Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

Ruzhansky, Michael; Sabitbek, Bolys; Сураган, Дурвудхан

doi:10.48550/arxiv.1806.06226

“…The approach to prove the main results is based on the works [11] and [12] (see, also [13]- [14]). For a vector field g ∈ C ∞ (Ω) we compute…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

Chapter 8 Geometric Hardy Inequalities on Stratified Groups

Ruzhansky

¹

,

Сураган

²

2019

Hardy Inequalities on Homogeneous Groups

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In this paper, we present the geometric Hardy inequalities on the starshaped sets in the Carnot groups. Also, we obtain the geometric Hardy inequalities on half-spaces for general vector fields. Sec. 1: We give a brief overview of the sub-Riemannian manifolds, Grushin plane, Carnot groups, Heisenberg groups, and Engel groups. Sec. 2: We obtain the geometric Hardy inequalities on the starshaped sets in the Carnot groups and provide some examples. Sec. 3: We obtain the geometric Hardy inequalities on the half-spaces for general vector fields and provide some examples. Sec. 4: We give the proofs of main results. 1991 Mathematics Subject Classification. 35A23, 35H20, 35R03.

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“…The Hardy inequality was intensively studied. For example, the Hardy inequalities were considered on the Euclidean space in [19], [20], on the Heisenberg group in [6], [7], on stratified groups in [3], [31,32,33], [36], [42], for the vector fields in [34] and [38], on homogeneous groups in [37], [39]. In [37] (see e.g., [35]), authors showed the Hardy inequality with radial derivative on homogeneous groups in the following form: Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups

Kassymov

¹

,

Ruzhansky

²

,

Сураган

³

2022

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In this note, we prove the reverse Stein–Weiss inequality on general homogeneous Lie groups. The results obtained extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we prove reverse Hardy, Hardy–Littlewood–Sobolev, L p {L^{p}} -Sobolev and L p {L^{p}} -Caffarelli–Kohn–Nirenberg inequalities on homogeneous Lie groups.

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“…The Hardy inequality was intensively studied. For example, the Hardy inequalities were considered on the Euclidean space in [19], [20], on the Heisenberg group in [6], [7], on stratified groups in [3], [31,32,33], [36], [42], for the vector fields in [34] and [38], on homogeneous groups in [37], [39]. In [37] (see e.g., [35]), authors showed the Hardy inequality with radial derivative on homogeneous groups in the following form: Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%