scholarly journals Marcinkiewicz integrals associated with Schrödinger operator on generalized Morrey spaces

2014 ◽  
pp. 791-801 ◽  
Author(s):  
Ali Akbulut ◽  
Okan Kuzu
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Schrodinger Operator ◽  
Marcinkiewicz Integrals ◽  
Generalized Morrey Spaces

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.

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

2020 ◽  
Vol 57 (4) ◽  
pp. 465-507
Author(s):  
Hua Wang
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Heat Semigroup ◽  
Schrodinger Operator ◽  
Heisenberg Groups ◽  
Area Integral ◽  
Weak Type Estimates ◽  
Homogeneous Dimension ◽  
Lusin Area Integral ◽  
Reverse Hölder

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

scholarly journals Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Hua Wang
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Fractional Integrals ◽  
Laplacian Operator ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Reverse Hölder

Let ℒ=−Δ+V be a Schrödinger operator on ℝd, d≥3, where Δ is the Laplacian operator on ℝd, and the nonnegative potential V belongs to the reverse Hölder class RHs with s≥d/2. For given 0<α<d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐα=ℒ−α/2. Suppose that b is a locally integrable function on ℝd and the commutator generated by b and ℐα is defined by b.ℐαfx=bx⋅ℐαfx−ℐαbfx. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. Then, we will establish the boundedness properties of the fractional integrals ℐα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b,ℐα in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMOℝd, and Lp,κμ,ν corresponding to the classical case (that is V≡0).

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