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

- Fritz Gesztesy fritz_gesztesy@baylor.edu
- Isaac Michael imichael@lsu.edu
- Michael M. H. Pang pangm@missouri.edu

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https://doi.org/10.4067/S0719-06462022000100115## 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.

## Keywords

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*CUBO*, vol. 24, no. 1, pp. 115–165, Apr. 2022.