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   https://doi.org/10.3842/SIGMA.2021.014  

   Baudoin, Fabrice; Cho, Gunhee (February 2021, Symmetry, Integrability and Geometry: Methods and Applications)

   null (Ed.)

   In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space OH¹. We also obtain two integral representations for the corresponding subelliptic heat kernel. 

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2. Davies’ method for anomalous diffusions

   https://doi.org/10.1090/proc/13324  

   Murugan, Mathav; Saloff-Coste, Laurent (April 2017, Proceedings of the American Mathematical Society)

   Davies’ method of perturbed semigroups is a classical technique to obtain off-diagonal upper bounds on the heat kernel. However Davies’ method does not apply to anomalous diffusions due to the singularity of energy measures. In this note, we overcome the difficulty by modifying the Davies’ perturbation method to obtain sub-Gaussian upper bounds on the heat kernel. 

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3. Sub-Riemannian Limit of the Differential Form Heat Kernels of Contact Manifolds

   https://doi.org/10.1093/imrn/rnaa270  

   Albin, Pierre; Quan, Hadrian (October 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the $$\eta $$-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative $$\eta $$-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological. 

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4. Sobolev and Schatten estimates for the complex Green operator on spheres

   Kim, Elena; Ogden, W. Jacob; Reerink, Tommie; Zeytuncu, Yunus E. (February 2020, New York journal of mathematics)

   The complex Green operator $$\mathcal{G}$$ on CR manifolds is the inverse of the Kohn-Laplacian $$\square_b$$ on the orthogonal complement of its kernel. In this note, we prove Schatten and Sobolev estimates for $$\mathcal{G}$$ on the unit sphere $$\mathbb{S}^{2n-1}\subset \mathbb{C}^n$$. We obtain these estimates by using the spectrum of $$\boxb$$ and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator. 

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5. Logarithmic Sobolev Inequalities on Homogeneous Spaces

   https://doi.org/10.1093/imrn/rnae205  

   Gordina, Maria; Luo, Liangbing (September 2024, International Mathematics Research Notices)

   Abstract We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a sub-Riemannian structure induced by a transitive action by a Lie group. Then the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such spaces. We show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we show that the constant is independent of the dimension of the underlying spaces in several examples. 

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