An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

Bhowmik, Mithun; Pusti, Sanjoy

doi:10.1016/j.jfa.2022.109413

Cited by 8 publications

(5 citation statements)

References 40 publications

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“…Observe that due to the subordination (3.1) to the heat kernel, Q σ t is a positive, bi-K-invariant and symmetric (in the sense that Q σ t (g) = Q σ t (g −1 ) for all g ∈ G) function on G. We next recall some large-time upper and lower bounds for the kernel Q σ t proved in [BP22]. Theorem 3.2.…”

Section: The Fractional Laplacian and The Extension Problemmentioning

confidence: 98%

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

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Papageorgiou

Zhang

2023

Journal of Functional Analysis

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Section: The Fractional Laplacian and The Extension Problemmentioning

confidence: 98%

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

Anker

Papageorgiou

Zhang

2023

Journal of Functional Analysis

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“…We next recall some large-time upper and lower bounds for the kernel Q σ t proved in [10]. Theorem 3.2.…”

Section: The Fractional Laplacian and The Extension Problemmentioning

confidence: 99%

“…The extension problem has drawn much attention. Since the associated literature is enormous, we shall refer indicatively to [3,9,10,12,16,24,25,27] and the references therein. From a probabilistic point of view, the extension problem corresponds to the property that all symmetric stable processes can be obtained as traces of degenerate Bessel diffusion processes, see [26].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Papageorgiou

2024

J. Evol. Equ.

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This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi-K-invariance.

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“…The high-frequency features of cracks can be well expressed by edge detection [13,14], and an improved Gaussian Laplacian operator is proposed. The core idea of the Gaussian Laplacian operator [15,16] is to combine the Laplacian operator and the Gaussian smoothing filter to detect the edge of the image. For the pavement crack image, the Gaussian smoothing filter can effectively suppress the impact noise and other signals, which plays a good auxiliary role in the expression of low-frequency information of the crack-free image.…”

Section: Non/crack Image Filteringmentioning

confidence: 99%

Detection Method of Cracks in Expressway Asphalt Pavement Based on Digital Image Processing Technology

Fang,

2023

Applied Sciences

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Considering the limitations of the current pavement crack damage detection methods, this study proposes a method based on digital image processing technology for detecting highway asphalt pavement crack damage. Firstly, a non-subsampled contourlet transform is used to enhance the image of highway asphalt pavement. Secondly, the non-crack regions in the image are screened, and the crack extraction is completed by obtaining and enhancing the crack intensity map. Finally, the features of cracks are extracted and input into the support vector machine for classification and recognition to complete the detection of cracks in highway asphalt pavement. The experimental results show that the proposed method can effectively enhance the quality of a pavement image and precisely extract a crack area from the image with a high level of damage detection accuracy.

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An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

Cited by 8 publications

References 40 publications

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Detection Method of Cracks in Expressway Asphalt Pavement Based on Digital Image Processing Technology

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