On two-sided and asymptotic estimates for the norms of embedding operators of W̊2 n(-1, 1) into Lq(dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of W̊2 n(-1, 1), n ∈ ℕ, in Lq(dμ), 0 < q < ∞. Conditions on the measure μ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ1, n, β, β > 0, of the boundary value problems (-d2/dx2)nu(x) = λ{pipe}x{pipe}β-1u(x), x ∈ (-1, 1), u(k)(±1) = 0, k ∈ {0, 1,..., n - 1}. © 2014 Pleiades Publishing, Ltd