Improved Hardy and Hardy–Rellich type inequalities with Bessel pairs via factorizations

Hardy-Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

“…|x| − V rr (x) ≥ 0 on (0, R), then the above is true for non radial function as well (we refer [17, Theorem 3.1-3.3] for more insight). We also refer to [14,15,26] for recent results on Hardy-Rellich inequalities and their improvements on the Euclidean space using the approach of Bessel pairs.…”

A Sequence of Weighted Birman-Hardy-Rellich Inequalities with Logarithmic Refinements

“…Since then there has been enormous activity in this context and we mention, for instance, [1,2,3,4,5,6,7,8,9,10,11,12], [14,Chs. 3,5], [16,17,18,21,24,26,27,28,29,30,31,32,33,34,35,36,38,40,47,49,50], [51,Chs. 2,6,7], [59,60,70,71,72,73,75,76,80,82,83], [87, Sect.…”

Some Hardy identities on half‐spaces

“…\end{equation}The sharp constant $$\left(\frac{N-2}{2}\right)^{2}$$ is never attained by nontrivial functions u . This fact can be seen from the following equality that has been set up in, for example, [3, 8]: for $$u\in C_{0}^{\infty }\!\left(\mathbb {R}^{N}\right)$$: $$\begin{equation} {\int _{\mathbb {R}^{N}}}\vert \nabla u\vert ^{2}\,dx={\left(\frac{N-2}{2}\right)}^{2}\! {\int _{\mathbb {R}^{N}}}\frac{\vert u\vert ^{2}}{\vert x\vert ^{2}}\,dx +{\int _{\mathbb {R}^{N}}}\frac{{\left|\!$$…”

“…In [8, 33], the following identity has been established: for $$u\in C_{0}^{\infty }\!\left(\mathbb {R}^{N}\right)$$: $$\begin{equation} {\int _{\mathbb {R}^{N}}}\vert \mathcal {R}u\vert ^{2}\,dx={\left(\frac{N-2}{2}\right)}^{2}\,{\int _{\mathbb {R}^{N}}} \frac{\vert u\vert ^{2}}{\vert x\vert ^{2}}\,dx+{\int _{\mathbb {R}^{N}}}\frac{{\left|\mathcal {R}{\left(\vert x \vert ^{\frac{N-2}{2}}u\right)}\right|}^{2}}{\vert x\vert ^{N-2}}\,dx. \end{equation}$$As a consequence, we obtain $$\begin{equation} {\int _{\mathbb {R}^{N}}}\vert \mathcal {R}u\vert ^{2}\,dx\ge {\left(\frac{N-2}{2}\right)}^{2}\,{\int _{\mathbb {R}^{N}}}\frac{\vert u\vert ^{2}}{\vert x\vert ^{2}}\,dx \end{equation}$$which in turn implies the Hardy inequality (1.2) since …”

“…\end{equation}As a consequence, we obtain $$\begin{equation} {\int _{\mathbb {R}^{N}}}\vert \mathcal {R}u\vert ^{2}\,dx\ge {\left(\frac{N-2}{2}\right)}^{2}\,{\int _{\mathbb {R}^{N}}}\frac{\vert u\vert ^{2}}{\vert x\vert ^{2}}\,dx \end{equation}$$which in turn implies the Hardy inequality (1.2) since $$\begin{equation*} {\int _{\mathbb {R}^{N}}}\vert \nabla u\vert ^{2}\,dx\ge {\int _{\mathbb {R}^{N}}}\vert \mathcal {R}u\vert ^{2}\,dx. \end{equation*}$$In fact, a version of Theorem B with exact remainder terms and radial derivative operator $$\mathcal {R}$$ has also been set up in [8]: Theorem Let $$0<R\le \infty$$, V and W be positive C 1 ‐functions on (0, R ) such that $$\int _{0}^{R} \frac{1}{r^{N-1}V(r) }\,dr=\infty$$ and $${\int _{0}^{R}}r^{N-1}V(r)\,dr<\infty$$. Assume that …”