zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Commutators of Riesz transforms related to Schrödinger operators. (English) Zbl 1213.42075
The authors consider the Schrödinger operator $$\mathfrak{L}=-\Delta+V$$ on $\mathbb{R}^d$, $d\ge 3$, where the nonnegative potential $V(x)$ belongs to $RH_{q}$, which is a class satisfying the following reverse Hölder inequality $$ \left ( \frac{1}{|B|} \int_B V(x)^q \, dx \right )^{\frac{1}{q}} \leq C \left ( \frac{1}{|B|} \int_B V(x) \, dx \right )\tag {e1}$$ for every ball $B$ in $\mathbb{R}^n$. The auxiliary function $\rho(x)$ related to $V(x)$ is defined as $$ \rho(x)=\frac{1}{m(x,V)}\dot{=} \sup_{r>0}\, \bigg \{ r:\; \frac{1}{r^{n-2}}\int_{B(x, r)}V(y)\, dy \leq 1 \bigg \}, \qquad x\in \mathbb{R}^n. $$ For $\theta>0$, a locally integrable function $b$ is said to be in the class $BMO_{\theta}(\rho) $ if $$\frac{1}{|B(x,r)|}\int_{B(x,r)}|b(y)-b_B|dy\le C(1+\frac{r}{\rho(x)})^{\theta},$$ for all $x\in\mathbb{R}^d$ and $r>0$, where $b_B=\frac{1}{|B|}\int_Bb(y)dy.$ Define $BMO_{\infty}(\rho)=\bigcup_{\theta>0}BMO_{\theta}(\rho) $. Clearly $BMO\subseteq BMO_{\infty}(\rho)$. For $\theta>0$, a locally integrable function $b$ is said to be in the class $BMO^{\log}_{\theta}(\rho) $ if $$\frac{1}{|B(x,r)|}\int_{B(x,r)}|b(y)-b_B|dy\le C\frac{(1+\frac{r}{\rho(x)})^{\theta}}{1+\log^{+}(\rho(x)/r)},$$ for all $x\in\mathbb{R}^d$ and $r>0$. Denote $\mathcal {R}=\nabla(-\Delta+V)^{-\frac{1}{2}}$. For a locally integrable function $b$, the commutators in this paper are defined as $$\mathcal {R}_bf(x)=\mathcal {R}(bf)(x)-b(x)\mathcal {R}f(x)$$ and $$\mathcal {R}^{*}_bf(x)=\mathcal {R}^{*}(bf)(x)-b(x)\mathcal {R}^{*}f(x),$$ under the assumption that $V\in RH_{q_0}$ for $q_0>d/2$ and $b\in BMO_{\infty}(\rho)$. The authors study the $L^p$ boundedness of the commutators $ \mathcal {R}_b $ and $\mathcal {R}^{*}_b $. Especially, they prove that $\mathcal {R}^{*}_b:L^{\infty}\rightarrow BMO_{\mathfrak{L}}$ is bounded if and only if $b\in BMO^{\log}_{\theta}(\rho).$ Moreover, this conclusion holds true for $\mathcal {R}_b$ when $V\in RH_{d}$.
Reviewer: Liu Yu (Beijing)

42B35Function spaces arising in harmonic analysis
35J10Schrödinger operator
Full Text: DOI
[1] Bongioanni, B., Harboure, E., Salinas, O.: Riesz transforms related to Schrödinger operators acting on BMO type spaces. J. Math. Anal. Appl. 357(1), 115--131 (2009) · Zbl 1180.42013 · doi:10.1016/j.jmaa.2009.03.048
[2] Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611--635 (1976) · Zbl 0326.32011 · doi:10.2307/1970954
[3] 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
[4] Dziubański, J., Garrigós, G., Martínez, T., Torrea, J., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249(2), 329--356 (2005) · Zbl 1136.35018 · doi:10.1007/s00209-004-0701-9
[5] Gehring, F.W.: The L p -integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265--277 (1973) · Zbl 0258.30021 · doi:10.1007/BF02392268
[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 · doi:10.1016/j.jmaa.2007.05.024
[7] Harboure, E., Segovia, C., Torrea, J.L.: Boundedness of commutators of fractional and singular integrals for the extreme values of p. Ill. J. Math. 41(4), 676--700 (1997) · Zbl 0892.42009
[8] John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415--426 (1961) · Zbl 0102.04302 · doi:10.1002/cpa.3160140317
[9] Pérez, C.: Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J. Fourier Anal. Appl. 3(6), 743--756 (1997) · Zbl 0894.42006 · doi:10.1007/BF02648265
[10] Pradolini, G., Salinas, O.: Commutators of singular integrals on spaces of homogeneous type. Czechoslov. Math. J. 57(1), 75--93 (2007) · Zbl 1174.42322 · doi:10.1007/s10587-007-0045-9
[11] Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45(2), 513--546 (1995) · Zbl 0818.35021