Improved Hardy and Rellich inequalities on Riemannian manifolds

Kömbe, Ismail; Özaydın, Murad

doi:10.1090/s0002-9947-09-04642-x

Cited by 97 publications

(121 citation statements)

References 26 publications

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“…Under the same geometric assumptions on the weight function ρ, Kombe and Özaydın obtained in an

L^{p}

version of (1.1):

\int_{M} ρ^{α} {| \nabla ϕ |}^{p} d V \geq {((), \frac{C + α + 1 - p}{p})}^{p} \int_{M} ρ^{α - p} {(||, ϕ)}^{p} d V

where

ϕ \in C_{0}^{\infty} (M ∖ ρ^{- 1} {0})

and

C + 1 + α - p > 0

1 < p < \infty

.…”

Section: Introductionmentioning

confidence: 91%

“…Examples of Cartan-Hadamard manifolds are the Euclidean space R n with the usual metric and the n-dimensional hyperbolic space H n . Under the same geometric assumptions on the weight function ρ, Kombe andÖzaydın obtained in [19] an L p version of (1.1):…”

Section: Introductionmentioning

confidence: 97%

“…In the article , Kombe and Özaydın obtained the following weighted Rellich type inequality

\int_{M} ρ^{α} {| Δ ϕ |}^{2} d V \geq \frac{{(C + α - 3)}^{2} {(C - α + 1)}^{2}}{16} \int_{M} ρ^{α - 4} ϕ^{2} d V

where

ϕ \in C_{0}^{\infty} (M ∖ ρ^{- 1} {0})

and

C + α - 3 > 0,

α < 2

and the weight function ρ satisfies

| \nabla ρ | = 1,

Δ ρ \geq \frac{C}{ρ}

in the sense of distribution. In this paper, we prove weighted

L^{p}

Rellich type inequality involving two weight functions with remainder terms.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

Kömbe

Yener

2015

Math. Nachr.

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“…Under the same geometric assumptions on the weight function ρ, Kombe and Özaydın obtained in an

L^{p}

version of (1.1):

\int_{M} ρ^{α} {| \nabla ϕ |}^{p} d V \geq {((), \frac{C + α + 1 - p}{p})}^{p} \int_{M} ρ^{α - p} {(||, ϕ)}^{p} d V

where

ϕ \in C_{0}^{\infty} (M ∖ ρ^{- 1} {0})

and

C + 1 + α - p > 0

1 < p < \infty

.…”

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 97%

“…In the article , Kombe and Özaydın obtained the following weighted Rellich type inequality

\int_{M} ρ^{α} {| Δ ϕ |}^{2} d V \geq \frac{{(C + α - 3)}^{2} {(C - α + 1)}^{2}}{16} \int_{M} ρ^{α - 4} ϕ^{2} d V

where

ϕ \in C_{0}^{\infty} (M ∖ ρ^{- 1} {0})

and

C + α - 3 > 0,

α < 2

and the weight function ρ satisfies

| \nabla ρ | = 1,

Δ ρ \geq \frac{C}{ρ}

in the sense of distribution. In this paper, we prove weighted

L^{p}

Rellich type inequality involving two weight functions with remainder terms.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

Kömbe

Yener

2015

Math. Nachr.

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“…If (M, F) = (R N , F) is a Minkowski space, the constant 4 (N −2) 2 is optimal, but no minimizers exist; see Van Schaftingen [9]. Note further that Hardy-type inequalities on Riemannian manifolds (M, F) = (M, g) have been studied in the papers of Carron [3], and Kombe and Özaydin [4]. for all u ∈ C ∞ 0 (M).…”

Section: An Example and Two Limiting Cases Of (Sii)x 0 K Ab Pmentioning

confidence: 99%

“…By means of a parameter-depending Hardy inequality, Kombe and Özaydin [4] established a non-sharp version of the uncertainty principle on certain Riemannian manifolds (e.g., on hyperbolic spaces).…”

Section: An Example and Two Limiting Cases Of (Sii)x 0 K Ab Pmentioning

confidence: 99%

A Sharp Sobolev Interpolation Inequality on Finsler Manifolds

Kristály

2014

J Geom Anal

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In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative Ricci curvature which supports the sharp Sobolev interpolation inequality is isometric to a Minkowski space. The proofs are based on properties of the Finsler-Laplace operator and on the Finslerian Bishop-Gromov volume comparison theorem.

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