Hardy’s inequalities in finite dimensional Hilbert spaces

Abstract

We study the behavior of the smallest possible constants d(a, b) and dn in Hardy's inequalities b a 1 x x a f (t)dt 2 dx ≤ d(a, b) b a f 2 (x)dx and n k=1 1 k k j=1 a j 2 ≤ dn n k=1 a 2 k .The exact constant d(a, b) and the exact rate of convergence of dn are established and the extremal function and the "almost extremal" sequence are found.

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

“…It is this simple observation that adds considerable complexity to sharpness proofs for the space C ∞ 0 ((0, ρ)). (The issue of dependence of optimal constants on the underlying function space is nicely illustrated in [30].) By the same token, optimality proofs obtained for C ∞ 0 function spaces automatically hold for larger function spaces as long as the inequalities have already been established for the larger function spaces with the same constants A(m, α), B(m, α).…”