A sharp subelliptic Sobolev embedding theorem with weights

Yung, Po-Lam

doi:10.1112/blms/bdv010

Cited by 16 publications

(7 citation statements)

References 32 publications

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“…The generalized Métivier's index is also known as the non-isotropic dimension (see [21,22,74]), which plays an important role in the geometry and functional settings associated with vector fields X. Note that n + max x∈S r(x) − 1 ≤ νS < nα n for α n > 1, and νS = ν if the closure of S is equiregular and connected.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Chen¹,

Chen²,

Li³

2022

Preprint

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We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators △ X = m j=1 X 2 j on a bounded connected open domain containing the origin, where X 1 , X 2 , . . . , X m are linearly independent smooth vector fields in R n satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that Ω is a bounded connected open domain in R n containing the origin, and its boundary ∂Ω is smooth and non-characteristic for X. Combining the subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry and some refined analysis involving convex geometry, we establish the explicit asymptotic behaviourwhere λ k denotes the k-th Dirichlet eigenvalue of △ X on Ω, Q 0 is a positive rational number, and d 0 is a non-negative integer. Furthermore, we also give the optimal bounds of index Q 0 , which depends on the homogeneous dimension associated with vector fields X 1 , X 2 , . . . , X m .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Chen¹,

Chen²,

Li³

2022

Preprint

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“…After that, the Poincaré inequality, Sobolev embedding theorem, estimation of Green kernel and heat kernel have also been studied by Jerison, Sanchez-Calle, Capogna, Danielli, Garofalo, Yung, etc. One can refer to [6,7,21,22,31,42] as well as the reference therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation

Chen¹,

Chen²,

Yuan³

2022

Preprint

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“…Therefore, we need to introduce the following generalized Métivier's index which is also called the non‐isotropic dimension of

normalΩ

related to

X

(cf. [15, 40, 52]). With the same notations as before, we set here

ν_{j} false(x false) = dim V_{j} false(x false)

and then

ν (x)

, the pointwise homogeneous dimension at

x

, is given by

ν false(x false) : = \sum_{j = 1}^{Q} j false(ν_{j} (x) - ν_{j - 1} (x) false), ν_{0} false(x false) : = 0 .

Then we define

\tilde{ν} : = \max_{x \in \bar{normalΩ}} ν false(x false)

as the generalized Métivier index of

normalΩ

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Estimates of Dirichlet eigenvalues for a class of sub‐elliptic operators

Chen

2020

Proc. London Math. Soc.

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Let normalΩ be a bounded connected open subset in Rn with smooth boundary ∂Ω. Suppose that we have a system of real smooth vector fields X=(X1,X2,…,Xmfalse) defined on a neighborhood of Ω¯ that satisfies the Hörmander's condition. Suppose further that ∂Ω is non‐characteristic with respect to X. For a self‐adjoint sub‐elliptic operator ▵X=−∑i=1mXi∗Xi on normalΩ, we denote its kth Dirichlet eigenvalue by λk. We will provide a uniform upper bound for the sub‐elliptic Dirichlet heat kernel. We will also give an explicit sharp lower bound estimate for λk, which has a polynomially growth in k of the order related to the generalized Métivier index. We will establish an explicit asymptotic formula of λk that generalizes the Métivier's results in 1976. Our asymptotic formula shows that under a certain condition, our lower bound estimate for λk is optimal in terms of the growth of k. Moreover, the upper bound estimate of the Dirichlet eigenvalues for general sub‐elliptic operators will also be given, which, in a certain sense, has the optimal growth order.

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A sharp subelliptic Sobolev embedding theorem with weights

Cited by 16 publications

References 32 publications

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation

Estimates of Dirichlet eigenvalues for a class of sub‐elliptic operators

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