Abstract
Let a function b belong to the space \(\operatorname{BMO}_{\theta }(\rho )\), which is larger than the space \(\operatorname{BMO}(\mathbb{R}^{n})\), and let a nonnegative potential V belong to the reverse Hölder class \(\mathit{RH}_{s}\) with \(n/2< s< n\), \(n\geq 3\). Define the commutator \([b,T_{\beta }]f=bT_{ \beta }f-T_{\beta }(bf)\), where the operator \(T_{\beta }=V^{\alpha } \nabla \mathcal{L}^{-\beta }\), \(\beta -\alpha =\frac{1}{2}\), \(\frac{1}{2}< \beta \leq 1\), and \(\mathcal{L}=-\Delta +V\) is the Schrödinger operator. We have obtained the \(L^{p}\)-boundedness of the commutator \([b,T_{\beta }]f\) and we have proved that the commutator is bounded from the Hardy space \(H^{1}_{\mathcal{L}}(\mathbb{R}^{n})\) into weak \(L^{1}(\mathbb{R}^{n})\).
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1 Introduction and results
Let \(\mathcal{L}=-\Delta +V\) be the Schrödinger operator, where the nonnegative potential V belongs to the reverse Hölder class \(\mathit{RH}_{s}\) with \(s> n/2\), \(n\geq 3\). Many papers related to Schrödinger operator have appeared (see [1,2,3,4,5]). In recent years, some researchers have studied the boundedness of the commutators generated by the operators associated with \(\mathcal{L}\) and the BMO type space (see [6,7,8,9]). In this paper, we investigated the boundedness of the commutator \([b,T_{\beta}]\), where \(T_{\beta }=V^{\alpha}\nabla\mathcal{L}^{-\beta} \) and the function \(b\in \operatorname{BMO}_{\theta }(\rho )\). We note that the space \(\operatorname{BMO}_{\theta }(\rho )\) is larger than the space \(\operatorname{BMO}(\mathbb{R}^{n})\).
For \(s>1\), a nonnegative locally \(L^{s}\)-integrable function V is said to belong to \(\mathit{RH}_{s}\) if there exists a constant \(C>0\) such that the reverse Hölder inequality
holds for every ball \(B\subset \mathbb{R}^{n}\). It is obvious that \(\mathit{RH}_{s_{1}}\subseteq \mathit{RH}_{s_{2}}\) for \(s_{1}\geq s_{2}\).
As in [2], for a given potential \(V\in \mathit{RH}_{s}\) with \(s>n/2\), we will use the auxiliary function \(\rho (x)\) defined as
It is well known that \(0<\rho (x)<\infty \) for any \(x\in \mathbb{R} ^{n}\).
Let \(\mathcal{L}=-\Delta +V\) be the Schrödinger operator on \(\mathbb{R}^{n}\), where \(V\in \mathit{RH}_{s}\) with \(s>n/2\) and \(n\geq 3\). We know \(\mathcal{L}\) generates a \((C_{0})\) semigroup \(\{e^{-t \mathcal{L}}\}_{t>0}\). The maximal function with respect to the semigroup \(\{e^{-t\mathcal{L}}\}_{t>0}\) is defined by \(M^{\mathcal{L}}f(x)= \sup_{t>0}|e^{-t\mathcal{L}}f(x)|\). The Hardy space associated with \(\mathcal{L}\) is defined as follows (see [3, 4]).
Definition 1
We say that f is an element of \(H_{\mathcal{L}}^{1}(\mathbb{R}^{n})\) if the maximal function \(M^{\mathcal{L}}f\) belongs to \(L^{1}( \mathbb{R}^{n})\). The quasi-norm of f is defined by
Definition 2
Let \(1< q\leq \infty \). A measurable function a is called an \((1,q)_{\rho }\)-atom related to the ball \(B(x_{0},r)\) if \(r<\rho (x _{0})\) and the following conditions hold:
-
(1)
\(\operatorname{supp} a\subset B(x_{0},r)\);
-
(2)
\(\|a\|_{L^{q}(\mathbb{R}^{n})}\leq |B(x_{0},r)|^{1/q-1}\);
-
(3)
\(\int _{B(x_{0},r)}a(x)\,dx=0\) if \(r<\rho (x_{0})/4\).
The space \(H^{1}_{\mathcal{L}}(\mathbb{R}^{n})\) admits the following atomic decomposition (see [3, 4]).
Proposition 1
Let \(f\in L^{1}(\mathbb{R}^{n})\). Then \(f\in H_{\mathcal{L}}^{1}( \mathbb{R}^{n})\) if and only if f can be written as \(f=\sum_{j} \lambda _{j}a_{j}\), where \(a_{j}\) are \((1,q)_{\rho }\)- atoms, \(\sum_{j}|\lambda _{j}|<\infty \), and the sum converges in the \(H_{\mathcal{L}}^{1}(\mathbb{R}^{n})\) quasi-norm. Moreover
where the infimum is taken over all atomic decompositions of f into \((1,q)_{\rho }\)- atoms.
Following [10], the space \(\operatorname{BMO}_{\theta }(\rho )\) with \(\theta \geq 0\) is defined as the set of all locally integrable functions b such that
for all \(x\in \mathbb{R}^{n}\) and \(r>0\), where \(b_{B}=\frac{1}{|B|} \int _{B}b(y)\,dy\). A norm for \(b\in \operatorname{BMO}_{\theta }(\rho )\), denoted by \([b]_{\theta }\), is given by the infimum of the constants in the inequalities above. Clearly, \(\operatorname{BMO}\subset \operatorname{BMO}_{\theta }(\rho )\).
We consider the operator
The boundedness of operator \(T_{1/2}\) and its commutator have been researched under the condition \(V\in \mathit{RH}_{s}\) for \(n/2< s< n\). In [2], Shen showed that \(T_{1/2}\) is bounded on \(L^{p}(\mathbb{R} ^{n}) \) for \(1< p< p_{0}\), \(\frac{1}{p_{0}}=\frac{1}{s}-\frac{1}{n}\). For \(b\in \operatorname{BMO}(\mathbb{R}^{n})\), Guo, Li and Peng [11] investigated the \(L^{p}\)-boundedness of commutator \([b,T_{1/2}]\) for \(1< p< p_{0}\); Li and Peng [12] studied the boundedness of \([b, T_{1/2}]\) from \(H_{\mathcal{L}}^{1}(\mathbb{R}^{n})\) into weak \(L^{1}(\mathbb{R}^{n})\). When \(b\in \operatorname{BMO}_{\theta }(\rho ) \), Bongioanni, Harboure and Salinas [10] obtained the \(L^{p}\)-boundedness of \([b,T_{1/2}]\) and Liu, Sheng and Wang [13] proved that \([b,T_{1/2}]\) is bounded from \(H_{\mathcal{L}}^{1}(\mathbb{R}^{n})\) to weak \(L^{1}(\mathbb{R}^{n})\). More boundedness of commutator \([b,T_{1/2}]\) can be found in [14] and [15].
For \(1/2<\beta \leq 1\), \(\beta -\alpha =1/2\), \(n/2< s< n\), Sugano [5] established the estimate for \(T^{*}_{\beta }\) (the adjoint operator of \(T_{\beta }\)), and proved that there exists a constant C such that
for all \(f\in C_{0}^{\infty }(\mathbb{R}^{n})\), where \(\frac{1}{{p} _{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}\), and \(\frac{1}{{p}_{ \alpha }}+ \frac{1}{{p}'_{\alpha }}=1\). Then, by the boundedness of maximal function, we get
Theorem 1
Suppose \(V\in \mathit{RH}_{s}\) with \(n/2< s< n\). Let \(1/2< \beta \leq 1\), \(\frac{1}{p _{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}\). Then
for \(p'_{\alpha }< p\leq \infty \), and by duality we get
for \(1\leq p< p_{\alpha }\).
Inspired by the above results, in the present work, we are interested in the boundedness of \([b,T_{\beta }]\). Our main results are as follows.
Theorem 2
Suppose \(V\in \mathit{RH}_{s}\) with \(n/2< s< n\). Let \(1/2< \beta \leq 1\), \(b \in \operatorname{BMO}_{\theta }(\rho )\). Then,
for \(p'_{\alpha }< p< \infty \), and
for \(1< p< p_{\alpha }\), where \(\frac{1}{p_{\alpha }}= \frac{\alpha +1}{s}-\frac{1}{n}\).
Theorem 3
Suppose \(V\in \mathit{RH}_{s}\) with \(n/2< s< n\). Let \(1/2<\beta \leq 1\), \(b\in \operatorname{BMO} _{\theta }(\rho )\). Then,
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 \(A\leq cB\). \(A\sim B\) means that \(A\lesssim B\) and \(B\lesssim A\).
2 Some preliminaries
We recall some important properties concerning the auxiliary function \(\rho (x)\) which have been proved by Shen [2]. Throughout this section we always assume \(V\in \mathit{RH}_{s}\) with \(n/2< s< n\).
Proposition 2
There exist constants C and \(k_{0}\geq 1\) such that
for all \(x,y \in \mathbb{R}^{n}\).
Assume that \(Q=B(x_{0},\rho (x_{0}))\), for any \(x\in Q \), then Proposition 2 tells us that \(\rho (x)\sim \rho (y)\), if \(|x-y|< C\rho (x)\). It is easy to get the following result from Proposition 2.
Lemma 1
Let \(k\in \mathbb{N}\) and \(x\in 2^{k+1}B(x_{0},r)\setminus 2^{k}B(x _{0},r)\). Then we have
Lemma 2
There exists a constant \(l_{0}>0\) such that
The following finite overlapping property was given by Dziubański and Zienkiewicz in [3].
Proposition 3
There exists a sequence of points \(\{x_{k}\}_{k=1}^{\infty }\) in \(\mathbb{R}^{n}\), so that the family of critical balls \(Q_{k}=B(x_{k}, \rho (x_{k}))\), \(k\geq 1\), satisfies
-
(i)
\(\bigcup_{k} Q_{k}=\mathbb{R}^{n}\).
-
(ii)
There exists \(N=N(\rho )\) such that for every \(k\in N\), \(\operatorname{card}\{j: 4Q_{j}\cap 4Q_{k}\}\leq N\).
For \(\alpha >0\), \(g\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\) and \(x\in \mathbb{R}^{n}\), we introduce the following maximal functions:
and
where \(\mathcal{B}_{\rho ,\alpha }=\{B(z,r): z\in \mathbb{R}^{n} \text{ and } r\leq \alpha \rho (y)\}\).
The following Fefferman–Stein type inequality can be found in [10].
Proposition 4
For \(1< p<\infty \), then there exist δ and γ such that if \(\{Q_{k}\}_{k}\) is a sequence of balls as in Proposition 3 then
for all \(g\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\).
We have the following result for the function \(b\in \operatorname{BMO}_{\theta }( \rho )\).
Lemma 3
([10])
Let \(1\leq s<\infty \), \(b\in \operatorname{BMO}_{\theta }(\rho )\), and \(B=B(x,r)\). Then
for all \(k\in \mathbb{N} \), with \(r>0\), where \(\theta '=(k_{0}+1) \theta \) and \(k_{0}\) is the constant appearing in Proposition 2.
We give an estimate of fundamental solutions; this result can be found in [2]. We denote by \(\varGamma (x,y,\lambda )\) the fundamental solution of \(-\Delta +(V(x)+i\lambda )\), and then \(\varGamma (x,y,\lambda )=\varGamma (y,x,-\lambda )\).
Lemma 4
Assume that \(-\Delta u+(V(x)+i\lambda )u=0\) in \(B(x_{0},2R)\) for some \(x_{0}\in \mathbb{R}^{n}\). Then, there exists a \(k'_{0}\) such that
where \(1/t=1/s-1/n\).
Suppose \(\mathcal{W}_{\beta }= \nabla \mathcal{L}^{-\beta }\). Let \(\mathcal{W}_{\beta }^{*}\) be the adjoint operator of \(\mathcal{W} _{\beta }\), K and \(K^{*}\) be the kernels of \(\mathcal{W}_{\beta }\) and \(\mathcal{W}_{\beta }^{*}\) respectively, then \(K(x,z)=K^{*}(z,x)\), and we have the following estimates.
Lemma 5
Suppose \(1/2<\beta \leq 1\).
-
(i)
For every N there exists a constant \(C_{N}\) such that
$$ \bigl\vert K^{*}(x,z) \bigr\vert \leq \frac{C_{N}}{ (1+\frac{ \vert x-z \vert }{\rho (x)} )^{N}} \frac{1}{ \vert x-z \vert ^{n-2\beta }} \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi +\frac{1}{ \vert x-z \vert } \biggr). $$Moreover, the inequality above also holds with \(\rho (x)\) replaced by \(\rho (z)\).
-
(ii)
For every N and \(0<\delta <\min \{1,2-n/q_{0}\}\) there exists a constant \(C_{N}\) such that
$$\begin{aligned} \bigl\vert K^{*}(x,z) -K^{*}(y,z) \bigr\vert &\leq \frac{C_{N}}{ (1+ \frac{ \vert x-z \vert }{\rho (x)} )^{N}} \\ &\quad {} \times \frac{ \vert x-y \vert ^{\delta }}{ \vert x-z \vert ^{n-2\beta +\delta }} \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi +\frac{1}{ \vert x-z \vert } \biggr) \end{aligned}$$whenever \(|x-y|<\frac{1}{16}|x-z|\). Moreover, the inequality above also holds with \(\rho (x)\) replaced by \(\rho (z)\).
Proof
The proof of (i) can be found in [5], page 449. Let us prove (ii). By (6) of [5] we know
Then
for \(\frac{1}{2}<\beta <1\) and
for \(\beta =1\).
Fix \(x,z\in \mathbb{R}^{n}\) and let \(R=|x-z|/8\), \(1/t=1/s-1/n\), \(\delta =2-n/s>0\). For any \(|x-y|< R/2\), it follows from the Morrey embedding theorem (see [16]) and Lemma 4 that
It follows from [11, p. 428] that
Then, by the fact that \(6R\leq |z-u|\leq 10R\), we get
Thus, for \(\beta =1\),
Note that
Then, for \(\frac{1}{2}<\beta <1\), we have
By Lemma 2, we know that the inequality above also holds with \(\rho (x)\) replaced by \(\rho (z)\). □
3 Proof of main results
Before proving Theorem 2, we need to give some necessary lemmas.
Lemma 6
Let \(V\in \mathit{RH}_{s}\) with \(n/2< s< n\), \(\frac{1}{{p}_{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}\), and \(b\in \operatorname{BMO}_{\theta }(\rho )\). Then, for any \({p}'_{\alpha }< t<\infty \), we have
for all \(f\in L^{t}_{\mathrm{loc}}(\mathbb{R}^{n})\) and every ball \(Q=B(x_{0}, \rho (x_{0}))\).
Proof
Let \(f\in L^{t}_{\mathrm{loc}}(\mathbb{R}^{n})\) and \(Q=B(x_{0},\rho (x _{0}))\). We consider
By Hölder’s inequality with \(t>{p}'_{\alpha }\) and Lemma 3,
Write \(f=f_{1}+f_{2}\) with \(f_{1}=f\chi _{2Q}\). By Theorem 1, we know that \(T^{*}_{\beta }\) is bounded on \(L^{t}(\mathbb{R}^{n})\) with \(t> {p}'_{\alpha } \), and then
For \(x\in Q\), using (i) in Lemma 5, we get
where
and
To deal with \(I_{2}(x)\), note that \(\rho (x)\sim \rho (x_{0})\) and \(|x-z|\sim |x_{0}-z|\) for \(x\in Q\). We split \((2Q)^{c}\) into annuli to obtain
Observe that \(\frac{1}{{p}'_{\alpha }}+\frac{\alpha }{s}+\frac{1}{q _{1}}=1\), \(\frac{1}{q_{1}}=\frac{1}{s}-\frac{1}{n}\), \(t> {p}'_{\alpha }\), and \(\beta -\alpha =1/2\). Then by Hölder’s inequality and the boundedness of fractional integral \(\mathcal{I}_{1}: L^{s}\rightarrow L^{q_{1}}\) with \(\frac{1}{q_{1}}=\frac{1}{s}-\frac{1}{n}\), we get
Then, since \(V\in \mathit{RH}_{s}\), from Lemma 2 and \(2\beta +n(1/s-1/ {q_{1}})-2\alpha -2=0\), we get
For \(I_{1}(x)\), we split \((2Q)^{c}\) into annuli to obtain
By Hölder’s inequality with \(\frac{1}{{p}'_{\alpha }}+\frac{ \alpha }{s}+\frac{1}{q_{1}}=1\), \(t> {p}'_{\alpha }\), \(\beta -\alpha =1/2\), and Lemma 2, we get
To deal with the second term of (1), we write again \(f=f_{1}+f_{2}\). Choosing \({p}'_{\alpha }<\bar{t}<t\) and denoting \(\nu =\frac{\bar{t} t}{t-\bar{t}}\), using the boundedness of \(T_{\beta }^{*}\) on \(L^{\bar{t}}(\mathbb{R}^{n})\) and applying Hölder’s inequality,
For the remaining term, we have
and
Since \(1\leq {p}'_{\alpha }< t\), we can choose t̄ such that \({p}'_{\alpha }<\bar{t} <t\). Let \(\nu =\frac{\bar{t} t}{t-\bar{t}} \), and then by Hölder’s inequality and Lemma 3, we get
Then, similar to the estimate of (3), we get
By (4) and similar to the estimate of (2), we can get
This completes the proof of Lemma 6. □
Lemma 7
Let \(V\in \mathit{RH}_{s}\) for \(n/2< s< n\), \(\frac{1}{{p}_{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}\), and \(b\in \operatorname{BMO}_{\theta }(\rho )\). Then, for any \({p}'_{\alpha }< t<\infty \) and \(\gamma \geq 1\) we have
for all f and \(x,y\in B=B(x_{0},r)\) with \(r<\gamma \rho (x_{0})\).
Proof
Denote \(Q=B(x_{0},\gamma \rho (x_{0}))\). By Lemma 5 and since in our situation \(\rho (x)\sim \rho (x_{0})\) and \(|x-z|\sim |x _{0}-z|\), we need to estimate the following four terms:
and
Splitting into annuli, we have
where \(j_{0}\) is the least integer such that \(2^{j_{0}}\geq \gamma \rho (x_{0})/r\). By Hölder’s inequality with \(\frac{1}{{p}'_{ \alpha }}+\frac{\alpha }{s}+\frac{1}{q_{1}}=1\), \(t> {p}'_{\alpha }\), similar to the estimate of (4), we have
Then, using \(\beta -\alpha =1/2\), we get
To deal with \(I_{2}\), we split into annuli and get
Notice that
Then, taking \(N>\theta '+l_{0}\alpha \), we get
For \(J_{3}\), splitting into annuli, we obtain
By Hölder’s inequality with \(\frac{1}{{p}'_{\alpha }}+\frac{ \alpha }{s}+\frac{1}{q_{1}}=1\), similar to the estimate of (2), we get
Then
Finally, for \(J_{4}\) we have
Notice that
We choose N large enough such that \(N>\theta '+l_{0}(\alpha +1)\), and then
which finishes the proof of Lemma 7. □
Now we are in a position to give the proof of Theorem 2.
Proof of Theorem 2
We will prove part (i), and (ii) follows by duality. We start with a function \(f\in L^{p}(\mathbb{R} ^{n})\) with \(p'_{\alpha }< p<\infty \), and by Lemma 6 we have \([b,T_{\beta }^{*}]f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\).
By Proposition 3 and Lemma 6 with \(p'_{\alpha }< t< p< \infty \), we have
By Proposition 2 and the boundedness of \(M_{t}\) on \(L^{p}(\mathbb{R}^{n})\), the second term is controlled by \([b]^{p} _{\theta }\|f\|^{p}_{L^{p}(\mathbb{R}^{n})}\). Then, we only need to consider the first term.
Our goal is to find a point-wise estimate of \(M_{\rho ,\gamma }[b,T _{\beta }^{*}]f\). Let \(x\in \mathbb{R}^{n}\) and \(B=B(x_{0},r)\) with \(r<\gamma \rho (x_{0})\) such that \(x\in B\). Write \(f=f_{1}+f_{2}\) with \(f_{1}=f\chi _{2B}\), then
Then, we need to control the mean oscillation on B of each term that we call \(\mathcal{O}_{1}\), \(\mathcal{O}_{2}\) and \(\mathcal{O}_{3}\).
Let \(t>p'_{\alpha }\), then, by Hölder’s inequality and Lemma 3, we get
since \(r<\gamma \rho (x_{0})\).
To estimate \(\mathcal{O}_{2}\), let \(p'_{\alpha }<\bar{t}<t\) and \(\nu =\frac{\bar{t}t}{t-\bar{t}}\). Then
For \(\mathcal{O}_{3}\), note that \(\inf_{y\in B}M_{t}f(y)\leq M_{t}f(x _{0})\), and so by Lemma 7 we get
Thus, we have showed that
Since \(t< p\), we obtain the desired result. □
Proof of Theorem 3
Let \(f\in H_{\mathcal{L}}^{1}( \mathbb{R}^{n})\). By Proposition 1, we can write \(f= \sum_{j=-\infty }^{\infty }\lambda _{j}a_{j}\), where each \(a_{j}\) is a \((1,q)_{\rho }\)-atom with \(1< q< {p}_{\alpha }\), \(\frac{1}{{p}_{\alpha }}=\frac{ \alpha +1}{q_{0}}-\frac{1}{n}\) and \(\sum_{j=-\infty }^{\infty }| \lambda _{j}|\leq 2\|f\|_{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}\). Suppose \(\operatorname{supp} a_{j}\subset B_{j}=B(x_{j},r_{j})\) with \(r_{j}<\rho (x _{j})\). Write
Note that
By Hölder’s inequality, for \(1< q< {p}_{\alpha } \), and using Theorem 2 we get
Thus
And so
Since \(z\in B_{j}\), \(x\in 2^{k}B_{j}\setminus 2^{k-1}B_{j}\), we have \(|x-z|\sim |x-x_{j}|\sim 2^{k}r_{j}\), and by Lemma 1 we get
By Hölder’s inequality, Lemmas 2 and 3, we get
Note that \(\frac{1}{{p}'_{\alpha }}+\frac{\alpha }{s}+ \frac{1}{q_{1}}=1\), \(\frac{1}{q_{1}}=\frac{1}{s}-\frac{1}{n} \), so by Hölder’s and Hardy–Littlewood–Sobolev’s inequalities and using the fact that \(V\in \mathit{RH}_{s}\), we obtain
Recall \(\int _{B_{j}}|a_{j}(y)|\,dy\lesssim 1\), \(\beta -\alpha = \frac{1}{2} \) and \(r_{j}/\rho (x_{j})\geq 1/4\). Then, taking N large enough such that \(\frac{N}{k_{0}+1}>\theta '+l_{0}(\alpha +1)\), we get
Thus
Therefore
When \(x\in 2^{k}B_{j}\setminus 2^{k-1}B_{j}\), and \(z\in B_{j}\), by Lemmas 5 and 1, we have
where \(\delta =2-n/s>0\). Thus, by the vanishing condition of \(a_{j}\), together with (5) and (6), we have
So that
Now let us deal with the last part. Since \(r_{j}\leq \rho (x_{j})\), we get
Note that
By Theorem 1, we know \(T_{\beta }\) is bounded from \(L^{1}(\mathbb{R}^{n})\) into weak \(L^{1}(\mathbb{R}^{n})\). Then
Thus,
□
4 Conclusion
In this paper, we established the \(L^{p}\)-boundedness of commutator operators \([b,T_{\beta }]\) and \([b,T^{*}_{\beta }]\), where \(T_{ \beta }=V^{\alpha }\nabla \mathcal{L}^{-\beta }\), \(\frac{1}{2}< \beta \leq 1\), \(\beta -\alpha =\frac{1}{2} \), and \(b\in \operatorname{BMO}_{\theta }(\rho )\), which is larger than the space \(\operatorname{BMO}(\mathbb{R}^{n})\). At the endpoint, we show that the operator \([b,T_{\beta }]\) is bounded from Hardy space \(H^{1}_{\mathcal{L}}(\mathbb{R}^{n})\) continuously into weak \(L^{1}(\mathbb{R}^{n})\). These results enrich the theory of Schrödinger operator.
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Hu, Y., Wang, Y. Estimates for the commutators of operator \(V^{\alpha }\nabla (-\Delta +V)^{-\beta }\). J Inequal Appl 2019, 126 (2019). https://doi.org/10.1186/s13660-019-2081-z
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DOI: https://doi.org/10.1186/s13660-019-2081-z