Some new characterizations of BLO and Campanato spaces in the Schrödinger setting

The theory of function spaces has been a central topic in modern analysis, and new function spaces are now of increasing use in the fields such as harmonic analysis and partial differential equations. The main purpose of this paper is to give several characterizations for the new BLO and Campanato spaces in the Schrödinger setting. Another purpose of this paper is to establish a version of John–Nirenberg inequality for the new BLO space. Let $d\geq 3$ be a positive integer and $\mathbb{R}^{d}$ be the $d$-dimensional Euclidean space, and let $V:\mathbb{R}^{d}\rightarrow\mathbb{R}$, $d\geq 3$, be a nonnegative locally integrable function which belongs to the reverse Hölder class $RH_{s}(\mathbb{R}^{d})$ with $s\in(1,\infty]$. We recall that $V\in RH_{s}(\mathbb{R}^{d})$ means that there exists a positive constant $C=C(s,V)>0$ such that the following reverse Hölder inequality

holds for every ball $B$ in $\mathbb{R}^{d}$, with the usual modification made when $s=\infty$. In particular, if $V$ is a nonnegative polynomial, then $V\in RH_{\infty}(\mathbb{R}^{d})$. Let us consider the Schrödinger differential operator with the nonnegative potential $V$.

where $\Delta=\sum_{j=1}^{d}\frac{\partial^{2}}{\partial x_{j}^{2}}$ is the standard Laplace operator on $\mathbb{R}^{d}$. As in [36], for any given $V\in RH_{s}(\mathbb{R}^{d})$ with $s\geq d/2$ and $d\geq 3$, we introduce the critical radius function $\rho(x)=\rho(x;V)$ (determined by $V$), which is defined by

where $B(x,r)$ denotes the open ball with the center at $x$ and radius $r$. It is well known that this auxiliary function satisfies $0<\rho(x)<\infty$ for any $x\in\mathbb{R}^{d}$ under the above assumption on $V$ (see [36]).

Throughout this paper, we will always assume that $V\not\equiv 0$ and $V\in RH_{s}(\mathbb{R}^{d})$ with $s\geq d/2$.

The notation $\mathbf{X}\approx\mathbf{Y}$ means that there exists a positive constant $C>0$ such that $1/C\leq\mathbf{X}/\mathbf{Y}\leq C$.

We need the following known result concerning the critical radius function (1.1), which was proved by Shen in [36].

To state our main results, we first recall the definition of the classical BMO space and BLO space.

A locally integrable function $f$ on $\mathbb{R}^{d}$ is said to belong to $\mathrm{BMO}(\mathbb{R}^{d})$, the space of bounded mean oscillation, if

where the supremum is taken over all cubes $\mathcal{Q}$ in $\mathbb{R}^{d}$ and $f_{\mathcal{Q}}$ stands for the mean value of $f$ over $\mathcal{Q}$; that is,

The space of BMO functions was first introduced by John and Nirenberg in [25].

A locally integrable function $f$ on $\mathbb{R}^{d}$ is said to belong to $\mathrm{BLO}(\mathbb{R}^{d})$, the space of bounded lower oscillation, if there exists a constant $c_{1}>0$ such that for any cube $\mathcal{Q}\subset\mathbb{R}^{d}$,

The minimal constant $c_{1}$ as above is defined to be the BLO-constant of $f$ and denoted by $\|f\|_{\mathrm{BLO}}$. The space of BLO functions was first introduced by Coifman and Rochberg in [15]. It is easy to see that

Moreover, the above inclusion relations are both strict, see [22, 23, 34] for some examples. It is easy to verify that

In fact, for any cube $\mathcal{Q}\subset\mathbb{R}^{d}$,

as desired.

On the other hand, the classical Campanato space was studied extensively in the literature, and played an important role in the study of harmonic analysis and partial differential equations. Let $0<\beta\leq 1$. A locally integrable function $f$ is said to belong to the Campanato space $\mathcal{C}^{\beta}(\mathbb{R}^{d})$ if

where the supremum is taken over all balls $\mathcal{B}$ in $\mathbb{R}^{d}$ and $f_{\mathcal{B}}$ stands for the mean value of $f$ over $\mathcal{B}$. The Campanato space $\mathcal{C}^{\beta}(\mathbb{R}^{d})$ was first introduced by Campanato in [12]. In 2007, motivated by the definition of the space $\mathrm{BLO}(\mathbb{R}^{d})$, Hu–Meng–Yang introduced the following space $\mathcal{C}^{\beta,\ast}(\mathbb{R}^{d})$, which is a subspace of $\mathcal{C}^{\beta}(\mathbb{R}^{d})$. Let $0<\beta\leq 1$. A locally integrable function $f$ is said to belong to $\mathcal{C}^{\beta,\ast}(\mathbb{R}^{d})$ if there exists a positive constant $c_{2}>0$ such that for any ball $\mathcal{B}\subset\mathbb{R}^{d}$,

The minimal constant $c_{2}$ as above is defined to be the $\mathcal{C}^{\beta,\ast}$-constant of $f$ and denoted by $\|f\|_{\mathcal{C}^{\beta,\ast}}$.

In 2011, Bongioanni–Harboure–Salinas [3] introduced a new class of function spaces (see also [2]). According to [3], the new BMO space $\mathrm{BMO}_{\rho,\infty}(\mathbb{R}^{d})$ is defined by

where for any fixed $0<\theta<\infty$ the space $\mathrm{BMO}_{\rho,\theta}(\mathbb{R}^{d})$ is defined to be the set of all locally integrable functions $f$ satisfying

for all $x_{0}\in\mathbb{R}^{d}$ and $r\in(0,\infty)$, $f_{Q}$ denotes the mean value of $f$ on $Q(x_{0},r)$. A norm for $f\in\mathrm{BMO}_{\rho,\theta}(\mathbb{R}^{d})$, denoted by $\|f\|_{\mathrm{BMO}_{\rho,\theta}}$, is given by the infimum of the constants satisfying (1.4), after identifying functions that differ by a constant, or equivalently,

where the supremum is taken over all cubes $\mathcal{Q}=Q(x_{0},r)$ with $x_{0}\in\mathbb{R}^{d}$ and $r\in(0,\infty)$. Note that if we let $\theta=0$ in (1.4), we obtain the classical (John–Nirenberg) BMO space. Define

With the above definition in mind, one has

whenever $0<\theta_{1}<\theta_{2}<\infty$, and hence

Moreover, it can be shown that the classical BMO space is properly contained in $\mathrm{BMO}_{\rho,\infty}(\mathbb{R}^{d})$ (see [2, 3, 43] for more examples).

A classical result due to John and Nirenberg in the 1960’s(also known as the John–Nirenberg inequality) states that there exist two positive constants $C_{1}$ and $C_{2}$ such that for every cube $\mathcal{Q}=Q(x_{0},r)$ in $\mathbb{R}^{d}$ and every $\lambda>0$, we have (see [25])

when $f\in\mathrm{BMO}(\mathbb{R}^{d})$.

In 2015, Tang proved a version of John–Nirenberg inequality suitable for the new BMO space $\mathrm{BMO}_{\rho,\theta}(\mathbb{R}^{d})$ with $\theta>0$. His proof can be found in [43, Proposition 4.2].

In 2014, Liu–Sheng introduced a new class of function spaces which is larger than the classical Campanato space. According to [28], for $0<\theta<\infty$ and $0\leq\beta\leq 1$, the space $\mathcal{\mathcal{C}}^{\beta}_{\rho,\theta}(\mathbb{R}^{d})$ is defined to be the set of all locally integrable functions $f$ satisfying

for all $x_{0}\in\mathbb{R}^{d}$ and $r\in(0,\infty)$, $f_{B}$ denotes the mean value of $f$ on $B(x_{0},r)$. The infimum of the constants satisfying (1.5) is defined to be the norm of $f\in\mathcal{\mathcal{C}}^{\beta}_{\rho,\theta}(\mathbb{R}^{d})$ and denoted by $\|f\|_{\mathcal{C}^{\beta}_{\rho,\theta}}$, or equivalently,

where the supremum is taken over all balls $\mathcal{B}=B(x_{0},r)$ with $x_{0}\in\mathbb{R}^{d}$ and $r\in(0,\infty)$.

Define

Since $[1+r/{\rho(x_{0})}]^{\theta}\geq 1$, it is obvious that

whenever $0<\theta_{1}<\theta_{2}<\infty$. Then we write

Motivated by the definition of $\mathrm{BLO}(\mathbb{R}^{d})$ and $\mathcal{C}^{\beta,\ast}(\mathbb{R}^{d})$, we now introduce the following spaces $\mathrm{BLO}_{\rho,\theta}(\mathbb{R}^{d})$ and $\mathcal{C}^{\beta,\ast}_{\rho,\theta}(\mathbb{R}^{d})$ related to Schrödinger operators.

Adapting the arguments in [18, p.123] and [20, p.124], we can also prove the following variant of John–Nirenberg inequality: if $f\in\mathrm{BLO}(\mathbb{R}^{d})$, then there exist two positive constants $C_{3}$ and $C_{4}$ such that for every cube $\mathcal{Q}$ and every $\lambda>0$, we have (see [44])

Inspired by these results, we will establish a version of John–Nirenberg inequality with precise constants for the space $\mathrm{BLO}_{\rho,\theta}(\mathbb{R}^{d})$ related to Schrödinger operators. That is, if $f\in\mathrm{BLO}_{\rho,\theta}(\mathbb{R}^{d})$ with $0<\theta<\infty$, then there exist two positive constants $\overline{C}_{3}$ and $\overline{C}_{4}$ such that for every cube $\mathcal{Q}=Q(x_{0},r)$ and every $\lambda>0$, we have

where $N_{0}$ is the constant appearing in Lemma 1.1. Based on this result, the goal of this paper is to give some new characterizations for the space $\mathrm{BLO}_{\rho,\theta}(\mathbb{R}^{d})$ with $0<\theta<\infty$. Moreover, we also establish similar results for the space $\mathcal{C}^{\beta,\ast}_{\rho,\theta}(\mathbb{R}^{d})$ with $0<\beta\leq 1$ and $0<\theta<\infty$.

A weight will always mean a non-negative function $\omega$ on $\mathbb{R}^{d}$ which is locally integrable. For a Lebesgue measurable set $E\subset\mathbb{R}^{d}$ and a weight $\omega$, we use the notation $|E|$ to denote the Lebesgue measure of $E$ and $\omega(E)$ to denote the weighted measure of $E$,

In the sequel, for any positive number $\gamma>0$, we denote $\omega^{\gamma}(x):=\omega(x)^{\gamma}$ by convention. For a measurable set $E\subset\mathbb{R}^{d}$, we let

For any given ball $B=B(x_{0},r)$ and $\lambda\in(0,\infty)$, we will write $\lambda B$ for the $\lambda$-dilate ball, which is the ball with the same center $x_{0}$ and radius $\lambda r$; that is $\lambda B=B(x_{0},\lambda r)$. Similarly, $Q(x_{0},r)$ denotes the cube centered at $x_{0}$ and with the sidelength $r$. Here and in what follows, only cubes with sides parallel to the coordinate axes are considered, and $\lambda Q=Q(x_{0},\lambda r)$. Let us recall two classes of weights that are given in terms of the critical radius function (1.1). As in [1] (see also [2]), we say that a weight $\omega$ belongs to the class $A^{\rho,\theta}_{p}(\mathbb{R}^{d})$ for $1<p<\infty$ and $0<\theta<\infty$, if there is a positive constant $C>0$ such that for all balls $B=B(x_{0},r)\subset\mathbb{R}^{d}$ with $x_{0}\in\mathbb{R}^{d}$ and $r\in(0,\infty)$,

where $p^{\prime}=p/{(p-1)}$ denotes the conjugate exponent of $p$, namely, $1/p+1/{p^{\prime}}=1$. For $p=1$ and $0<\theta<\infty$, we also say that a weight $\omega$ belongs to the class $A^{\rho,\theta}_{1}(\mathbb{R}^{d})$, if there is a positive constant $C>0$ such that for all balls $B=B(x_{0},r)\subset\mathbb{R}^{d}$ with $x_{0}\in\mathbb{R}^{d}$ and $r\in(0,\infty)$,

Since

whenever $0<\theta_{1}<\theta_{2}<\infty$, then for given $p$ with $1\leq p<\infty$, by definition, we have

where $A_{p}(\mathbb{R}^{d})$ denotes the classical Muckenhoupt class (see [30] and [13]). For any given $1\leq p<\infty$, as the classes $A^{\rho,\theta}_{p}(\mathbb{R}^{d})$ increase with respect to $\theta$, it is natural to define

Consequently, one has the inclusion relation

However, the converse is not true, it is easy to check that the above inclusion is strict. In fact, if $\omega\in A_{p}(\mathbb{R}^{d})$ for some $p\geq 1$, then $\omega(x)\,dx$ is a doubling measure (see [18] and [19]), i.e., there exists a universal constant $C>0$ such that for any ball $B$ in $\mathbb{R}^{d}$,

If $\omega\in A^{\rho,\theta}_{p}(\mathbb{R}^{d})$ for some $p\geq 1$ and $\theta>0$, then $\omega(x)\,dx$ may not be a doubling measure. For example, the weight

provided that $V\equiv 1$ and $\rho(\cdot)\equiv 1$. It is easy to see that such choice of $\omega_{\gamma}$ yields $\omega_{\gamma}(x)\,dx$ is not a doubling measure, hence it does not belong to $A_{q}(\mathbb{R}^{d})$ for any $1\leq q<\infty$. The situation is more complicated. We can define (generalized) doubling classes of weights adapted to the Schrödinger context, see [9] and [10], for example. In addition, for some fixed $\theta>0$, we have the following inclusion relations (see [43])

whenever $1\leq p_{1}<p_{2}<\infty$. As in the classical Muckenhoupt theory, we define the $A^{\rho,\theta}_{p}$ characteristic constants of $\omega$ as follows:

for $1<p<\infty,$ and

for $p=1$, where the supremum is taken over all balls $B=B(x_{0},r)$ in $\mathbb{R}^{d}$. In view of (2.1), we can see that if $\omega\in A^{\rho,\theta_{1}}_{p}(\mathbb{R}^{d})$ with $1\leq p<\infty$ and $0\leq\theta_{1}<\infty$, then for any $\theta_{1}<\theta_{2}<\infty$, we have

Hence, for any given $\omega\in A^{\rho,\infty}_{p}(\mathbb{R}^{d})$ with $1\leq p<\infty$, we let

Now define the $A^{\rho,\infty}_{p}$ characteristic constant of $\omega$ by

As in [43] (see also [11] and [46]), we say that a weight $\omega$ is in the class $A^{\rho,\theta}_{p,q}(\mathbb{R}^{d})$ for $1<p,q<\infty$ and $0<\theta<\infty$, if there exists a positive constant $C>0$ such that

holds for all balls $B=B(x_{0},r)\subset\mathbb{R}^{d}$. For the case $p=1$, we also say that a weight $\omega$ is in the class $A^{\rho,\theta}_{1,q}(\mathbb{R}^{d})$ for $1<q<\infty$ and $0<\theta<\infty$, if there exists a positive constant $C>0$ such that