Isaac Michael | Searcharxiv

arXiv2020

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

The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand side of the inequality as well. Our new technique of proof for this sequence of inequalities relies on a combination of transforms originally due to Hartman and M\"uller-Pfeiffer. The results obtained considerably improve on prior results in the literature.

math.CA↗

arXiv2019

On Weighted Hardy-Type Inequalities

We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.

math.CA↗

arXiv2017

On Birman's sequence of Hardy-Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_{n}\},$ for $n\in\mathbb{N},$ valid for functions in $C_0^{n}((0,\infty))\subset L^{2}((0,\infty)).$ In particular, $I_{1}$ is the classical (integral) Hardy inequality and $I_{2}$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_{n}([0,\infty))$ of functions defined on $[0,\infty).$ Moreover, $f\in H_{n}([0,\infty))$ implies $f^{\prime}\in H_{n-1}([0,\infty));$ as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0,$ these inequalities hold on the standard Sobolev space $H_0^{n}((0,b))$. Furthermore, in all cases, the Birman constants $[(2n-1)!!]^{2}/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^{2}((0,\infty))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Ces\`aro averaging operator whose spectral properties we determine in detail.

math.SP↗