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Weighted inequalities for commutators of Schrödinger-Riesz transforms. (English) Zbl 1246.42018

Let $V:{ℝ}^{d}↦ℝ$, $d\ge 3$, be a non-negative locally integrable function that belongs to a reverse Hölder class $R{H}_{q}\phantom{\rule{4pt}{0ex}}$ for some $q>\frac{d}{2}$, that is, there exists $C>0$ such that the reverse Hölder inequality

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

holds for every ball $B$ in ${ℝ}^{d}$. The critical radius function $\rho \left(x\right)$ is defined as follows:

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

Let $ℒ=-{\Delta }+V$ be a Schrödinger operator. And let $ℛ=\nabla {ℒ}^{-\frac{1}{2}}$ be the associated Riesz transform vector and ${ℛ}^{*}$ be its adjoint operator.

The space $BM{O}_{\theta }\left(\rho \right)$ is the set of locally integrable functions $b$ satisfying

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

for all $x\in 𝒳$ and $r>0$, where $\theta >0$ and ${b}_{B}=\frac{1}{\mu \left(B\right)}{\int }_{B}b\left(y\right)d\mu \left(y\right)$. Denote $BM{O}_{\infty }\left(\rho \right)={\bigcup }_{\theta >0}BM{O}_{\theta }\left(\rho \right)$. Moreover, the authors introduce classes of weights that are given in terms of the critical radius function $\rho \left(x\right)$ and they are larger than the classical ${A}_{p}$ weights.

For $b\in BM{O}_{\infty }\left(\rho \right)$, the authors consider the commutator

${T}_{b}\left(f\right)\left(x\right)=T\left(bf\right)\left(x\right)-b\left(x\right)Tf\left(x\right),\phantom{\rule{4pt}{0ex}}x\in 𝒳,$

where $T=ℛ$ or ${ℛ}^{*}$.

Their main results are to obtain weighted ${L}^{p}$, $1, and weak $L\mathrm{log}L$ estimates for the commutator ${T}_{b}$.

Reviewer: Yu Liu (Beijing)
##### MSC:
 42B20 Singular and oscillatory integrals, several variables 47F05 Partial differential operators 47B47 Commutators, derivations, elementary operators, etc.
##### Keywords:
Schrödinger operator; Riesz transforms; commutators; weights.
##### References:
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