# Optimality of constants in power-weighted Birman–Hardy–Rellich-Type inequalities with logarithmic refinements

• Fritz Gesztesy Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S., 4th Street, Waco, TX 76706, USA.
• Isaac Michael Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA.
• Michael M. H. Pang Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.
Keywords: Birman-Hardy-Rellich inequalities, logarithmic refinements

### Abstract

The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants $$A(m, \alpha)$$ and $$B(m, \alpha)$$, $$m \in \mathbb N$$, $$\alpha \in \mathbb R$$, of the form  \begin{align*} &A(m, \alpha) = 4^{-m} \prod_{j=1}^{m} (2j - 1 -\alpha)^2, \\ &B(m, \alpha) = 4^{-m} \sum_{k=1}^{m} \ \prod_{\substack{j = 1\\ j \ne k}}^{m} ( 2j - 1 - \alpha )^{2}, \end{align*} in the power-weighted Birman--Hardy--Rellich-type integral inequalities with logarithmic refinement terms recently proved in [41], namely, \begin{align*} &\int_0^{\rho} dx \, x^{\alpha} \big| f^{(m )}(x) \big|^{2} \geq A(m, \alpha) \int_0^{\rho} dx \,  x^{\alpha - 2m} \big|f(x)\big|^{2}  \\ &\quad+ B(m, \alpha) \sum_{k=1}^{N} \int_0^{\rho} dx \, x^{\alpha - 2m}\prod_{p=1}^{k} [\ln_{p}(\gamma/x)]^{-2} \big|f(x)\big|^{2},   \\ & \, f \in C_{0}^{\infty}((0, \rho)), \; m, {N} \in \mathbb N, \; \alpha \in \mathbb R, \; \rho, \gamma \in (0,\infty), \; \gamma \geq e_{N} \rho. \end{align*} Here the iterated logarithms are given by $\ln_{1}( \, \cdot \,) = \ln(\, \cdot \,), \quad \ln_{j+1}( \, \cdot \,) = \ln( \ln_{j}(\, \cdot \,)), \quad j \in \mathbb N,$ and the iterated exponentials are defined via $e_{0} = 0, \quad e_{j+1} = e^{e_{j}}, \quad j \in \mathbb N_{0} = \mathbb N \cup \{0\}.$ Moreover, we prove the analogous sequence of inequalities on the exterior interval $$(r,\infty)$$ for $$f \in C_{0}^{\infty}((r,\infty))$$, $$r \in (0,\infty)$$, and once again prove optimality of the constants involved.

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Published
2022-04-12
How to Cite
[1]
F. Gesztesy, I. Michael, and M. M. H. Pang, “Optimality of constants in power-weighted Birman–Hardy–Rellich-Type inequalities with logarithmic refinements”, CUBO, vol. 24, no. 1, pp. 115–165, Apr. 2022.
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