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Commutator of fractional integral with Lipschitz functions associated with Schrödinger operator on local generalized Morrey spaces

Abstract

Let \(L=-\Delta+V\) be a Schrödinger operator on \(\mathbb{R}^{n}\), where \(n\ge3\) and the nonnegative potential V belongs to the reverse Hölder class \(RH_{q_{1}}\) for some \(q_{1} > n/2\). Let b belong to a new Campanato space \(\Lambda_{\nu}^{\theta}(\rho)\) and \(\mathcal {I}_{\beta}^{L}\) be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators \([b,\mathcal {I}_{\beta}^{L}]\) with \(b \in\Lambda_{\nu}^{\theta}(\rho)\) on local generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\), generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}\) and vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha,V}\) associated with Schrödinger operator, respectively. When b belongs to \(\Lambda_{\nu}^{\theta}(\rho)\) with \(\theta >0\), \(0<\nu<1\) and \((\varphi_{1},\varphi_{2})\) satisfies some conditions, we show that the commutator operator \([b,\mathcal {I}_{\beta}^{L}]\) are bounded from \(LM_{p,\varphi_{1}}^{\alpha,V,\{ x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha,V,\{x_{0}\}}\), from \(M_{p,\varphi_{1}}^{\alpha,V}\) to \(M_{q,\varphi_{2}}^{\alpha,V}\) and from \(VM_{p,\varphi_{1}}^{\alpha,V}\) to \(VM_{q,\varphi_{2}}^{\alpha ,V}\), \(1/p-1/q=(\beta+\nu)/n\).

Introduction and main results

Let us consider the Schrödinger operator

$$\begin{aligned} L=-\Delta+V\quad \mbox{on } {\mathbb{R}^{n}}, n \geq3, \end{aligned}$$

where V is a nonnegative, \(V \neq0\), and belongs to the reverse Hölder class \(RH_{q}\) for some \(q\geq n/2\), i.e., there exists a constant \(C>0\) such that the reverse Hölder inequality

$$\begin{aligned} \biggl(\frac{1}{ \vert B(x,r) \vert } \int_{B(x,r)}V^{q}(y)\,dy \biggr)^{1/q}\leq \frac{C}{ \vert B(x,r) \vert } \int_{B(x,r)}V(y)\,dy \end{aligned}$$
(1.1)

holds for every \(x \in{\mathbb{R}^{n}}\) and \(0< r<\infty \), where \(B(x,r)\) denotes the ball centered at x with radius r. In particular, if V is a nonnegative polynomial, then \(V \in RH_{\infty }\).

As in [29], for a given potential \(V \in RH_{q}\) with \(q\geq n/2\), we define the auxiliary function

$$\begin{aligned} \rho(x):= \frac{1}{m_{V}(x)} = \sup_{r>0} \biggl\{ r : \frac {1}{r^{n-2}} \int_{B(x,r)} V(y)\,dy \le1 \biggr\} . \end{aligned}$$

It is well known that \(0<\rho(x)<\infty \) for any \(x\in{\mathbb{R}}^{n}\).

Let \(\theta>0\) and \(0<\nu<1\), in view of [22], the Campanato class, associated with the Schrödinger operator \(\Lambda_{\nu }^{\theta}(\rho)\) consists of the locally integrable functions b such that

$$\begin{aligned} \frac{1}{ \vert B(x,r) \vert ^{1+\nu/n}} \int_{B(x,r)} \bigl\vert b(y)-b_{B} \bigr\vert \,dy \le C \biggl(1+\frac{r}{\rho(x)} \biggr)^{\theta} \end{aligned}$$
(1.2)

for all \(x\in{\mathbb{R}}^{n}\) and \(r>0\). A seminorm of \(b \in\Lambda _{\nu }^{\theta}(\rho)\), denoted by \([b]_{\beta}^{\theta}\), is given by the infimum of the constants in the inequality above.

Note that if \(\theta=0\), \(\Lambda_{\nu}^{\theta}(\rho)\) is the classical Campanato space; if \(\nu=0\), \(\Lambda_{\nu}^{\theta}(\rho)\) is exactly the space \(BMO_{\theta}(\rho)\) introduced in [5].

We now present the definition of generalized Morrey spaces \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) (including the weak version) associated with a Schrödinger operator, which was introduced by the first author in [18].

The classical Morrey spaces \(L_{p,\lambda}({\mathbb{R}}^{n})\) was introduced by Morrey in [24] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the reader to [912, 24, 35]. The generalized Morrey spaces are defined with \(r^{\lambda}\) replaced by a general nonnegative function \(\varphi(x,r)\) satisfying some assumptions (see, for example, [15, 23, 25, 30]).

For brevity, in the sequel we use the notations

$$\mathfrak{A}_{p,\varphi}^{\alpha ,V}(f;x,r):= \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } r^{-n/p} \varphi(x,r)^{-1} \Vert f \Vert _{L_{p}(B(x,r))} $$

and

$$\mathfrak{A}_{\Phi,\varphi}^{W,\alpha ,V}(f;x,r):= \biggl(1+\frac {r}{\rho(x)} \biggr)^{\alpha } r^{-n/p} \varphi(x,r)^{-1} \Vert f \Vert _{WL_{p}(B(x,r))}. $$

Definition 1.1

Let \(\varphi(x,r)\) be a positive measurable function on \(\mathbb{R}^{n}\times(0,\infty)\), \(1\le p<\infty\), \(\alpha\ge0\), and \(V\in RH_{q}\), \(q\ge1\). For any fixed \(x_{0} \in{\mathbb{R}^{n}}\) we denote by \(LM_{p,\varphi}^{\alpha,V,\{ x_{0}\} }=LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) the local generalized Morrey space associated with Schrödinger operator, the space of all functions \(f\in L^{loc}_{p}(\mathbb{R}^{n})\) with finite norm

$$\begin{aligned} \Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}}= \sup_{r>0} \mathfrak {A}_{p,\varphi}^{\alpha,V}(f;x_{0},r). \end{aligned}$$

Also \(WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}=WLM_{p,\varphi}^{\alpha ,V,\{x_{0}\}}(\mathbb{R}^{n})\) we denote the weak local generalized Morrey space associated with Schrödinger operator, the space of all functions \(f\in WL^{loc}_{p} (\mathbb{R}^{n})\) with

$$\begin{aligned} \Vert f \Vert _{WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}}= \sup_{r>0} \mathfrak {A}_{p,\varphi}^{W,\alpha,V}(f;x_{0},r)< \infty . \end{aligned}$$

The local spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) and \(WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) are Banach spaces with respect to the norm

$$\begin{aligned} \Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}} & = \sup_{r>0} \mathfrak {A}_{p,\varphi}^{\alpha,V}(f;x_{0},r),\qquad \Vert f \Vert _{WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}} = \sup_{r>0} \mathfrak {A}_{p,\varphi}^{W,\alpha,V}(f;x_{0},r), \end{aligned}$$

respectively.

Remark 1.1

  1. (i)

    When \(\alpha=0\), and \(\varphi(x,r)=r^{(\lambda-n)/p}\), \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) is the local (central) Morrey space \(LM_{p,\lambda}^{\{0\}}(\mathbb {R}^{n})\) studied in [4].

  2. (ii)

    When \(\alpha=0\), \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\} }(\mathbb{R}^{n})\) is the local generalized Morrey space \(VM_{p,\varphi}^{\{x_{0}\}}(\mathbb{R}^{n})\) were introduced by the first author in [13]; see also [14, 16, 21] etc.

Definition 1.2

The vanishing generalized Morrey space associated with the Schrödinger operator \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) is defined as the spaces of functions \(f\in M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) such that

$$\begin{aligned} \lim_{r\rightarrow0} \sup_{x \in{\mathbb{R}^{n}}} \mathfrak {A}_{p,\varphi }^{\alpha ,V}(f;x,r) = 0. \end{aligned}$$
(1.3)

The vanishing weak generalized Morrey space associated with the Schrödinger operator \(VWM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) is defined as the spaces of functions \(f\in WM_{p,\varphi}^{\alpha ,V}({\mathbb {R}}^{n})\) such that

$$\begin{aligned} \lim_{r\rightarrow0} \sup_{x \in{\mathbb{R}^{n}}} \mathfrak {A}_{p,\varphi }^{W,\alpha ,V}(f;x,r) = 0. \end{aligned}$$

The vanishing spaces \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) and \(VWM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) are Banach spaces with respect to the norm

$$\begin{aligned} &\Vert f \Vert _{VM_{p,\varphi}^{\alpha ,V}} \equiv \Vert f \Vert _{M_{p,\varphi }^{\alpha ,V}} = \sup_{x\in{\mathbb{R}^{n}}, r>0} \mathfrak{A}_{p,\varphi}^{\alpha ,V}(f;x,r), \\ &\Vert f \Vert _{VWM_{p,\varphi}^{\alpha ,V}} \equiv \Vert f \Vert _{WM_{p,\varphi }^{\alpha ,V}} = \sup_{x\in{\mathbb{R}^{n}}, r>0} \mathfrak{A}_{W,p,\varphi }^{\alpha ,V}(f;x,r), \end{aligned}$$

respectively.

In the case \(\alpha=0\), and \(\varphi(x,r)=r^{(\lambda-n)/p}\) \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) is the vanishing Morrey space \(VM_{p,\lambda}\) introduced in [33], where applications to PDE were considered.

We refer to [3, 20, 27, 28] for some properties of vanishing generalized Morrey spaces.

Definition 1.3

Let \(L=-\Delta+V\) with \(V\in RH_{q_{1}}\), \(q_{1}>n/2\). The fractional integral associated with L is defined by

$$\begin{aligned} \mathcal{I}_{\beta}^{L} f(x)=L^{-\beta/2} f(x)= \int_{0}^{\infty}e^{-tL}(f) (x) t^{\beta/2-1} \,dt \end{aligned}$$

for \(0<\beta<n\). The commutator of \(\mathcal{I}_{\beta}^{L}\) is defined by

$$\begin{aligned} \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x)=b(x) \mathcal{I}_{\beta}^{L} f(x)-\mathcal{I}_{\beta}^{L}(bf) (x). \end{aligned}$$

Note that, if \(L=-\Delta\) is the Laplacian on \({\mathbb{R}}^{n}\), then \(\mathcal {I}_{\beta}^{L}\) and \([b,\mathcal{I}_{\beta}^{L}]\) are the Riesz potential \(I_{\beta}\) and the commutator of the Riesz potential \([b,I_{\beta}]\), respectively, that is,

$$\begin{aligned} I_{\beta}f(x)= \int_{{\mathbb{R}}^{n}} \frac{f(y)}{ \vert x-y \vert ^{n}} \,dy, \qquad [b,I_{\beta}]f(x)= \int_{{\mathbb{R}}^{n}} \frac{b(x)-b(y)}{ \vert x-y \vert ^{n}} f(y) \,dy. \end{aligned}$$

When \(b\in BMO\), Chanillo proved in [8] that \([b,I_{\beta }]\) is bounded from \(L_{p}({\mathbb{R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) with \(1/q=1/p-\beta /n\), \(1< p< n/\beta\). When b belongs to the Campanato space \(\Lambda _{\nu}\), \(0<\nu<1\), Paluszynski in [26] showed that \([b,I_{\beta}]\) is bounded from \(L_{p}({\mathbb{R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) with \(1/q=1/p-(\beta+\nu)/n\), \(1< p< n/(\beta+\nu)\). When \(b\in BMO_{\theta}(\rho)\), Bui in [6] obtained the boundedness of \([b,\mathcal{I}_{\beta}^{L}]\) from \(L_{p}({\mathbb {R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) with \(1/q=1/p-\beta/n\), \(1< p< n/\beta\).

Inspired by the above results, we are interested in the boundedness of \([b,\mathcal{I}_{\beta}^{L}]\) on generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\) and the vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\), when b belongs to the new Campanato class \(\Lambda_{\nu}^{\theta}(\rho)\).

In this paper, we consider the boundedness of the commutator of \(\mathcal{I}_{\beta}^{L}\) on the local generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\), the generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}({\mathbb {R}}^{n})\) and the vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}} ^{n})\). When b belongs to the new Campanato space \(\Lambda_{\nu }^{\theta}(\rho)\), \(0<\nu<1\), we show that \([b,\mathcal{I}_{\beta }^{L}]\) are bounded from \(LM_{p,\varphi_{1}}^{\alpha ,V,\{x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha ,V,\{x_{0}\}}\), from \(M_{p,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\) to \(M_{q,\varphi }^{\alpha ,V}({\mathbb{R}}^{n})\) and from \(VM_{p,\varphi}^{\alpha,V}({\mathbb {R}}^{n})\) to \(VM_{q,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\) with \(1/q=1/p-(\beta +\nu)/n\), \(1< p< n/(\beta+\nu)\).

Our main results are as follows.

Theorem 1.1

Let \(x_{0} \in{\mathbb{R}^{n}}\), \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(V\in RH_{q_{1}}\), \(q_{1}>n/2\), \(0<\nu<1\), \(\alpha\ge0\), \(1\le p< n/(\beta+\nu )\), \(1/q=1/p-(\beta+\nu)/n\) and let \(\varphi_{1}, \varphi_{2}\in\Omega _{p,loc}^{\alpha ,V}\) satisfy the condition

$$\begin{aligned} \int_{r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi _{1}(x_{0},s)s^{\frac{n}{p}}}{ t^{\frac{n}{q}}}\,\frac{dt}{t} \le c_{0} \varphi_{2}(x_{0},r), \end{aligned}$$
(1.4)

where \(c_{0}\) does not depend on r. Then the operator \([b,\mathcal {I}_{\beta}^{L}]\) is bounded from \(LM_{p,\varphi_{1}}^{\alpha,V,\{x_{0}\} }\) to \(LM_{q,\varphi_{2}}^{\alpha,V,\{x_{0}\}}\) for \(p>1\) and from \(LM_{1,\varphi_{1}}^{\alpha,V,\{x_{0}\}}\) to \(WLM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V,\{x_{0}\}}\). Moreover, for \(p>1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{LM_{q,\varphi_{2}}^{\alpha,V,\{x_{0}\} }}\le C [b]_{\nu}^{\theta} \Vert f \Vert _{LM_{p,\varphi_{1}}^{\alpha,V,\{ x_{0}\}}}, \end{aligned}$$

and for \(p=1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WLM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V,\{x_{0}\}}}\le C [b]_{\nu}^{\theta} \Vert f \Vert _{LM_{1,\varphi_{1}}^{\alpha,V,\{x_{0}\}}}, \end{aligned}$$

where C does not depend on f.

Corollary 1.1

Let \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(V\in RH_{q_{1}}\), \(q_{1}>n/2\), \(0<\nu<1\), \(\alpha\ge0\), \(1\le p< n/(\beta+\nu)\), \(1/q=1/p-(\beta+\nu)/n\) and let \(\varphi_{1}\in\Omega_{p}^{\alpha ,V}\), \(\varphi_{2} \in\Omega_{q}^{\alpha ,V}\) satisfy the condition

$$\begin{aligned} \int_{r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi _{1}(x,s)s^{\frac{n}{p}}}{ t^{\frac{n}{q}}}\,\frac{dt}{t} \le c_{0} \varphi_{2}(x,r), \end{aligned}$$
(1.5)

where \(c_{0}\) does not depend on x and r. Then the operator \([b,\mathcal{I}_{\beta}^{L}]\) is bounded from \(M_{p,\varphi_{1}}^{\alpha,V}\) to \(M_{q,\varphi_{2}}^{\alpha,V}\) for \(p>1\) and from \(M_{1,\varphi_{1}}^{\alpha,V}\) to \(WM_{\frac{n}{n-\beta-\nu},\varphi _{2}}^{\alpha,V}\). Moreover, for \(p>1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{M_{q,\varphi_{2}}^{\alpha,V}}\le C [b]_{\theta} \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}}, \end{aligned}$$

and for \(p=1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V}}\le C \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha,V}}, \end{aligned}$$

where C does not depend on f.

Theorem 1.2

Let \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(V\in RH_{q_{1}}\), \(q_{1}>n/2\), \(0<\nu<1\), \(\alpha\ge0\), \(b \in\Lambda_{\nu}^{\theta }(\rho)\), \(1< p<n/(\beta+\nu)\), \(1/q=1/p-(\beta+\nu)/n\), and let \(\varphi_{1}\in\Omega_{p,1}^{\alpha ,V}\), \(\varphi_{2} \in\Omega _{q,1}^{\alpha ,V}\) satisfy the conditions

$$\begin{aligned} c_{\delta}:= \int_{\delta}^{\infty}\sup_{x\in{\mathbb {R}^{n}}} \varphi_{1}(x,t) \,\frac{dt}{t} < \infty \end{aligned}$$

for every \(\delta>0\), and

$$\begin{aligned} \int_{r}^{\infty} \varphi_{1}(x,t)\, \frac{dt}{t^{1-\beta-\nu}} \leq C_{0} \varphi_{2}(x,r), \end{aligned}$$
(1.6)

where \(C_{0}\) does not depend on \(x\in{\mathbb{R}^{n}}\) and \(r>0\). Then the operator \([b,\mathcal{I}_{\beta}^{L}]\) is bounded from \(VM_{p,\varphi _{1}}^{\alpha ,V}\) to \(VM_{q,\varphi_{2}}^{\alpha ,V}\) for \(p>1\) and from \(VM_{1,\varphi_{1}}^{\alpha ,V}\) to \(VWM_{\frac{n}{n-\beta-\nu },\varphi _{2}}^{\alpha ,V}\).

Remark 1.2

Note that, in the case of \(V \equiv0\), \(\nu=0\) Corollary 1.1 and Theorem 1.2 were proved in [19, Corollary 5.5 and 7.5] and in the case of \(\varphi(x,r)=r^{(\lambda-n)/p}\), \(\nu=0\) in [32, Theorems 1.3 and 1.4].

In this paper, we shall use the symbol \(A\lesssim B\) to indicate that there exists a universal positive constant C, independent of all important parameters, such that \(A\le CB\). \(A \approx B\) means that \(A\lesssim B\) and \(B\lesssim A\).

Some technical lemmas and propositions

We would like to recall the important properties concerning the critical function.

Lemma 2.1

([29])

Let \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\). For the associated function ρ there exist C and \(k_{0}\ge1\) such that

$$\begin{aligned} C^{-1}\rho(x) \biggl(1+\frac{ \vert x-y \vert }{\rho(x)} \biggr)^{-k_{0}}\le\rho (y)\le C\rho(x) \biggl(1+\frac{ \vert x-y \vert }{\rho(x)} \biggr)^{\frac{k_{0}}{1+k_{0}}} \end{aligned}$$
(2.1)

for all \(x, y\in{\mathbb{R}}^{n}\).

Lemma 2.2

([2])

Suppose \(x\in B(x_{0},r)\). Then for \(k\in N\) we have

$$\begin{aligned} \frac{1}{ (1+\frac{2^{k} r}{\rho(x)} )^{N}}\lesssim \frac{1}{ (1+\frac{2^{k} r}{\rho(x_{0})} )^{N/(k_{0}+1)}}. \end{aligned}$$

According to [5], the new BMO space \(BMO_{\theta}(\rho)\) with \(\theta\ge0\) is defined as a set of all locally integrable functions b such that

$$\begin{aligned} \frac{1}{ \vert B(x,r) \vert } \int_{B(x,r)} \bigl\vert b(y)-b_{B} \bigr\vert \,dy \le C \biggl(1+\frac {r}{\rho(x)} \biggr)^{\theta} \end{aligned}$$

for all \(x\in\mathbb{R}^{n}\) and \(r>0\), where \(b_{B}=\frac{1}{|B|}\int _{B} b(y)\,dy\). A norm for \(b \in BMO_{\theta}(\rho)\), denoted by \([b]_{\theta}\), is given by the infimum of the constants in the inequalities above. Clearly, \(BMO\subset BMO_{\theta}(\rho)\).

Let \(\theta>0\) and \(0<\nu<1\), a seminorm on the Campanato class \(\Lambda_{\nu}^{\theta}(\rho)\) is denoted by \([b]_{\nu}^{\theta}\),

$$\begin{aligned}{} [b]_{\nu}^{\theta}:=\sup_{x\in{\mathbb{R}^{n}}, r>0} \frac{\frac {1}{ \vert B(x,r) \vert ^{1+\nu/n}}\int_{B(x,r)} \vert b(y)-b_{B} \vert \,dy}{(1+\frac {r}{\rho(x)})^{\theta}}< \infty. \end{aligned}$$

The Lipschitz space, associated with the Schrödinger operator (see [22]), consists of the functions f satisfying

$$\begin{aligned} \Vert f \Vert _{\mathrm{Lip}_{\nu}^{\theta}(\rho)}:=\sup_{x\in{\mathbb{R}^{n}}, r>0} \frac{ \vert f(x)-f(y) \vert }{ \vert x-y \vert ^{\nu} (1+\frac{ \vert x-y \vert }{\rho(x)}+\frac { \vert x-y \vert }{\rho(y)} )^{\theta}}< \infty. \end{aligned}$$

It is easy to see that this space is exactly the Lipschitz space when \(\theta=0\).

Note that if \(\theta=0\) in (1.2), \(\Lambda_{\nu}^{\theta }(\rho)\) is exactly the classical Campanato space; if \(\nu=0\), \(\Lambda_{\nu}^{\theta}(\rho)\) is exactly the space \(BMO_{\theta}(\rho)\); if \(\theta=0\) and \(\nu=0\), it is exactly the John–Nirenberg space \(BMO\).

The following relations between \(\mathrm{Lip}_{\nu}^{\theta}(\rho)\) and \(\Lambda_{\nu}^{\theta}(\rho)\) were proved in [22, Theorem 5].

Lemma 2.3

([22])

Let \(\theta>0\) and \(0<\nu<1\). Then following embedding is valid:

$$\begin{aligned} \Lambda_{\nu}^{\theta}(\rho) \subseteq\mathrm{Lip}_{\nu}^{\theta }( \rho) \subseteq\Lambda_{\nu}^{(k_{0}+1)\theta}(\rho), \end{aligned}$$

where \(k_{0}\) is the constant appearing in Lemma 2.1.

We give some inequalities about the Campanato space, associated with the Schrödinger operator \(\Lambda_{\nu}^{\theta}(\rho)\).

Lemma 2.4

([22])

Let \(\theta>0\) and \(1\le s <\infty\). If \(b\in\Lambda_{\nu}^{\theta}(\rho)\), then there exists a positive constant C such that

$$\begin{aligned} \biggl(\frac{1}{ \vert B \vert } \int_{B} \bigl\vert b(y)-b_{B} \bigr\vert ^{s} \,dy \biggr)^{1/s} \le C [b]_{\nu}^{\theta} r^{\nu} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\theta'} \end{aligned}$$

for all \(B=B(x,r)\), with \(x\in{\mathbb{R}}^{n}\) and \(r>0\), where \(\theta '=(k_{0}+1)\theta\) and \(k_{0}\) is the constant appearing in (2.1).

Let \(K_{\beta}\) be the kernel of \(\mathcal{I}_{\beta}^{L}\). The following result gives the estimate on the kernel \(K_{\beta}(x,y)\).

Lemma 2.5

([6])

If \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\), then, for every N, there exists a constant C such that

$$\begin{aligned} \bigl\vert K_{\beta}(x,y) \bigr\vert \le \frac{C}{ (1+\frac{ \vert x-y \vert }{\rho(x)} )^{N}}\frac{1}{ \vert x-y \vert ^{n-\beta}}. \end{aligned}$$
(2.2)

Finally, we recall a relationship between an essential supremum and an essential infimum.

Lemma 2.6

([34])

Let f be a real-valued nonnegative function and measurable on E. Then

$$\begin{aligned} \Bigl(\mathop{\operatorname{ess\inf}}\limits _{x\in E} f(x) \Bigr)^{-1}= \mathop{ \operatorname {ess\sup}}\limits _{x\in E}\frac{1}{f(x)}. \end{aligned}$$

It is natural, first of all, to find conditions ensuring that the spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\) and \(M_{p,\varphi }^{\alpha,V}\) are nontrivial, that is, consist not only of functions equivalent to 0 on \({\mathbb{R}^{n}}\).

Lemma 2.7

Let \(x_{0} \in{\mathbb{R}^{n}}\), \(\varphi(x,r)\) be a positive measurable function on \(\mathbb{R}^{n}\times(0,\infty)\), \(1\le p<\infty\), \(\alpha\ge0\), and \(V\in RH_{q}\), \(q\ge1\). If

$$\begin{aligned} \sup_{t< r< \infty} \biggl(1+\frac{r}{\rho(x_{0})} \biggr)^{\alpha} \frac{r^{-\frac{n}{p}}}{\varphi(x_{0},r)}=\infty\quad\textrm{for some } t>0, \end{aligned}$$
(2.3)

then \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}({\mathbb{R}^{n}})=\Theta\), where Θ is the set of all functions equivalent to 0 on \({\mathbb{R}^{n}}\).

Proof

Let (2.4) be satisfied and f be not equivalent to zero. Then \(\Vert f \Vert _{L_{p}(B(x_{0},t))}>0 \), hence

$$\begin{aligned} \Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}}&\geq\sup_{ t< r< \infty} \biggl(1+ \frac{r}{\rho(x_{0})} \biggr)^{\alpha} \varphi (x_{0},r)^{-1} r^{-\frac{n}{p}} \Vert f \Vert _{L_{p}(B(x_{0},r))} \\ & \geq \Vert f \Vert _{L_{p}(B(x_{0},t))}\sup_{ t< r< \infty} \biggl(1+ \frac {r}{\rho(x_{0})} \biggr)^{\alpha} \varphi(x_{0},r)^{-1} r^{-\frac{n}{p}}. \end{aligned}$$

Therefore \(\Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\} }}=\infty\). □

Remark 2.1

We denote by \(\Omega_{p,loc}^{\alpha ,V}\) the sets of all positive measurable functions φ on \({\mathbb{R}^{n}}\times(0,\infty)\) such that, for all \(t>0\),

$$\sup_{x\in{\mathbb{R}^{n}}} \biggl\Vert \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac{r^{-\frac{n}{p}}}{\varphi(x,r)} \biggr\Vert _{L_{\infty }(t,\infty )} < \infty . $$

In what follows, keeping in mind Lemma 2.7, for the non-triviality of the space \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in\Omega_{p,loc}^{\alpha ,V}\).

Lemma 2.8

([2])

Let \(\varphi(x,r) \) be a positive measurable function on \({\mathbb{R}^{n}}\times(0,\infty )\), \(1\le p<\infty\), \(\alpha\ge0\), and \(V\in RH_{q}\), \(q\ge1\).

  1. (i)

    If

    $$\begin{aligned} \sup_{t< r< \infty } \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac {r^{-\frac{n}{p}}}{\varphi(x,r)}=\infty\quad\textit{for some } t>0 \textit{ and for all } x\in{\mathbb{R}^{n}}, \end{aligned}$$
    (2.4)

    then \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})=\Theta\).

  2. (ii)

    If

    $$\begin{aligned} \sup_{ 0< r< \tau} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \varphi (x,r)^{-1} = \infty\quad\textit{ for some } \tau>0\textit{ and for all } x\in{\mathbb{R}^{n}}, \end{aligned}$$
    (2.5)

    then \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})=\Theta\).

Remark 2.2

We denote by \(\Omega_{p}^{\alpha ,V}\) the sets of all positive measurable functions φ on \({\mathbb{R}^{n}}\times(0,\infty)\) such that, for all \(t>0\),

$$\begin{aligned} &\sup_{x\in{\mathbb{R}^{n}}} \biggl\Vert \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac{r^{-\frac{n}{p}}}{\varphi(x,r)} \biggr\Vert _{L_{\infty }(t,\infty )} < \infty , \quad \mbox{and} \\ & \sup_{x\in{\mathbb{R}^{n}}} \biggl\Vert \biggl(1+ \frac{r}{\rho(x)} \biggr)^{\alpha } \varphi(x,r)^{-1} \biggr\Vert _{L_{\infty }(0, t)}< \infty , \end{aligned}$$

respectively. In what follows, keeping in mind Lemma 2.8, for the non-triviality of the space \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in \Omega_{p}^{\alpha ,V}\).

Remark 2.3

We denote by \(\Omega_{p,1}^{\alpha ,V}\) the sets of all positive measurable functions φ on \({\mathbb{R}^{n}}\times(0,\infty )\) such that

$$\begin{aligned} \inf_{x\in{\mathbb{R}^{n}}} \inf_{r>\delta} \biggl(1+\frac{r}{\rho (x)} \biggr)^{-\alpha } \varphi(x,r)>0, \quad \mbox{for some } \delta>0, \end{aligned}$$
(2.6)

and

$$\begin{aligned} \lim_{r \to0} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac {r^{n/p}}{\varphi(x,r)} = 0. \end{aligned}$$

For the non-triviality of the space \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in \Omega_{p,1}^{\alpha ,V}\).

Proof of Theorem 1.1

We first prove the following conclusions.

Lemma 3.1

Let \(0<\nu<1\), \(0< \beta+\nu<n\) and \(b\in\Lambda_{\nu}^{\theta }(\rho)\), then the following pointwise estimate holds:

$$\bigl\vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x) \bigr\vert \lesssim[b]_{\nu }^{\theta} I_{\beta+\nu} \bigl( \vert f \vert \bigr) (x). $$

Proof

Note that

$$\begin{aligned} \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x) &= b(x) \mathcal{I}_{\beta }^{L}(f) (x) - \mathcal{I}_{\beta}^{L}(bf) (x) \\ &= \int_{{\mathbb{R}^{n}}} \bigl[b(x)-b(y) \bigr] K_{\beta}(x,y) f(y) \,dy. \end{aligned}$$

If \(b\in\Lambda_{\nu}^{\theta}(\rho)\), then from Lemma 2.5 we have

$$\begin{aligned} \begin{aligned} \bigl\vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x) \bigr\vert & \le \int_{{\mathbb{R}^{n}}} \bigl\vert b(x)-b(y) \bigr\vert \bigl\vert K_{\beta}(x,y) \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\ & \lesssim[b]_{\nu}^{\theta} \int_{{\mathbb{R}^{n}}} \vert x-y \vert ^{\nu} \bigl\vert K_{\beta }(x,y) \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\ &= [b]_{\nu}^{\theta} I_{\beta+\nu} \bigl( \vert f \vert \bigr) (x). \end{aligned} \end{aligned}$$

 □

From Lemma 3.1 we get the following.

Corollary 3.1

Suppose \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\) and \(b\in\Lambda_{\nu }^{\theta}(\rho)\) with \(0<\nu<1\). Let \(0< \beta+\nu<n\) and let \(1\le p< q<\infty\) satisfy \(1/q=1/p-(\beta+\nu)/n\). Then for all f in \(L_{p}({\mathbb{R}^{n}})\) we have

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}({\mathbb{R}^{n}})} \lesssim \Vert f \Vert _{L_{p}({\mathbb{R}^{n}})} \end{aligned}$$

when \(p>1\), and also

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WL_{q}({\mathbb{R}^{n}})} \lesssim \Vert f \Vert _{L_{1}({\mathbb{R}^{n}})} \end{aligned}$$

when \(p=1\).

In order to prove Theorem 1.1, we need the following.

Theorem 3.1

Suppose \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\), \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(\theta>0\), \(0<\nu<1\). Let \(0< \beta+\nu<n\) and let \(1\le p< q<\infty\) satisfy \(1/q=1/p-(\beta+\nu)/n\) then the inequality

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}(B(x_{0},r))} &\lesssim \bigl\Vert I_{\beta +\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{q}(B(x_{0},r))}\\ &\lesssim r^{\frac{n}{q}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \end{aligned}$$

holds for any \(f\in L_{loc}^{p}({\mathbb{R}}^{n})\). Moreover, for \(p=1\) the inequality

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L}f \bigr] \bigr\Vert _{WL_{\frac{n}{n-\beta-\nu }}(B(x_{0},r))} & \lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WL_{\frac {n}{n-\beta-\nu}}(B(x_{0},r))} \\ & \lesssim r^{n-\beta} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t} \end{aligned}$$

holds for any \(f\in L_{loc}^{1}({\mathbb{R}}^{n})\).

Proof

For arbitrary \(x_{0}\in{\mathbb{R}}^{n}\), set \(B=B(x_{0},r)\) and \(\lambda B=B(x_{0},\lambda r)\) for any \(\lambda>0\). We write f as \(f=f_{1}+f_{2}\), where \(f_{1}(y)=f(y)\chi_{{B(x_{0},2r)}}(y)\), and \(\chi_{{B(x_{0},2r)}}\) denotes the characteristic function of \(B(x_{0},2r)\). Then

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}(B(x_{0},r))} & \lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{q}(B(x_{0},r))} \\ & \le \Vert I_{\beta+\nu}f_{1} \Vert _{L_{q}(B(x_{0},r))}+ \Vert I_{\beta+\nu}f_{2} \Vert _{L_{q}(B(x_{0},r))}. \end{aligned}$$

Since \(f_{1}\in L_{p}({\mathbb{R}^{n}})\) and from the boundedness of \(I_{\beta+\nu}\) from \(L_{p}({\mathbb{R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) (see [31]) it follows that

$$\begin{aligned} \Vert I_{\beta+\nu}f_{1} \Vert _{L_{q}(B(x_{0},r))} &\lesssim \Vert f \Vert _{L_{p}(B(x_{0},2r))} \\ & \lesssim r^{\frac{n}{q}} \Vert f \Vert _{L_{p}(B(x_{0},2r))} \int_{2r}^{\infty}\frac{dt}{t^{\frac{n}{q}+1}} \lesssim r^{\frac{n}{q}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}} \, \frac{dt}{t}. \end{aligned}$$
(3.1)

To estimate \(\|I_{\beta+\nu}f_{2}\|_{L_{p}(B(x_{0},r))}\), the obverse of \(x\in B\), \(y\in(2B)^{c}\) implies \(|x-y|\approx|x_{0}-y|\). Then by (2.2) we have

$$\begin{aligned} \sup_{x\in B} \bigl\vert I_{\beta+\nu}f_{2}(x) \bigr\vert & \lesssim \int_{(2B)^{c}}\frac { \vert f(y) \vert }{ \vert x_{0}-y \vert ^{n-\beta-\nu}}\,dy \lesssim\sum _{k=1}^{\infty}\bigl(2^{k+1} r \bigr)^{-n+\beta} \int_{2^{k+1}B} \bigl\vert f(y) \bigr\vert \,dy. \end{aligned}$$

By Hölder’s inequality we get

$$\begin{aligned} \sup_{x\in B} \bigl\vert I_{\beta+\nu}f_{2}(x) \bigr\vert & \lesssim\sum_{k=1}^{\infty} \Vert f \Vert _{L_{p}(2^{k+1}B)} \bigl(2^{k+1} r \bigr)^{-1-\frac{n}{p}+\beta} \int_{2^{k} r}^{2^{k+1}r} \,dt \\ & \lesssim\sum_{k=1}^{\infty}\int_{2^{k} r}^{2^{k+1}r} \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \lesssim \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac {n}{q}}}\, \frac{dt}{t}. \end{aligned}$$
(3.2)

Then

$$\begin{aligned} \Vert I_{\beta+\nu}f_{2} \Vert _{L_{q}(B(x_{0},r))} \lesssim r^{\frac{n}{q}} \int _{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \end{aligned}$$
(3.3)

holds for \(1\le p< n/\beta\). Therefore, by (3.1) and (3.3) we get

$$\begin{aligned} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{q}(B(x_{0},r))}\lesssim r^{\frac{n}{q}} \int _{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \end{aligned}$$
(3.4)

for \(1< p<n/\beta\).

When \(p=1\), by the boundedness of \(I_{\beta+\nu}\) from \(L_{1}({\mathbb{R}}^{n})\) to \(WL_{\frac{n}{n-\beta-\nu}}({\mathbb{R}}^{n})\), we get

$$\begin{aligned} \Vert I_{\beta+\nu}f_{1} \Vert _{WL_{\frac{n}{n-\beta-\nu}}(B(x_{0},r))} & \lesssim \Vert f \Vert _{L_{1}(B(x_{0},2r))} \lesssim r^{n-\beta-\nu} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t}. \end{aligned}$$

By (3.3) we have

$$\begin{aligned} \Vert I_{\beta+\nu}f_{2} \Vert _{WL_{\frac{n}{n-\beta-\nu}}(B(x_{0},r))} & \le \Vert I_{\beta+\nu}f_{2} \Vert _{L_{\frac{n}{n-\beta-\nu}} (B(x_{0},2r))} \\ & \lesssim r^{n-\beta-\nu} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t}. \end{aligned}$$

Then

$$\begin{aligned} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WL_{\frac{n}{n-\beta-\nu}}(B(x_{0},r))} \lesssim r^{n-\beta-\nu} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t}. \end{aligned}$$

 □

Proof of Theorem 1.1

From Lemma 2.6, we have

$$\begin{aligned} \frac{1}{\operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}} =\mathop{\operatorname{ess\sup}}_{t< s< \infty}\frac{1}{\varphi _{1}(x,s)s^{\frac{n}{p}}}. \end{aligned}$$

Note the fact that \(\|f\|_{L_{p}(B(x_{0},t))}\) is a nondecreasing function of t, and \(f\in M_{p,\varphi_{1}}^{\alpha,V}\), then

$$\begin{aligned} \frac{ (1+\frac{t}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},t))}}{ \operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}}& \lesssim\operatorname{ess\sup}_{t< s< \infty} \frac{ (1+\frac {t}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},t))}}{ \varphi_{1}(x,s)s^{\frac{n}{p}}} \\ & \lesssim\sup_{0< s< \infty} \frac{ (1+\frac{s}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},s))}}{ \varphi_{1}(x,s)s^{\frac{n}{p}}} \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}}. \end{aligned}$$

Since \(\alpha\ge0\), and \((\varphi_{1},\varphi_{2})\) satisfies the condition (1.5),

$$\begin{aligned} & \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}} \, \frac{dt}{t} \\ &\quad = \int_{2r}^{\infty}\frac{ (1+\frac{t}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},t))}}{ \operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac {n}{p}}}\frac{\operatorname{ess\inf} _{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}}{ (1+\frac{t}{\rho(x_{0})} )^{\alpha}t^{\frac{n}{q}}} \,\frac{dt}{t} \\ &\quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}} \int_{2r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}}{ (1+\frac{t}{\rho(x_{0})} )^{\alpha}t^{\frac{n}{q}}}\,\frac{dt}{t} \\ &\quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{-\alpha} \int_{r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi _{1}(x,s)s^{\frac{n}{p}}}{ t^{\frac{n}{q}}}\,\frac{dt}{t} \\ &\quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{-\alpha} \varphi_{2}(x_{0},r). \end{aligned}$$
(3.5)

Then by Theorem 3.1 we get

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{M_{q,\varphi_{2}}^{\alpha,V}} & \lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{M_{q,\varphi_{2}}^{\alpha,V}} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} r^{-n/q} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{p}(B(x_{0},r))} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \\ & \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}}. \end{aligned}$$

Let \(q=\frac{n}{n-\beta-\nu}\), similar to the estimates of (3.5) we have

$$\begin{aligned} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\,\frac{dt}{t} \lesssim \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha,V}} \biggl(1+\frac{r}{\rho(x_{0})} \biggr)^{-\alpha} \varphi_{2}(x_{0},r). \end{aligned}$$

Thus by Theorem 3.1 we get

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V}} &\lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WM_{\frac {n}{n-\beta-\nu},\varphi_{2}}^{\alpha,V}} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} r^{\beta-n} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WL_{\frac {n}{n-\beta-\nu}}(B(x_{0},r))} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t} \\ & \lesssim \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha,V}}. \end{aligned}$$

 □

Proof of Theorem 1.2

The statement is derived from the estimate (3.4). The estimation of the norm of the operator, that is, the boundedness in the non-vanishing space, immediately follows by Theorem 1.1. So we only have to prove that

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{p,\varphi_{1}}^{\alpha ,V}(f;x,r)=0\quad \Rightarrow\quad \lim _{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{q,\varphi_{2}}^{\alpha ,V} \bigl( \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f;x,r \bigr)=0 \end{aligned}$$
(4.1)

and

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{1,\varphi_{1}}^{\alpha ,V}(f;x,r)=0\quad \Rightarrow\quad \lim _{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{n/(n-\beta),\varphi_{2}}^{W,\alpha ,V} \bigl( \bigl[b,\mathcal {I}_{\beta}^{L} \bigr]f;x,r \bigr)=0. \end{aligned}$$
(4.2)

To show that \(\sup_{x\in{\mathbb{R}^{n}}} (1+\frac{r}{\rho (x)} )^{\alpha } \varphi_{2}(x,r)^{-1} r^{-n/p} \|[b,\mathcal{I}_{\beta}^{L}]f\| _{L_{q}(B(x,r))}<\varepsilon\) for small r, we split the right-hand side of (3.4):

$$\begin{aligned} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \varphi_{2}(x,r)^{-1} r^{-n/p} \bigl\Vert \bigl[b, \mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}(B(x,r))}\leq C \bigl[I_{\delta_{0}}(x,r)+J_{\delta_{0}}(x,r) \bigr], \end{aligned}$$
(4.3)

where \(\delta_{0}>0\) (we may take \(\delta_{0}>1\)), and

$$\begin{aligned} I_{\delta_{0}}(x,r):=\frac{ (1+\frac{r}{\rho(x)} )^{\alpha }}{\varphi_{2}(x,r)} \int_{r}^{\delta_{0}} t^{-\frac{n}{q}-1} \Vert f \Vert _{L_{p}(B(x,t))} \,dt \end{aligned}$$

and

$$\begin{aligned} J_{\delta_{0}}(x,r):=\frac{ (1+\frac{r}{\rho(x)} )^{\alpha }}{\varphi_{2}(x,r)} \int_{\delta_{0}}^{\infty } t^{-\frac{n}{q}-1} \Vert f \Vert _{L_{p}(B(x,t))} \,dt \end{aligned}$$

and it is supposed that \(r<\delta_{0}\). We use the fact that \(f \in VM_{p,\varphi_{1}}^{\alpha ,V}({\mathbb{R}^{n}})\) and choose any fixed \(\delta_{0}>0\) such that

$$\begin{aligned} \sup_{x\in{\mathbb{R}^{n}}} \biggl(1+\frac{t}{\rho(x)} \biggr)^{\alpha } \varphi_{1}(x,t)^{-1} t^{-n/p} \Vert f \Vert _{L_{p}(B(x,t))}< \frac{\varepsilon}{2CC_{0}}, \end{aligned}$$

where C and \(C_{0}\) are constants from (1.6) and (4.3). This allows one to estimate the first term uniformly in \(r\in (0,\delta_{0})\):

$$\begin{aligned} \sup_{x\in{\mathbb{R}^{n}}}CI_{\delta_{0}}(x,r)< \frac{\varepsilon}{2},\quad 0< r< \delta_{0}. \end{aligned}$$

The estimation of the second term now my be made already by the choice of r sufficiently small. Indeed, thanks to the condition (2.6) we have

$$\begin{aligned} J_{\delta_{0}}(x,r)\leq c_{\sigma_{0}} \frac{ (1+\frac{r}{\rho (x)} )^{\alpha }}{\varphi_{1}(x,r)} \Vert f \Vert _{VM_{p,\varphi _{1}}^{\alpha ,V}}, \end{aligned}$$

where \(c_{\sigma_{0}}\) is the constant from (1.3). Then by (2.6) it suffices to choose r small enough such that

$$\begin{aligned} \sup_{x\in{\mathbb{R}^{n}}}\frac{ (1+\frac{r}{\rho(x)} )^{\alpha }}{\varphi_{2}(x,r)} \leq\frac{\varepsilon}{2c_{\sigma_{0}} \Vert f \Vert _{VM_{p,\varphi_{1}}^{\alpha ,V}}}, \end{aligned}$$

which completes the proof of (4.1).

The proof of (4.2) is similar to the proof of (4.1).

Conclusions

In this paper, we study the boundedness of the commutators \([b,\mathcal {I}_{\beta}^{L}]\) with \(b \in\Lambda_{\nu}^{\theta}(\rho)\) on local generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\), generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}\) and vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha,V}\) associated with the Schrödinger operator, respectively. When b belongs to \(\Lambda_{\nu}^{\theta}(\rho)\) with \(\theta >0\), \(0<\nu<1\) and \((\varphi_{1},\varphi_{2})\) satisfies some conditions, we show that the commutator operator \([b,\mathcal {I}_{\beta}^{L}]\) are bounded from \(LM_{p,\varphi_{1}}^{\alpha ,V,\{ x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha ,V,\{x_{0}\}}\), from \(M_{p,\varphi_{1}}^{\alpha ,V}\) to \(M_{q,\varphi_{2}}^{\alpha ,V}\) and from \(VM_{p,\varphi_{1}}^{\alpha ,V}\) to \(VM_{q,\varphi_{2}}^{\alpha ,V}\), \(1/p-1/q=(\beta+\nu)/n\).

Our results about the boundedness of \([b,\mathcal{I}_{\beta}^{L}]\) with \(b \in\Lambda_{\nu}^{\theta}(\rho)\) from \(LM_{p,\varphi _{1}}^{\alpha ,V,\{x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha ,V,\{x_{0}\}}\) (Theorem 1.1) are based on the local estimate for the commutators \([b,\mathcal {I}_{\beta}^{L}]\) (Theorem 3.1).

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Acknowledgements

The authors thank the referees for careful reading the paper and useful comments.

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Funding

The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.17.020). The research of V.S. Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.17.008), by the grant of 1st Azerbaijan–Russia Joint Grant Competition (the Agreement number No. 18-51-06005) and by the Ministry of Education and Science of the Russian Federation (Agreement number: 02.a03.21.0008).

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This work was carried out in collaboration between all authors. VSG raised these interesting problems in the research. VSG and AA proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.

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Correspondence to Ali Akbulut.

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Guliyev, V.S., Akbulut, A. Commutator of fractional integral with Lipschitz functions associated with Schrödinger operator on local generalized Morrey spaces. Bound Value Probl 2018, 80 (2018). https://doi.org/10.1186/s13661-018-1002-2

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MSC

  • 42B35
  • 35J10
  • 47H50

Keywords

  • Schrödinger operator
  • Fractional integral
  • Commutator
  • Lipschitz function
  • Local generalized Morrey space