Bessel-Type Operators and a Refinement of Hardy’s Inequality

On domain properties of Bessel-type operators

“…We emphasize that all constants in (1.2)- (1.4) are optimal and all inequalities are strict in the sense that equality holds in them if and only if f ≡ 0. We also stress that inequality (1.4) (for a = 0, b = π) was first proved by Avkhadiev [6, Lemma 1], a fact that was unfortunately missed in [26].…”

“…Due to the enormity of the literature on Bessel-type operators, an exhaustive survey of the literature on this subject is an insurmountable task. Hence, we conclude this introduction with a brief discussion of the literature in the immediate vicinity of the circle of ideas discussed in this paper (for additional comments in this direction we refer the reader to [26]). For some background on Bessel operators and their spectral properties we refer, for instance, to [2, p. 544-552], [15], [16, p. 1532-1538], [18], [25], [43, p. 142-144], [53, p. 81-90], and the literature cited therein.…”

Self‐adjointness for the MIT bag model on an unbounded cone

“…Proof To show (48), we exploit the following Hardy inequality: for all $$g \in H_0^1((0,\pi ))$$ $$$$\begin{equation} \int _0^{\pi } |g^{\prime }(\theta )|^2 \,{{d}}\theta \ge \frac{1}{4}\int _0^{\pi } \frac{|g(\theta )|^2}{\sin ^2 \theta }\,{{d}}\theta + \frac{1}{4}\int _0^{\pi } |g(\theta )|^2 \,{{d}}\theta . \end{equation}$$$$Such inequality can be derived from the inequality $$$$\begin{equation*} 0 \le \int _0^{\pi } \Big \vert g^{\prime }(\theta ) - \frac{\cos \theta }{2\sin {\theta }}g(\theta )\Big \vert ^2 \, {{d}}\theta \end{equation*}$$$$expanding the square and integrating by parts: for a detailed proof and interesting details on how this inequality is related to Bessel‐type operators we refer to [33]. Let $$f \in C_0^{\infty }((0,\omega ])$$ and define …”

“…expanding the square and integrating by parts: for a detailed proof and interesting details on how this inequality is related to Bessel-type operators we refer to [33]. Let 𝑓 ∈ 𝐶 ∞ 0 ((0, 𝜔]) and define…”

On the domains of Bessel operators

“…The analysis of Bessel forms in the self-adjoint case, that is for real m > −1, is well known-it is essentially equivalent to the famous Hardy inequality. We will not discuss the literature on this subject, except that we want to mention a recent interesting paper [10] about a refinement of Hardy's inequality. This paper contains many references about the Hardy inequality and factorizations of the Bessel operators in the self-adjoint case.…”