scholarly journals Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators

2019 ◽  
Vol 9 (1) ◽  
pp. 1291-1314
Author(s):  
Jizheng Huang ◽  
Pengtao Li ◽  
Yu Liu
Keyword(s):  
Hardy Space ◽  
Schrödinger Operators ◽  
Heat Semigroup ◽  
Schrodinger Operators ◽  
Reverse Hölder Class ◽  
Area Integral ◽  
Muckenhoupt Class ◽  
Degenerate Schrödinger Operator ◽  
Reverse Hölder

Abstract Let $$\begin{array}{} \displaystyle Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}^{}\partial_{i}(a_{ij}(\cdot)\partial_{j}f)(x)+V(x)f(x) \end{array}$$ be the degenerate Schrödinger operator, where ω is a weight from the Muckenhoupt class A2, V is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure ω(x)dx. For such an operator we define the area integral $\begin{array}{} \displaystyle S^{L}_h \end{array}$ associated with the heat semigroup and obtain the area integral characterization of $\begin{array}{} \displaystyle H^{1}_{L} \end{array}$, which is the Hardy space associated with L.

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 The Boundedness of Marcinkiewicz Integrals Associated with Schrödinger Operator on Morrey Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Dongxiang Chen ◽  
Fangting Jin
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Marcinkiewicz Integrals ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereVbelongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.

scholarly journals Variation inequalities related to Schrödinger operators on weighted Morrey spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 813-827
Author(s):  
Jing Zhang
Keyword(s):  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Variation Operators ◽  
Reverse Hölder

Abstract This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

2015 ◽  
Vol 67 (5) ◽  
pp. 1161-1200 ◽  
Author(s):  
Junqiang Zhang ◽  
Jun Cao ◽  
Renjin Jiang ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic Operator ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Function Characterization

AbstractLet w be either in the Muckenhoupt class of A2(ℝn) weights or in the class of QC(ℝn) weights, and let be the degenerate elliptic operator on the Euclidean space ℝn, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with , and when with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .

scholarly journals A characterization of singular Schrödinger operators on the half-line

2020 ◽  
pp. 1-19
Author(s):  
Raffaele Scandone ◽  
Lorenzo Luperi Baglini ◽  
Kyrylo Simonov
Keyword(s):  
Schrödinger Operators ◽  
Half Line ◽  
Schrodinger Operators
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