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Commutators of Riesz transforms related to Schrödinger operators. (English) Zbl 1213.42075

The authors consider the Schrödinger operator

$𝔏=-{\Delta }+V$

on ${ℝ}^{d}$, $d\ge 3$, where the nonnegative potential $V\left(x\right)$ belongs to $R{H}_{q}$, which is a class satisfying the following reverse Hölder inequality

${\left(\frac{1}{|B|}{\int }_{B}V{\left(x\right)}^{q}\phantom{\rule{0.166667em}{0ex}}dx\right)}^{\frac{1}{q}}\le C\left(\frac{1}{|B|}{\int }_{B}V\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\right)\phantom{\rule{2.em}{0ex}}\left(\mathrm{e}1\right)$

for every ball $B$ in ${ℝ}^{n}$. The auxiliary function $\rho \left(x\right)$ related to $V\left(x\right)$ is defined as

$\rho \left(x\right)=\frac{1}{m\left(x,V\right)}\stackrel{˙}{=}\underset{r>0}{sup}\phantom{\rule{0.166667em}{0ex}}\left\{r:\phantom{\rule{0.277778em}{0ex}}\frac{1}{{r}^{n-2}}{\int }_{B\left(x,r\right)}V\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy\le 1\right\},\phantom{\rule{2.em}{0ex}}x\in {ℝ}^{n}·$

For $\theta >0$, a locally integrable function $b$ is said to be in the class $BM{O}_{\theta }\left(\rho \right)$ if

$\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}|b\left(y\right)-{b}_{B}|dy\le C{\left(1+\frac{r}{\rho \left(x\right)}\right)}^{\theta },$

for all $x\in {ℝ}^{d}$ and $r>0$, where ${b}_{B}=\frac{1}{|B|}{\int }_{B}b\left(y\right)dy·$ Define $BM{O}_{\infty }\left(\rho \right)={\bigcup }_{\theta >0}BM{O}_{\theta }\left(\rho \right)$. Clearly $BMO\subseteq BM{O}_{\infty }\left(\rho \right)$. For $\theta >0$, a locally integrable function $b$ is said to be in the class $BM{O}_{\theta }^{log}\left(\rho \right)$ if

$\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}|b\left(y\right)-{b}_{B}|dy\le C\frac{{\left(1+\frac{r}{\rho \left(x\right)}\right)}^{\theta }}{1+{log}^{+}\left(\rho \left(x\right)/r\right)},$

for all $x\in {ℝ}^{d}$ and $r>0$.

Denote $ℛ=\nabla {\left(-{\Delta }+V\right)}^{-\frac{1}{2}}$. For a locally integrable function $b$, the commutators in this paper are defined as

${ℛ}_{b}f\left(x\right)=ℛ\left(bf\right)\left(x\right)-b\left(x\right)ℛf\left(x\right)$

and

${ℛ}_{b}^{*}f\left(x\right)={ℛ}^{*}\left(bf\right)\left(x\right)-b\left(x\right){ℛ}^{*}f\left(x\right),$

under the assumption that $V\in R{H}_{{q}_{0}}$ for ${q}_{0}>d/2$ and $b\in BM{O}_{\infty }\left(\rho \right)$. The authors study the ${L}^{p}$ boundedness of the commutators ${ℛ}_{b}$ and ${ℛ}_{b}^{*}$. Especially, they prove that ${ℛ}_{b}^{*}:{L}^{\infty }\to BM{O}_{𝔏}$ is bounded if and only if $b\in BM{O}_{\theta }^{log}\left(\rho \right)·$ Moreover, this conclusion holds true for ${ℛ}_{b}$ when $V\in R{H}_{d}$.

Reviewer: Liu Yu (Beijing)
##### MSC:
 42B35 Function spaces arising in harmonic analysis 35J10 Schrödinger operator
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