Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized Schechter classes

Cao, Jun; Li, Yu; Yang, Dachun; Zhang, Chao

doi:10.1112/jlms.12620

Cited by 9 publications

(3 citation statements)

References 62 publications

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“…for any t ∈ (0, ∞) and x, y ∈ R n , see [14,17]. Inspiring the results in [12], we can consider the Lipschitz spaces associated with L as in the Schrödinger setting.…”

Section: A Pointwisementioning

confidence: 99%

Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators

Chen

Zhang

2022

Mathematics

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Let be the high-order Schrödinger operator (−Δ)2+V2, where V is a non-negative potential satisfying the reverse Hölder inequality (RHq), with q>n/2 and n≥5. In this paper, we prove that when 0<α≤2−n/q, the adapted Lipschitz spaces Λα/4 we considered are equivalent to the Lipschitz space CLα associated to the Schrödinger operator L=−Δ+V. In order to obtain this characterization, we should make use of some of the results associated to (−Δ)2. We also prove the regularity properties of fractional powers (positive and negative) of the operator , Schrödinger Riesz transforms, Bessel potentials and multipliers of the Laplace transforms type associated to the high-order Schrödinger operators.

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“…for any t ∈ (0, ∞) and x, y ∈ R n , see [14,17]. Inspiring the results in [12], we can consider the Lipschitz spaces associated with L as in the Schrödinger setting.…”

Section: A Pointwisementioning

confidence: 99%

Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators

Chen

Zhang

2022

Mathematics

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“…is small enough (see [6,43,45] for more details on this class of potentials). Moreover, by letting 𝜆, 𝑏 → ∞, the above condition becomes the Kato class condition that…”

Section: Introductionmentioning

confidence: 99%

Local potential operator and uniform resolvent estimate for generalized Schrödinger operator in Orlicz spaces

Cao

Dou

Gao

et al. 2023

Mathematische Nachrichten

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The local potential operator with integral kernel restricted in a ball of radius less than some fixed number 𝑟 ∈ (0, ∞) has appeared frequently in the spectral estimates of the Schrödinger operator. In this paper, we establish a good-𝜆 inequality for this operator and characterize its uniform boundedness on the weighted Orlicz space in both strong and weak senses. The uniformity in 𝑟 of this boundedness enables us to recover the classical boundedness of "global" potential operator, by letting 𝑟 → ∞. As an application, we establish uniform estimate for the resolvent (𝜆 − ) −1 of some generalized Schrödinger operator  ∶=  0 + 𝑉 on the Orlicz space. An explicit representation in its operator norm on the dependence of 𝜆 ∈ (0, ∞) is also given.

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“…The study of fundamental solutions is central in the theory of linear parabolic PDEs. For more results on heat kernel estimates for higher-order operators we refer to [6,7,9,10,11,13,15,16]. See also [14,17] for related results specific to fourth-order operators.…”

Section: Introductionmentioning

confidence: 99%

Heat kernel estimates for fourth-order non-uniformly elliptic operators with non-strongly convex symbols

Barbatis¹,

Panagiotis²

2022

ejde

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We obtain heat-kernel estimates for fourth-order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This entails certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are inducedby the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequality and an interpolation inequality, which are related to the singularity or degeneracy of the coefficients.

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Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized Schechter classes

Cited by 9 publications

References 62 publications

Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators

Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators

Local potential operator and uniform resolvent estimate for generalized Schrödinger operator in Orlicz spaces

Heat kernel estimates for fourth-order non-uniformly elliptic operators with non-strongly convex symbols

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