Besov Spaces for the Schrödinger Operator with Barrier Potential

Benedetto, John J.; Zheng, Shijun

doi:10.1007/s11785-009-0011-7

Cited by 11 publications

(17 citation statements)

References 32 publications

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“…As to (i-a), let us take ψ = ψ (k) , φ j = φ (k) j (k = 1, 2) satisfying (2.1), (2.2) and (2.9). Since ψ (1) and φ…”

Section: Proof Of Theorem 25mentioning

confidence: 99%

Besov spaces on open sets

Iwabuchi¹,

Matsuyama²,

Taniguchi³

2016

Preprint

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This paper is devoted to giving definitions of Besov spaces on an arbitrary open set of R n via the spectral theorem for the Schrödinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on Ω. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schrödinger operators.

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“…As to (i-a), let us take ψ = ψ (k) , φ j = φ (k) j (k = 1, 2) satisfying (2.1), (2.2) and (2.9). Since ψ (1) and φ…”

Section: Proof Of Theorem 25mentioning

confidence: 99%

Besov spaces on open sets

Iwabuchi¹,

Matsuyama²,

Taniguchi³

2016

Preprint

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“…A counterexample can be found in [33] for V = −ν(ν + 1)sech 2 x. For non-smooth potentials, in [4] we are able to obtain an appropriate variant of the kernel decay…”

Section: 5mentioning

confidence: 92%

“…We are mainly concerned with the following three interrelated problems. The decay estimate in (1.2) has been known to be fundamental and useful in function space theory and spectral multiplier problem [19,15,4,33,34]. We will see that it can be applied to characterize B(H) and F (H) spaces with full ragne of parameters 0 < p, q < ∞ and show Mihlin-Hörmander type multiplier result on L p , B(H) and F (H) spaces; for the multiplier problem we actually formulate a more general condition as in (1.5).…”

Section: Introductionmentioning

confidence: 99%

Harmonic analysis related to Schrödinger operators

Ólafsson¹,

Zheng²

2008

Contemporary Mathematics

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In this article we give an overview on some recent development of Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our approach by considering potential in C ∞ 0 or Schwartz class in one dimension. In particular the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with H. We then prove a spectral multiplier theorem on these spaces and derive Strichartz estimates for the wave equation with a potential. We also consider similar problem for the unbounded potentials in the Hermite and Laguerre cases, whose V = a|x| 2 + b|x| −2 are known to be critical in the study of perturbation of nonlinear dispersive equations. This improves upon the previous results when we apply the upper Gaussian bound for the heat kernel and its gradient.

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“…There are a lot of literatures on characterization of Besov spaces (see, e.g., Triebel [18][19][20]). We are concerned with Besov spaces characterized by differential operators via the spectral approach (see [1][2][3]6,7,[10][11][12]14] and the references therein). The purpose of this paper is to give a definition of Besov spaces generated by the Neumann Laplacian on a domain, and prove their fundamental properties; completeness and embedding relations etc.…”

Section: Introductionmentioning

confidence: 99%