Remarks on a limiting case of Hardy type inequalities

Abstract

The classical Hardy inequality holds in Sobolev spaces W 1,p 0 when 1 p < N . In the limiting case where p = N , it is known that by introducing a logarithmic weight function in the Hardy potential, some inequality which is called the critical Hardy inequality holds in W 1,N 0 . In this note, in order to give an explanation of the appearance of the logarithmic function in the potential, we derive the logarithmic function from the classical Hardy inequality with best constant via some limiting procedure as p N . We show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W 2,p 0 with p ∈ (1, N2 ) and the Poincaré inequality.

Hardy inequality for domains with a geometric boundary condition and applications

“…Although this, it is showed by Solomyak in [27] that there exists a constant $$C&gt;0$$ such that $$$$\begin{equation*} \int _{\mathbb {R}^2}\frac{|u|^2}{|x|^2(1+\log ^2(|x|))}dx\leqslant C\int _{\mathbb {R}^2}|\nabla u|^2dx, \end{equation*}$$$$for any $$u\in C^\infty _0(\mathbb {R}^2)$$ satisfying the mean zero condition $$\int _{\partial B_1(0)}u(x) d\sigma =0$$. For Hardy inequality in the borderline case $$p=N$$ and $$\Omega$$ the unit ball, we refer the reader [14, 25] and references therein.…”

“…We start quoting that Janssen [15] and Pfluger [23] obtained, for any $$1&lt;p&lt;N$$, a constant $$C_0&gt;0$$ such that $$$$\begin{equation} \int_{\mathrm{\Omega}}\frac{{|u|}^{p}}{{(1+|x|)}^{p}}\textit{dx}\leqslant {C}_{0}\left(\int_{\mathrm{\Omega}}{|\nabla u|}^{p}\textit{dx}+\int_{\partial \mathrm{\Omega}}\frac{|x\cdot \nu |}{{(1+|x|)}^{p}}{|u|}^{p}d\sigma \right),u\in {C}_{\delta}^{\infty}(\mathrm{\Omega}), \end{equation}$$$$where $$\nu$$ denotes the unit outward normal vector on $$\partial \Omega$$. For more results concerning Hardy inequalities in the limiting case we refer to Ioku–Ishiwata [14], Laptev [16], Sano–Sobukawa [25], Wang‐Zhu [29] and its references.…”

“…Although this, it is showed by Solomyak in [27] that there exists a constant 𝐶 > 0 such that for any 𝑢 ∈ 𝐶 ∞ 0 (ℝ 2 ) satisfying the mean zero condition ∫ 𝜕𝐵 1 (0) 𝑢(𝑥)𝑑𝜎 = 0. For Hardy inequality in the borderline case 𝑝 = 𝑁 and Ω the unit ball, we refer the reader [14,25] and references therein.…”

Minimization problem associated with an improved Hardy-Sobolev type inequality

“…Indeed, Ioku [10] showed that the improved inequality (2) with s = 0 implies Alvino's inequality [1] which implies the optimal embedding of W 1,N 0 (B R ) into Orlicz space, and also the improved inequality (2) with s = p implies the critical Hardy inequality which implies the embedding of W 1,N 0 (B R ) into the Lorentz-Zygmund space L ∞,N (log L) −1 which is smaller than the Orlicz space. For indirect limiting procedures for the classical Hardy-Sobolev inequalities, see [21], [4], [15]. Based on the transformation (4), the improved inequality (2) on B R equivalently connects to the classical one (1) on the whole space R N .…”

Critical Hardy inequality on the half-space via the harmonic transplantation

“…For the subcritical Rellich inequality, a part of this argument still holds, see §2 in [28]. A limit of the Hardy-Sobolev and the Poincaré inequalities (in some sense) can be considered, see [20] and [6] for taking a limit p ր N or N ր ∞ in the Sobolev inequality respectively, [33] for p ր N in the Hardy inequality, and |Ω| ց 0 in the Poincaré inequality. Also see [32] for a survey.…”