \ldots, Q_{2^{d}}^{1}$. Repeat this process for any sub-cubes $Q_{j}^{1}$ of the second generation to produce the third generation $Q_{1}^{2}, Q_{2}^{2}, \ldots, Q_{2^{2d}}^{2}$. Continuing this process, we obtain the collection of all dyadic cubes of Q, to say $\mathcal{D}(Q)$, and $\mathcal{F}(Q)$ is the collection of all the first cubes $Q' \in \mathcal{D}(Q)$ such that

$$ -\!\!\!\!\!\!\int _{Q'} \alpha \bigl( \bigl\vert f(x) - f_{Q} \bigr\vert \bigr) \,dx > \alpha (s), $$

where the “first $Q '$” means that, for any cube $\tilde{Q} \in \mathcal{D}(Q)$ with $Q' \subset \tilde{Q}$, $\frac{1}{|\tilde{Q}|} \int _{\tilde{Q}} \alpha (|f(x) - f_{Q} |) \,dx < \alpha (s)$. Then we note that

$$ \biggl\{ x \in Q: \sup_{x \in Q' \in \mathcal{D}(Q)} -\!\!\!\!\!\!\int _{Q'} \alpha \bigl( \bigl\vert f(x) - f_{Q} \bigr\vert \bigr) \,dx > \alpha (s) \biggr\} = \bigcup _{Q' \in \mathcal{F}(Q)} Q'. $$

Also, by virtue of the Lebesgue differentiation theorem, for almost every $x \in E_{Q}( s)$, there exists $Q' \in \mathcal{D}(Q)$ (indeed, $Q' \in \mathcal{F}(Q)$) such that $x \in Q'$ and

$$ \alpha (s) < -\!\!\!\!\!\!\int _{Q'} \alpha \bigl( \bigl\vert f(y) - f_{Q} \bigr\vert \bigr) \,dy. $$

This means that the statement

$$\begin{aligned} {``} x \in E_{Q}( s) \quad\text{implies}\quad x \in \bigcup _{Q' \in \mathcal{F}(Q)} Q ' {''} \end{aligned}$$

(3.6)

holds for almost every x. For $Q' \in \mathcal{F}(Q)$, we take the parent Q̃ of $Q'$, that is, the smallest cube $\tilde{Q} \in \mathcal{D}(Q) \setminus \mathcal{F}(Q)$ containing $Q'$. Then Jensen’s inequality for the convex function α or the Hölder type inequality (2.10) for the Hölder function α leads to

$$\begin{aligned} \frac{1}{ \vert Q' \vert } \int _{Q'} \bigl\vert f(x) - f_{Q} \bigr\vert \,dx &\leq \biggl( \frac{ \vert \tilde{Q} \vert }{ \vert Q' \vert } \biggr) \alpha ^{-1} \biggl( \frac{1}{ \vert \tilde{Q} \vert } \int _{\tilde{Q}} \alpha \bigl( \bigl\vert f(x) - f_{Q} \bigr\vert \bigr) \,dx \biggr) \\ &\leq 2^{d} s. \end{aligned}$$

(3.7)

Hence from the fact that for $x \in E_{Q}( t)$ with $t > 2^{d} s$,

$$\begin{aligned} t < \bigl\vert f(x) - f_{Q} \bigr\vert &\leq \bigl\vert f(x) - f_{Q'} \bigr\vert + \vert f_{Q'} - f_{Q} \vert \\ &\leq \bigl\vert f(x) - f_{Q'} \bigr\vert + \frac{1}{ \vert Q' \vert } \int _{Q'} \bigl\vert f(x) - f_{Q} \bigr\vert \,dx \\ &\leq \bigl\vert f(x) - f_{Q'} \bigr\vert + 2^{d} s, \end{aligned}$$

we have

$$ t - 2^{d} s < \bigl\vert f(x) - f_{Q'} \bigr\vert . $$

By (3.6), we obtain that for $t > 2^{d} s$,

$$\begin{aligned} \mu _{Q} (t) &= \bigl\vert E_{Q}( s) \cap E_{Q}( t) \bigr\vert \\ &\leq \biggl\vert \bigcup_{Q' \in \mathcal{F}(Q)} Q' \cap E_{Q}( t) \biggr\vert \\ &\leq \sum_{Q' \in \mathcal{F}(Q)} \bigl\vert Q' \cap E_{Q}( t) \bigr\vert \\ &\leq \sum_{Q' \in \mathcal{F}(Q)} \frac{ \vert \{ x \in Q': \vert f(x) - f_{Q'} \vert > t - 2^{d} s \} \vert }{ \vert Q' \vert } \bigl\vert Q' \bigr\vert \\ &\leq \varTheta \bigl(t - 2^{d} s\bigr) \sum _{Q' \in \mathcal{F}(Q)} \bigl\vert Q' \bigr\vert . \end{aligned}$$

Therefore, by (3.7) and (3.5), we get

$$\begin{aligned} \mu _{Q}(t) &\leq \varTheta \bigl(t - 2^{d} s\bigr) \sum _{Q' \in \mathcal{F}(Q)} \frac{1}{\alpha (s)} \int _{Q'} \alpha \bigl( \bigl\vert f(x) - f_{Q} \bigr\vert \bigr) \,dx \\ &= \frac{1}{\alpha (s)} \varTheta \bigl(t - 2^{d} s\bigr) \int _{\cup Q'} \alpha \bigl( \bigl\vert f(x) - f_{Q} \bigr\vert \bigr) \,dx \\ &\leq \frac{1}{\alpha (s)} \varTheta \bigl(t - 2^{d} s\bigr) \cdot \vert Q \vert \cdot \alpha \bigl( \Vert f \Vert _{\mathrm{BMO}_{\alpha }}\bigr) \end{aligned}$$

for $t > 2^{d} s$. That is,

$$ \frac{1}{ \vert Q \vert }\mu _{Q}(t) \leq \frac{\alpha ( \Vert f \Vert _{\mathrm{BMO}_{\alpha }}) }{\alpha (s)} \varTheta \bigl(t - 2^{d} s\bigr). $$

We take the supremum over Q to have

$$\begin{aligned} \varTheta (t) \leq \frac{\varTheta (t - T )}{\lambda } \quad(t > T), \end{aligned}$$

(3.8)

where we put $\lambda = \frac{\alpha (s)}{\alpha ( \| f \|_{\mathrm{BMO}_{\alpha }}) } > 1$ and $T:= 2^{d} s = 2^{d} \alpha ^{-1}( \lambda \alpha ( \| f \|_{\mathrm{BMO}_{ \alpha }}))$.

We revisit (3.4) to have

$$\begin{aligned} \varTheta (t) \leq \chi _{ (0, T]}(t) \quad\text{for } 0< t \leq T. \end{aligned}$$

(3.9)

Applying (3.9) into (3.8), we have

$$\begin{aligned} \varTheta (t) \leq \frac{\varTheta (t - T)}{\lambda } &\leq \frac{1}{ \lambda } \chi _{ (0, T]}(t- T)\quad (T < t \leq 2T). \end{aligned}$$

Continue this process to get, for $t >0$,

$$\begin{aligned} \varTheta (t) \leq \sum_{k=1}^{\infty } \frac{1}{ \lambda ^{k-1}} \chi _{ (0, T]}\bigl(t-(k-1)T\bigr). \end{aligned}$$

In order to find a smooth dominating function for $F_{\lambda }(t)$, consider (3.4) to have

$$\begin{aligned} \chi _{ (0, T]}(t) \leq \exp \biggl(b- b \frac{t}{T} \biggr) \quad\text{for } 0< t \leq T= 2^{d} \alpha ^{-1}\bigl( \lambda \alpha \bigl( \Vert f \Vert _{\mathrm{BMO}_{\alpha }}\bigr)\bigr). \end{aligned}$$

(3.10)

Then we have that, for $kT < t \leq (k+1)T$,

$$\begin{aligned} \frac{1}{ \lambda ^{k}} \chi _{ (0, T]}(t- kT) &\leq \frac{1}{ \lambda ^{k}} \exp \biggl( b- b \frac{t - kT}{T} \biggr) = \frac{1}{ \lambda ^{k}} e^{kb} \exp \biggl(b- b \frac{t}{T} \biggr). \end{aligned}$$

Choosing $b= \ln \lambda$, we obtain that, for $t >0$,

$$\begin{aligned} F_{\lambda }(t) \leq \lambda \exp \biggl( - \frac{t \ln \lambda }{2^{d} \alpha ^{-1} ( \lambda \alpha ( \Vert f \Vert _{\mathrm{BMO}_{\alpha }}))} \biggr) \end{aligned}$$

for every $\lambda >0$. The proof is now completed. □

By Theorem 4, we can see that

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO}_{\alpha }} &= \alpha ^{-1} \biggl( \sup _{Q} \frac{1}{ \vert Q \vert } \int _{0}^{\infty } \alpha ' (t) \mu _{Q}(t) \,dt \biggr) \\ &\leq \alpha ^{-1} \biggl( \int _{0}^{\infty } \alpha ' (t) F_{ \lambda }(t) \,dt \biggr) \\ &= \alpha ^{-1} \Biggl( \sum_{k=0}^{\infty } \biggl( \frac{1}{ \lambda } \biggr)^{k} \int _{kT}^{(k+1)T} \alpha ' (t) \,dt \Biggr) \\ &= \alpha ^{-1} \Biggl( \sum_{k=0}^{\infty } \frac{\alpha ((k+1)T) - \alpha (kT)}{ \lambda ^{k}} \Biggr). \end{aligned}$$

(3.11)

When α is a convex pre-Hölder function or the Hölder function, Jensen’s inequality or the Hölder type inequality (2.10), respectively, yields

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO} } \leq \Vert f \Vert _{\mathrm{BMO}_{\alpha }}. \end{aligned}$$

(3.12)

We can argue that for a convex Hölder pair $(\alpha, \beta )$, the classical $\mathrm{BMO}(\mathbb{R}^{d})$-norm and the modified $\mathrm{BMO}_{\alpha }(\mathbb{R}^{d})$-norm are equivalent: For it, we will demonstrate that the identity function $\mathrm{id}: \mathrm{BMO} \to \mathrm{BMO}_{\alpha }$ is continuous at 0, which implies that

$$ \Vert f \Vert _{\mathrm{BMO}_{\alpha }} \leq C \Vert f \Vert _{\mathrm{BMO}} $$

for some $C >0$. In fact, for any sequence $\{ f_{n} \}$ converging to 0 in $\mathrm{BMO}(\mathbb{R}^{d})$, we have

$$ \alpha \bigl( \Vert f_{n} \Vert _{\mathrm{BMO}_{\alpha }}\bigr) = \sup _{Q} \int _{0}^{\infty } \alpha ' (t) \frac{\mu _{Q}(t)}{ \vert Q \vert } \,dt. $$

We apply Theorem 4 for the case when the base function α is the identity function so that we have

$$ \frac{1}{ \vert Q \vert }\mu _{Q} (t) \leq F_{2}(t) = \sum _{k=0}^{\infty } \biggl( \frac{1}{2} \biggr)^{k} \chi _{ (0, T]}(t- kT), $$

where $T:= 2^{d+1} \| f \|_{\mathrm{BMO}}$. Then, as in (3.11), we obtain

$$\begin{aligned} \alpha \bigl( \Vert f_{n} \Vert _{\mathrm{BMO}_{\alpha }}\bigr) &\leq \sum _{k=0}^{\infty } \frac{\alpha ((k+1)2^{d+1} \Vert f_{n} \Vert _{\mathrm{BMO}} ) - \alpha (k 2^{d+1} \Vert f_{n} \Vert _{\mathrm{BMO}})}{ 2^{k}} \\ &\leq \sum_{k=0}^{\infty } \frac{\alpha ((k+1)2^{d+1} \Vert f_{n} \Vert _{\mathrm{BMO}} )}{ 2^{k}}. \end{aligned}$$

Corollary 3 yields

$$\begin{aligned} \bigl( \Vert f_{n} \Vert _{\mathrm{BMO}_{\alpha }}\bigr)^{c_{1}} \leq \alpha \bigl( \Vert f_{n} \Vert _{\mathrm{BMO}_{ \alpha }}\bigr) &\leq \sum_{k=0}^{\infty } \frac{((k+1)2^{d+1} \Vert f_{n} \Vert _{\mathrm{BMO}} )^{c_{2}}}{ 2^{k}} \lesssim \Vert f_{n} \Vert _{\mathrm{BMO}}^{c_{2}} \end{aligned}$$

for some $1< c_{1} < c_{2}$. Therefore we obtain the equivalence between $\mathrm{BMO}(\mathbb{R}^{d})$-norm and $\mathrm{BMO}_{\alpha }(\mathbb{R}^{d})$-norm:

Corollary 5

Let$(\alpha, \beta )$be a convex Hölder pair. Then there exists a constant$C_{\alpha }$such that

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO}} \leq \Vert f \Vert _{\mathrm{BMO}_{\alpha }} \leq C_{\alpha } \Vert f \Vert _{\mathrm{BMO}}. \end{aligned}$$