Approximate Estimates for a Differential Operator in a Weighted Hilbert Space

Abstract

We consider the self-adjoint operator L (the Friedrichs extension) associated with the closable form ${a_m}[u,f] = \int_\Omega {\left( {\sum\nolimits_{\left| \alpha \right| = m} {{\rho ^2}(x){D^\alpha }u\overline {{D^\alpha }f} + v_m^2(x)u\bar f} } \right)dx}$ in the space L_2,ω where Ω ⊂ ℝⁿ is a domain, $f,u \in C_0^\infty (\Omega )$, and m ∈ ℕ. Here v_s(x) = ρ(x)h^-s(x), s > 0, and the positive functions ρ(·) and h(·) satisfy some special conditions. The space L_2,ω is a weighted Hilbert space with a locally integrable weight ω(x) positive almost everywhere in Ω. For s > 0 and 1 < p < ∞, we introduce the weighted space of potentials $H_p^s(\rho ,{v_s})$. For s = m ∈ ℕ and p = 2, the space $H_2^m(\rho ,{v_m})$ is the weighted Sobolev space $H_2^m(\rho ,{v_m})$ with the equivalent norm $\left\| {f;W_2^m(\rho ,{v_m})} \right\| = \sqrt {{a_m}[f,f]}$. Descriptions are given of the interpolation spaces H_s obtained by the complex and real interpolation methods from the pair $\left\{ {H_p^{{s_0}}(\rho ,{v_{{s_0}}}), H_p^{{s_1}}(\rho ,{v_{{s_1}}})} \right\}$, 0 < s₀ < s₁. Estimates are derived for the linear widths and Kolmogorov widths of the compact sets $\mathcal{F} = B{H_s}\bigcap {{L^{ - 1}}(B{L_{2,\omega }})}$, s > 0 (where BX is the unit ball in a space X).