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Weighted inequalities for commutators of Schrödinger-Riesz transforms. (English) Zbl 1246.42018
Let $V:\mathbb{R}^d\mapsto \mathbb{R}$, $d\ge 3$, be a non-negative locally integrable function that belongs to a reverse Hölder class $RH_q\ $ for some $q>\frac{d}{2}$, that is, there exists $C>0$ such that the reverse Hölder inequality $$ \left ( \frac{1}{|B|} \int_B V(x)^q \, dx \right )^{\frac{1}{q}} \leq \frac{C }{|B|} \int_B V(x) \, dx \tag1$$ holds for every ball $B$ in $\mathbb{R}^d$. The critical radius function $\rho(x)$ is defined as follows: $$ \rho(x)= \sup_{r>0}\, \bigg \{ r:\; \frac{1}{r^{d-2}}\int_{B(x, r)}V(y)\, dy \leq 1 \bigg \}, \qquad x\in \mathbb{R}^d. $$ Let $\mathcal{L}=-\Delta+V$ be a Schrödinger operator. And let $\mathcal{R}=\nabla \mathcal{L}^{-\frac{1}{2}}$ be the associated Riesz transform vector and $\mathcal{R}^{*}$ be its adjoint operator. The space $BMO_{\theta}(\rho)$ is the set of locally integrable functions $b$ satisfying $$\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|b(y)-b_B|d\mu(y)\le C\big(1+\frac{r}{\rho(x)}\big)^{\theta}, $$ for all $x\in\mathcal{X}$ and $r>0$, where $\theta>0$ and $b_B=\frac{1}{\mu(B)}\int_Bb(y)d\mu(y)$. Denote $BMO_{\infty}(\rho)=\bigcup_{\theta>0}BMO_{\theta}(\rho)$. Moreover, the authors introduce classes of weights that are given in terms of the critical radius function $\rho(x)$ and they are larger than the classical $A_p$ weights. For $b\in BMO_{\infty}(\rho)$, the authors consider the commutator $$ T_b(f)(x)=T(bf)(x)-b(x)Tf(x),\ x\in \mathcal{X}, $$ where $T=\mathcal{R}$ or $\mathcal{R}^{*}$. Their main results are to obtain weighted $L^p$, $1<p<\infty$, and weak $L \mathrm{log} L$ estimates for the commutator $T_b$.
Reviewer: Yu Liu (Beijing)

42B20Singular and oscillatory integrals, several variables
47F05Partial differential operators
47B47Commutators, derivations, elementary operators, etc.
Full Text: DOI arXiv
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