A note on spectral multipliers on Engel and Cartan groups

Chatzakou, Marianna

doi:10.1090/proc/15830

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…r = 1 in (2)) is called the shear group and it was used in a similar context in [3,4]. This nilpotent step 3 group is also known as the Engel group [9].…”

Section: Therefore the Transformation θmentioning

confidence: 99%

Metamorphism as a covariant transform for the SSR group

Taghreed¹,

Kisil²

2023

Preprint

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Metamorphism is a recently introduced integral transform, which is useful in solving partial differential equations. Basic properties of metamorphism can be verified by direct calculations. In this paper we present metamorphism as a sort of covariant transform and derive its most important features in this way. Our main result is a characterisation of metamorphism's image space. Reading this paper does not require advanced knowledge of group representations or theory of covariant transform.

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“…r = 1 in (2)) is called the shear group and it was used in a similar context in [3,4]. This nilpotent step 3 group is also known as the Engel group [9].…”

Section: Therefore the Transformation θmentioning

confidence: 99%

Metamorphism as a covariant transform for the SSR group

Taghreed¹,

Kisil²

2023

Preprint

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“…We want to note that the aforesaid family does not necessarily include all the Carnot group with a filiform Lie algebra, and therefore, in the present work one might extend the validity of global Poincaré inequalities to a bigger class of Carnot groups with a Lie algebra of filiform type as well. To be more precise let us first recall the definition of the Carnot groups of "Engel type", see [15], [14], as appeared in [16]; a group G is of this type if G = (R 𝑛+1 , •), with 𝔤 𝑛+1 = span{𝑋 1 , ..., 𝑋 𝑛+1 } and [𝑋 1 , 𝑋 𝑗 ] =𝑋 𝑗+1 , ∀ 𝑗 = 1, ..., 𝑛, [𝑋 𝑖 , 𝑋 𝑗 ] =0, 2 ⩽ 𝑖, 𝑗 ⩽ 𝑛 + 1 [𝑋 1 , 𝑋 𝑛+1 ] =0.…”

Section: Carnot Groups Of Step 𝒓 ⩾mentioning

confidence: 99%

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

Chatzakou

Federico

Zegarliński

2023

Preprint

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In this work we give a sufficient condition under which the global Poincaré inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. The density of such probability measure is given in terms of homogeneous quasi-norm on the group. We provide examples to which our condition applies including the most known families of Carnot groups. This, in particular, allows to extend the results in our previous work [16]. A consequence of our result is that the associated Schrödinger operators have a spectral gap.

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“…For the homogeneous dimension Q we have Q = 7. For a detailed description of the Engel group we refer to [8] and [9].…”

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confidence: 99%

Heat and wave type equations with non-local operators, II. Hilbert spaces and graded Lie groups

Chatzakou¹,

Restrepo²,

Ruzhansky³

2023

Preprint

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We study heat and wave type equations on a separable Hilbert space H by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the solution and analyse the time-decay rate in a scale of suitable Sobolev space. We perform similar analysis on multi-term heat and multi-wave type equations. The main tool here is the Fourier analysis which can be developed in a separable Hilbert space based on the linear operator involved. As an application, the same Cauchy problems are considered and analysed in the setting of a graded Lie group. In this case our analysis relies on the group Fourier analysis. An extra ingredient in this framework allows, in the case of heat type equations, to establish L p -L q estimates for 1 p 2 q < +∞ for the solutions on graded Lie group groups. Examples and applications of the developed theory are given, either in terms of self-adjoint operators on compact or non-compact manifolds, or in the case of particular settings of graded Lie groups. The results of this paper significantly extend in different directions the results of Part I ([40]), where operators on compact Lie groups were considered. We note that the results obtained in this paper are also new already in the Euclidean setting of R n .

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A note on spectral multipliers on Engel and Cartan groups

Cited by 5 publications

References 23 publications

Metamorphism as a covariant transform for the SSR group

Metamorphism as a covariant transform for the SSR group

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

Heat and wave type equations with non-local operators, II. Hilbert spaces and graded Lie groups

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