The higher order Riesz transform and BMO type space associated to Schrödinger operators

2011 ◽  
Vol 285 (4) ◽  
pp. 486-496 ◽  
Author(s):  
Jianfeng Dong ◽  
Yu Liu
Keyword(s):  
Schrödinger Operators ◽  
Riesz Transform ◽  
Higher Order ◽  
Schrodinger Operators ◽  
Type Space

scholarly journals Commutators of Higher Order Riesz Transform Associated with Schrödinger Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Yu Liu ◽  
Lijuan Wang ◽  
Jianfeng Dong
Keyword(s):  
Hardy Space ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Higher Order ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator onℝn(n≥3), whereV≢0is a nonnegative potential belonging to certain reverse Hölder classBsfors≥n/2. In this paper, we prove the boundedness of commutatorsℛbHf=bℛHf-ℛH(bf)generated by the higher order Riesz transformℛH=∇2(-Δ+V)-1, whereb∈BMOθ(ρ), which is larger than the spaceBMO(ℝn). Moreover, we prove thatℛbHis bounded from the Hardy spaceHL1(ℝn)into weakLweak1(ℝn).

THE ESTIMATES OF RIESZ TRANSFORMS ASSOCIATED TO SCHRÖDINGER OPERATORS

2016 ◽  
Vol 101 (3) ◽  
pp. 290-309 ◽  
Author(s):  
QINGQUAN DENG ◽  
YONG DING ◽  
XIAOHUA YAO
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Holder Class ◽  
Small Norm ◽  
Reverse Hölder

Let$H=-\unicode[STIX]{x1D6E5}+V$be a Schrödinger operator with some general signed potential$V$. This paper is mainly devoted to establishing the$L^{q}$-boundedness of the Riesz transform$\unicode[STIX]{x1D6FB}H^{-1/2}$for$q>2$. We mainly prove that under certain conditions on$V$, the Riesz transform$\unicode[STIX]{x1D6FB}H^{-1/2}$is bounded on$L^{q}$for all$q\in [2,p_{0})$with a given$2<p_{0}<n$. As an application, the main result can be applied to the operator$H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where$V_{+}$belongs to the reverse Hölder class$B_{\unicode[STIX]{x1D703}}$and$V_{-}\in L^{n/2,\infty }$with a small norm. In particular, if$V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$for some positive number$\unicode[STIX]{x1D6FE}$,$\unicode[STIX]{x1D6FB}H^{-1/2}$is bounded on$L^{q}$for all$q\in [2,n/2)$and$n>4$.

scholarly journals Long range scattering for higher order Schrödinger operators

2013 ◽  
Vol 254 (8) ◽  
pp. 3329-3351 ◽  
Author(s):  
Zhiwen Duan ◽  
Quan Zheng ◽  
Jing Feng
Keyword(s):  
Long Range ◽  
Schrödinger Operators ◽  
Higher Order ◽  
Schrodinger Operators

scholarly journals The Riesz transform for homogeneous Schrödinger operators on metric cones

2014 ◽  
Vol 30 (2) ◽  
pp. 477-522 ◽  
Author(s):  
Andrew Hassell ◽  
Peijie Lin
Keyword(s):  
Schrödinger Operators ◽  
Riesz Transform ◽  
Schrodinger Operators

scholarly journals Higher order eigenvalues for non-local Schrödinger operators

2018 ◽  
Vol 17 (1) ◽  
pp. 191-208 ◽  
Author(s):  
Niels Jacob ◽  
◽  
Feng-Yu Wang ◽  
◽  
Keyword(s):  
Schrödinger Operators ◽  
Higher Order ◽  
Schrodinger Operators ◽  
Non Local
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