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Michael M. H. Pang

Publications and source records attributed to Michael M. H. Pang.

8 recordsLinked to original sources
arXiv2025

Optimal Power-Weighted Birman--Hardy--Rellich-type Inequalities on Finite Intervals and Annuli

We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power-weighted Birman-Hardy-Rellich-type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one-dimensional Hardy-type result in differential form to derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman-Hardy-Rellich-type sequence of power-weighted higher-order Hardy-type inequalities for annuli. In the limit as the annulus approaches $\mathbb{R}^n\backslash\{0\}$, we recover well-known prior results on Rellich-type inequalities on $\mathbb{R}^n\backslash\{0\}$.

math.CA↗
arXiv2021

Bessel-Type Operators and a refinement of Hardy's inequality

The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^πdx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^πdx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^πdx \, |f(x)|^2,\quad f\in H_0^1 ((0,π)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrödinger operator associated with the differential expression \begin{align*} τ_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,π). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^πdx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^πdx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,π)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^πdx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^πdx \, \dfrac{|f(x)|^2}{d_{(0,π)}(x)^2}, \quad f\in H_0^1 ((0,π)), \end{align*} where $d_{(0,π)}(x)$ represents the distance from $x \in (0,π)$ to the boundary $\{0,π\}$ of $(0,π)$.

math.SP↗
arXiv2021

On domain properties of Bessel-type operators

Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,π)$, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $L^2((a,b); dx)$ associated with differential expressions of the form \[ ω_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} τ_{s_a,s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a,b),& \\ s_a, s_b \in [0,\infty), \; q \in L^{\infty}((a,b); dx), \; q \text{ real-valued~a.e.~on $(a,b)$,}& \end{align*} where $(a,b) \subset \mathbb{R}$ is a bounded interval. As an explicit illustration we describe the Krein-von Neumann extension of the minimal operator corresponding $ω_{s_a}$ and $τ_{s_a,s_b}$.

math.CA↗
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 Müller-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↗