An improved maximal inequality for 2D fractional order Schrödinger operators

Changxing Miao; Jianwei Yang; Jiqiang Zheng

Displaying similar documents to “An improved maximal inequality for 2D fractional order Schrödinger operators”

Trace inequalities for fractional integrals in grand Lebesgue spaces

Vakhtang Kokilashvili, Alexander Meskhi (2012)

Studia Mathematica

Similarity:

rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p), θ} (X, μ)$ to $L^{q), q θ / p} (X, ν)$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace...

Generalized Morrey spaces associated to Schrödinger operators and applications

Nguyen Ngoc Trong, Le Xuan Truong (2018)

Czechoslovak Mathematical Journal

Similarity:

We first introduce new weighted Morrey spaces related to certain non-negative potentials satisfying the reverse Hölder inequality. Then we establish the weighted strong-type and weak-type estimates for the Riesz transforms and fractional integrals associated to Schrödinger operators. As an application, we prove the Calderón-Zygmund estimates for solutions to Schrödinger equation on these new spaces. Our results cover a number of known results.

Multiplicity solutions of a class fractional Schrödinger equations

Li-Jiang Jia, Bin Ge, Ying-Xin Cui, Liang-Liang Sun (2017)

Open Mathematics

Similarity:

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, ${(- Δ)}^{s} u + V (x) u = λ f (x, u) in ℝ^{N},$ where [...] (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN ${(- Δ)}^{s} u (x) = 2 lim_{ε \to 0} \int_{ℝ^{N} ∖ B_{ε} (X)} \frac{u (x) - u (y)}{| x - y |^{N + 2 s}} d y, x \in ℝ^{N}$ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski (2008)

Studia Mathematica

Similarity:

We construct the fundamental solution of $\partial_{t} - Δ_{y} - q (t, y)$ for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure...

Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions

V. Kiryakova (1988)

Matematički Vesnik

Similarity:

Existence results for systems of conformable fractional differential equations

Bouharket Bendouma, Alberto Cabada, Ahmed Hammoudi (2019)

Archivum Mathematicum

Similarity:

In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L_{α}^{1}$ -carathéodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.

Square functions associated to Schrödinger operators

I. Abu-Falahah, P. R. Stinga, J. L. Torrea (2011)

Studia Mathematica

Similarity:

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^{p}$ and BMO of classical ℒ-square functions.

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

Celso Martínez, Antonia Redondo, Miguel Sanz (2011)

Studia Mathematica

Similarity:

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^{p} (ℝ)$ -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable...

lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

Peiluan Li, Youlin Shang (2017)

Open Mathematics

Similarity:

Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

A spatially sixth-order hybrid $L 1$ -CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang, Jun-Wei Jin, Hai-Wei Sun, Qin Sheng (2021)

Applications of Mathematics

Similarity:

We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L 1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L 1$ -CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2 - γ$ in time, where $0 < γ < 1$ is the order of the Caputo fractional derivative...

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space $H^{2}$

Mahsa Fatehi, Bahram Khani Robati (2012)

Czechoslovak Mathematical Journal

Similarity:

In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{ϕ}$ , when $ϕ$ is a linear-fractional self-map of $𝔻$ . In this paper first, we investigate the essential normality problem for the operator $T_{w} C_{ϕ}$ on the Hardy space $H^{2}$ , where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F (ϕ)$ , $ϕ \in 𝒮 (2)$ , and $T_{w}$ is the Toeplitz operator with symbol $w$ . Then we use these results and characterize the essentially...

Initial value problems for fractional functional differential inclusions with Hadamard type derivative

Nassim Guerraiche, Samira Hamani, Johnny Henderson (2016)

Archivum Mathematicum

Similarity:

We establish sufficient conditions for the existence of solutions of a class of fractional functional differential inclusions involving the Hadamard fractional derivative with order $α \in (0, 1]$ . Both cases of convex and nonconvex valued right hand side are considered.

Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

Kamil Kaleta (2012)

Studia Mathematica

Similarity:

We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator ${(- Δ)}^{α / 2} + V$ , α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant $C_{α}$ appearing in our estimate $λ ₂ - λ ₁ \geq C_{α} {(b - a)}^{- α}$ is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower...

Boundary value problems for Caputo-Hadamard fractional differential inclusions in Banach spaces

Amouria Hammou, Samira Hamani, Johnny Henderson (2022)

Archivum Mathematicum

Similarity:

In this article, we study the existence of solutions in a Banach space of boundary value problems for Caputo-Hadamard fractional differential inclusions of order $r \in (0, 1]$ .

Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak (2017)

Czechoslovak Mathematical Journal

Similarity:

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $σ$ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $σ$ -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

$L^{p}$ estimates for Schrödinger operators with certain potentials

Zhongwei Shen (1995)

Annales de l'institut Fourier

Similarity:

We consider the Schrödinger operators $- Δ + V (x)$ in $ℝ^{n}$ where the nonnegative potential $V (x)$ belongs to the reverse Hölder class $B_{q}$ for some $q \geq n / 2$ . We obtain the optimal $L^{p}$ estimates for the operators $(- Δ + V)^{i γ}, \nabla^{2} (- Δ + V)^{- 1}, \nabla (- Δ + V)^{- 1 / 2}$ and $\nabla (- Δ + V)^{- 1}$ where $γ \in ℝ$ . In particular we show that $(- Δ + V)^{i γ}$ is a Calderón-Zygmund operator if $V \in B_{n / 2}$ and $\nabla (- Δ + V)^{- 1 / 2}, \nabla (- Δ + V)^{- 1} \nabla$ are Calderón-Zygmund operators if $V \in B_{n}$ .