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Bessel-type operators and a refinement of Hardy’s inequality. (English) Zbl 07456826

Gesztesy, Fritz (ed.) et al., From operator theory to orthogonal polynomials, combinatorics, and number theory. A volume in honor of Lance Littlejohn’s 70th birthday. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 285, 143-172 (2021).
Summary: The principal aim of this paper is to employ Bessel-type operators in proving the inequality \[ \begin{aligned} \int_0^\pi dx |f'(x)|^2 \geq \frac{1}{4} \int_0^\pi dx \frac{|f(x)|^2}{\sin^2 (x)}+\frac{1}{4} \int_0^\pi dx |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{aligned} \] 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 \[ \begin{aligned} \tau_s=-\frac{d^2}{dx^2}+\frac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), x \in (0,\pi). \end{aligned} \] The new inequality represents a refinement of Hardy’s classical inequality \[ \begin{aligned} \int_0^\pi dx |f'(x)|^2 \geq \frac{1}{4}\int_0^\pi dx \frac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{aligned} \] and it also improves upon one of its well-known extensions in the form \[ \begin{aligned} \int_0^\pi dx |f'(x)|^2 \geq \frac{1}{4}\int_0^\pi dx \frac{|f(x)|{}^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{aligned} \] where \(d_{(0,\pi)}(x)\) represents the distance from \(x \in (0,\pi)\) to the boundary \(\{0,\pi\}\) of \((0,\pi)\).
For the entire collection see [Zbl 1479.47003].

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
47E05 General theory of ordinary differential operators
34L99 Ordinary differential operators

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