Anisotropic Hardy inequalities

Pietra, Francesco Della; Blasio, Giuseppina di; Gavitone, Nunzia

doi:10.1017/s0308210517000336

Cited by 11 publications

(8 citation statements)

References 20 publications

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“…in Ω, we have

- {normalΔ}_{H, p} d_{H} = - {normalΔ}_{H} d_{H} \geq 0

. Thus Theorem 4.2 is an improvement of the result proved in [19], which gives the inequality (4.4) when

p = 2

under the assumption

- {normalΔ}_{H} d_{H} \geq 0

.…”

Section: Hardy Inequality Of Geometric Typesupporting

confidence: 54%

“…Then we have

H (\nabla d_{H} false(x false)) = 1 normala . normale . in Ω

and

\frac{1}{β} d false(x false) \leq d_{H} false(x false) \leq \frac{1}{α} d false(x false)

where

d false(x false) = {trueprefixinf}_{y \in \partial Ω} false| x - y false|

is the Euclidean distance from the boundary

\partial Ω

. In [19], the authors studied the anisotropic Hardy inequality of geometric type as follows: Theorem Suppose

d_{H}

is a

Δ_{H}

‐superharmonic in Ω, i.e.,

- {normalΔ}_{H} d_{H} \geq 0

in the distribution sense. Then the inequality

\frac{1}{4} \int_{normalΩ} \frac{{false| u false|}^{2}}{{true(d_{H} (x) true)}^{2}} d x \leq \int_{normalΩ} {(H false(\nabla u false))}^{2} d x

holds true for any

u \in C_{0}^{\infty} false(normalΩ false)

.…”

Section: Hardy Inequality Of Geometric Typementioning

confidence: 99%

“…Note that if

Ω

is convex, the assumption

- {normalΔ}_{H} d_{H} \geq 0

holds true. In [19], it is shown that there exists a non‐convex domain Ω such that

d_{H}

Δ_{H}

‐superharmonic on Ω. For the Euclidean geometric type Hardy inequalities, see [16], [22], [26], [39], and references there in.…”

Section: Hardy Inequality Of Geometric Typementioning

confidence: 99%

“…Anisotropic Hardy inequalities are also known. For example (1.1) and (1.2) have been extended to the case where the Euclidean norm is replaced by a Finsler norm in [40], [19] when

p = 2

, and [8], [9], [15] when

p \neq 2

, respectively. The method in [15] and [8] is to use Picone type identities in Finsler setting.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finsler Hardy inequalities

Mercaldo

Sano

Takahashi

2020

Mathematische Nachrichten

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“…in Ω, we have

- {normalΔ}_{H, p} d_{H} = - {normalΔ}_{H} d_{H} \geq 0

. Thus Theorem 4.2 is an improvement of the result proved in [19], which gives the inequality (4.4) when

p = 2

under the assumption

- {normalΔ}_{H} d_{H} \geq 0

.…”

Section: Hardy Inequality Of Geometric Typesupporting

confidence: 54%

“…Then we have

H (\nabla d_{H} false(x false)) = 1 normala . normale . in Ω

and

\frac{1}{β} d false(x false) \leq d_{H} false(x false) \leq \frac{1}{α} d false(x false)

where

d false(x false) = {trueprefixinf}_{y \in \partial Ω} false| x - y false|

is the Euclidean distance from the boundary

\partial Ω

. In [19], the authors studied the anisotropic Hardy inequality of geometric type as follows: Theorem Suppose

d_{H}

is a

Δ_{H}

‐superharmonic in Ω, i.e.,

- {normalΔ}_{H} d_{H} \geq 0

in the distribution sense. Then the inequality

\frac{1}{4} \int_{normalΩ} \frac{{false| u false|}^{2}}{{true(d_{H} (x) true)}^{2}} d x \leq \int_{normalΩ} {(H false(\nabla u false))}^{2} d x

holds true for any

u \in C_{0}^{\infty} false(normalΩ false)

.…”

Section: Hardy Inequality Of Geometric Typementioning

confidence: 99%

“…Note that if

Ω

is convex, the assumption

- {normalΔ}_{H} d_{H} \geq 0

holds true. In [19], it is shown that there exists a non‐convex domain Ω such that

d_{H}

Δ_{H}

‐superharmonic on Ω. For the Euclidean geometric type Hardy inequalities, see [16], [22], [26], [39], and references there in.…”

Section: Hardy Inequality Of Geometric Typementioning

confidence: 99%

“…Anisotropic Hardy inequalities are also known. For example (1.1) and (1.2) have been extended to the case where the Euclidean norm is replaced by a Finsler norm in [40], [19] when

p = 2

, and [8], [9], [15] when

p \neq 2

, respectively. The method in [15] and [8] is to use Picone type identities in Finsler setting.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finsler Hardy inequalities

Mercaldo

Sano

Takahashi

2020

Mathematische Nachrichten

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“…if Ω = R n + := {x = (x 1 , ..., x n ) ∈ R n | x n > 0}, where we simply have d(x) = x n and so −∆d = 0, choosing α = (p − 1)(p + n − 1), the corresponding inequality R n + x p−1 n |∇v| p dx ≥ C(n, p) R n + x p−1 n |v| p(p+n−1)/(n−1) dx (n−1)/(p+n −1) had been established using different approaches for example in [22, §6] for p = 2, in [8,25] for general p > 1 and in [11] for the fractional case. Improved versions of the anisotropic Hardy inequality for p = 2 have been investigated in [7]. In the last section of our paper we adapt the (by now classical) method from [5] to our case and, once (1.3) has been established, we deduce a series of improved versions of the anisotropic Hardy inequality.…”

Section: Introductionmentioning

confidence: 99%

A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

2020

Self Cite

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A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities

Kajántó,

Kristály,

Peter

et al. 2024

Math. Ann.

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We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed.

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Anisotropic Hardy inequalities

Abstract: We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

Cited by 11 publications

References 20 publications

Finsler Hardy inequalities

Finsler Hardy inequalities

A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities

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