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

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.

show abstract

“…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

Сураган

2019

Hardy Inequalities on Homogeneous Groups

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, the sharp constants can be attained if they are strictly less than

\left(\frac{N}{2}\right)^{2}

. The interested reader is referred to, for instance, [2, 5–7, 11, 29, 32, 39, 45, 47] for more information and results on the Hardy type inequalities with boundary singularities.…”

Section: Introductionmentioning

confidence: 99%

Some Hardy identities on half‐spaces

Duy

Lam

Phi

2021

Mathematische Nachrichten

View full text Add to dashboard Cite

We prove some Hardy identities on the half‐space double-struckR+N$\mathbb {R}_{+}^{N}$. Our equalities imply correponding versions of the Hardy type inequalities with exact remainder terms on double-struckR+N$\mathbb {R}_{+}^{N}$. These equalities give straightforward understandings of the optimal constants as well as the nonexistence of nontrivial optimizers for various Hardy type inequalities on half‐spaces. These identities also provide the “virtual” ground state in the sense of Frank and Seiringer [13] for several Hardy type inequalities on double-struckR+N$\mathbb {R}_{+}^{N}$.

show abstract

“…We also mention here that the Hardy type inequalities with radial gradient have been intensively studied recently. See [13,14,15,16,27,30,31,39,41,42,43], for example.…”

Section: Introductionmentioning

confidence: 99%