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 ({\it i.e.}, sharpness) of the constants $A(m, \alpha)$ and $B(m, \alpha)$, $m \in \bbN$, $\alpha \in \bbR$, 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 \cite{GLMP20}, 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} \no \\&\quad+ B(m, \alpha) \sum_{k=1}^{\kk} \int_0^{\rho} dx \, x^{\alpha - 2m}\prod_{p=1}^{k} [\ln_{p}(\gamma/x)]^{-2} \big|f(x)\big|^{2}, \no \\ & \, f \in C_{0}^{\infty}((0, \rho)), \; m, \kk \in \bbN, \; \alpha \in \bbR, \; \rho, \gamma \in (0,\infty), \; \gamma \geq e_{\kk} \rho.\end{align*} Here the iterated logarithms are given by \[\ln_{1}( \, \cdot \,) = \ln(\, \cdot \,), \quad \ln_{j+1}( \, \cdot \,) = \ln( \ln_{j}(\, \cdot \,)), \quad j \in \bbN,\] and the iterated exponentials are defined via \[e_{0} = 0, \quad e_{j+1} = e^{e_{j}}, \quad j \in \bbN_{0} = \bbN \cup \{0\}. \] 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

• birman-hardy-rellich inequalities
• logarithmic refinements