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...

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:

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.

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...

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 $V \in B_{n}$ .

On the boundary convergence of solutions to the Hermite-Schrödinger equation

Peter Sjögren, J. L. Torrea (2010)

Colloquium Mathematicae

Similarity:

In the half-space $ℝ^{d} \times ℝ ₊$ , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on $ℝ^{d}$ . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

M. Magdziarz, A. Weron (2007)

Studia Mathematica

Similarity:

We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{κ} (t), t \geq 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{κ} (t)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{κ} (t)$ is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of...

Left general fractional monotone approximation theory

George A. Anastassiou (2016)

Applicationes Mathematicae

Similarity:

We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative...

Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman, Fabricio Macià (2014)

Journal of the European Mathematical Society

Similarity:

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality,...

Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi (2013)

Annales Polonici Mathematici

Similarity:

We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $- d / d t (1 / 2_{0} D_{t}^{- σ} (u^{'} (t)) + 1 / 2_{t} D_{T}^{- σ} (u^{'} (t))) - λ β (t) f (u (t)) - μ γ (t) g (u (t)) = 0$ , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where $_{0} D_{t}^{- σ}$ and $_{t} D_{T}^{- σ}$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

Localisation for non-monotone Schrödinger operators

Alexander Elgart, Mira Shamis, Sasha Sodin (2014)

Journal of the European Mathematical Society

Similarity:

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schrödinger operators with non-monotone random potentials, on the $d$ -dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts...

Dispersion Phenomena in Dunkl-Schrödinger Equation and Applications

Mejjaoli, H. (2009)

Serdica Mathematical Journal

Similarity:

2000 Mathematics Subject Classification: 35Q55,42B10. In this paper, we study the Schrödinger equation associated with the Dunkl operators, we study the dispersive phenomena and we prove the Strichartz estimates for this equation. Some applications are discussed.

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

Similarity:

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form $u (x, t) = t^{- α} f (| x | t^{- β})$ with suitable and $β$ . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov...

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

Similarity:

We prove that a function belonging to a fractional Sobolev space $L^{α, p} (ℝ ⁿ)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m, λ} (ℝ ⁿ)$ , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

Similarity:

Nonlinear Schrödinger equations (NLS) $_{a}$ with strongly singular potential ${a | x |}^{- 2}$ on a bounded domain $Ω$ are considered. If $Ω = ℝ^{N}$ and $a > - {(N - 2)}^{2} / 4$ , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a = - {(N - 2)}^{2} / 4$ is excluded because $D (P_{a (N)}^{1 / 2})$ is not equal to $H^{1} (ℝ^{N})$ , where $P_{a (N)} : = - Δ - {(N - 2)}^{2} / {(4 | x |}^{2})$ is nonnegative and selfadjoint in $L^{2} (ℝ^{N})$ . On the other hand, if $Ω$ is a smooth and bounded domain with $0 \in Ω$ , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...