A Sequence of Weighted Birman–Hardy–Rellich Inequalities with Logarithmic Refinements

Abstract

The principal aim of this paper is to extend Birman’s sequence of integral inequalities originally obtained in Mat. Sb. (N.S.) 55(97), 125–174, (1961), and containing Hardy’s and Rellich’s inequality as special cases, to a sequence of inequalities that incorporates power weights \(x^{\alpha }\) for x varying in intervals \((0,\rho ), \rho \in (0,\infty ) \cup \{\infty \}\), on either side and logarithmic refinements on the right-hand side of the inequality as well. Employing a new technique of proof relying on a combination of transforms originally due to Hartman and Müller-Pfeiffer, the parameter \(\alpha \in {{\mathbb {R}}}\) in the power weights is now unrestricted, considerably improving on prior results in the literature. We also discuss optimality of the constants in these inequalities. This continues a tradition of logarithmic refinements in connection with Hardy’s inequality, going back to work in oscillation theory by Kneser (Math. Ann. 42, 409–435, (1893)), Hartman (Am. J. Math. 70, 764-779 (1948)), Hille (Trans. Am. Math. Soc. 64, 234-252 (1948)), and Rellich (Math. Ann. 122, 343–368 (1951)), resulting in a sequence of sharp statements of boundedness from below by zero of a class of homogeneous 2mth order differential operators on \(C_0^{\infty }((0,\rho ))\). We also prove the analogous inequalities on exterior intervals, that is, for \(f \in C_0^{\infty }((\rho ,\infty ))\). Finally, we also indicate a vector-valued version of these inequalities, replacing complex-valued \(f(\,\cdot \,)\) by \(f(\,\cdot \,) \in {{\mathcal {H}}}\), with \({{\mathcal {H}}}\) a complex, separable Hilbert space.