Optimality of constants in power-weighted Birman–Hardy–Rellich-Type inequalities with 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.