The weighted Hardy inequality and self-adjointness of symmetric diffusion operators

Robinson, Derek W.

doi:10.1016/j.jfa.2021.109067

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…Here, W 1,2 0,α,β (Ω) is the completion of C ∞ 0 (Ω) with respect to the norm • α,β . For a generalized Hardy's inequality on general domains, refer to [2], [15], [16] and references therein. Some general results on the positivity and attainability of H α,β will be given in Proposition 2.4, Proposition A.2 and Proposition A.4.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

The Legendre-Hardy inequality on bounded domains

Byeon

Jin

2022

Trans. Amer. Math. Soc. Ser. B

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There have been numerous studies on Hardy’s inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C 2 C^2 -domain in R n \mathbb {R}^n of the following form ∫ Ω d β ( x ) | ∇ u ( x ) | 2 d x ≥ C ( α , β ) ∫ Ω | u ( x ) | 2 d α ( x ) d x with ∫ Ω u ( x ) d α ( x ) d x = 0 , \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d ( x ) d(x) is the distance from x ∈ Ω x \in \Omega to the boundary ∂ Ω \partial \Omega and α , β ∈ R \alpha ,\beta \in \mathbb {R} . We classify all ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 for which C ( α , β ) > 0 C(\alpha ,\beta ) > 0 . Then, we study whether an optimal constant C ( α , β ) C(\alpha ,\beta ) is attained or not. Our study on C ( α , β ) C(\alpha ,\beta ) for general ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {R}^2 shows that the (classical) Hardy inequality can be regarded as a special case of the Neumann version.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

The Legendre-Hardy inequality on bounded domains

Byeon

Jin

2022

Trans. Amer. Math. Soc. Ser. B

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“…Derek had never met Juha, but was thrilled to see him on Zoom at the meeting. In his penultimate paper [19], Derek included a little fractal picture.…”

Section: Michael Barnsleymentioning

confidence: 99%

“…The details were written in another book [4]. After that Derek got more and more interested in degenerate elliptic operators and the relation with Hardy inequalities [19].…”

Section: Afm Ter Elstmentioning

confidence: 99%