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

Let \(V:\mathbb{R}^d\mapsto \mathbb{R}\), \(d\geq 3\), 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 \[ \left ( \frac{1}{|B|} \int_B V(x)^q \, dx \right )^{\frac{1}{q}} \leq \frac{C }{|B|} \int_B V(x) \, dx \tag{1} \] 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 \[ \frac{1}{\mu(B(x,r))}\int_{B(x,r)}|b(y)-b_B|d\mu(y)\leq C\big(1+\frac{r}{\rho(x)}\big)^{\theta}, \] 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)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47F05 General theory of partial differential operators
47B47 Commutators, derivations, elementary operators, etc.
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References:

[1] Bongioanni, B.; Harboure, E.; Salinas, O., Classes of weights related to Schrödinger operators, J. Math. Anal. Appl., 373, 2, 563-579 (2011) · Zbl 1203.42029
[2] Bongioanni, B.; Harboure, E.; Salinas, O., Commutators of Riesz transforms related to Schrödinger operators, J. Fourier Anal. Appl., 17, 1, 115-134 (2011) · Zbl 1213.42075
[3] Coifman, R. R.; Rochberg, R.; Weiss, Guido, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2), 103, 3, 611-635 (1976) · Zbl 0326.32011
[4] Duoandikoetxea, Javier, Fourier Analysis, Grad. Stud. Math., vol. 29 (2001), American Mathematical Society: American Mathematical Society Providence, RI, Translated and revised from the 1995 Spanish original by David Cruz-Uribe · Zbl 0969.42001
[5] Dziubański, J.; Zienkiewicz, J., Hardy spaces \(H^1\) associated to Schrödinger operators with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoam., 15, 2, 279-296 (1999) · Zbl 0959.47028
[6] Guo, Z.; Li, P.; Peng, L., \(L^p\) boundedness of commutators of Riesz transforms associated to Schrödinger operator, J. Math. Anal. Appl., 341, 1, 421-432 (2008) · Zbl 1140.47035
[7] Krasnoselʼskiĭ, M. A.; Rutickiĭ, Ja. B., Convex Functions and Orlicz Spaces (1961), P. Noordhoff Ltd.: P. Noordhoff Ltd. Groningen, Translated from the first Russian edition by Leo F. Boron
[8] Pérez, Carlos, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128, 1, 163-185 (1995) · Zbl 0831.42010
[9] Pérez, Carlos; Pradolini, Gladis, Sharp weighted endpoint estimates for commutators of singular integrals, Michigan Math. J., 49, 1, 23-37 (2001) · Zbl 1010.42007
[10] Shen, Z., \(L^p\) estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45, 2, 513-546 (1995) · Zbl 0818.35021
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