Operator properties of Sobolev imbeddings over unbounded domains

Abstract

For an open set Ω ⊂ R N, 1 ⩽ p ⩽ ∞ and λ ∈ R +, let W ̊ p λ(Ω) denote the Sobolev-Slobodetzkij space obtained by completing C 0 ∞(Ω) in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “ r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971 , pp. 203–215). Choose a Banach ideal of operators U , 1 ⩽ p, q ⩽ ∞ and a quasibounded domain Ω ⊂ R N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map W ̊ p λ(Ω) λ L q(Ω) exists and belongs to the given Banach ideal U : Assume the quasibounded domain fulfills condition C k l for some l > 0 and 1 ⩽ k ⩽ N. Roughly this means that the distance of any x ϵ Ω to the boundary ∂Ω tends to zero as O(¦ x ¦ −l) for ¦ x ¦ → ∞, and that the boundary consists of sufficiently smooth ⩾( N − k)-dimensional manifolds. Take, furthermore, 1 ⩽ p, q ⩽ ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ N , μ > λ S( U ; p,q:N) and v > N/l · λ D( U ;p,q), one has that W ̊ p λ(Ω) λ L q(Ω) belongs to the Banach ideal U . Here λ D( U ;p,q;N)∈ R + and λ S( U ;p,q;N)∈ R + are the D-limit order and S-limit order of the ideal U , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l p n → l q n for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators ( p = q = 2) as well as results on imbeddings over bounded domains. Similar results over general unbounded domains are indicated for weighted Sobolev spaces. As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in L 2(Ω) , where Ω fulfills condition C 1 l . For an open set Ω in R N, let W ̊ p λ(Ω) denote the Sobolev-Slobodetzkij space obtained by completing C 0 ∞(Ω) in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ⩽ p, q ⩽ ∞, we consider quasibounded domains Ω in R N and give sufficient conditions on λ such that the Sobolev imbedding operator W ̊ p λ(Ω) λ L q(Ω) exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators ( p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in L 2(Ω) , where Ω is a quasibounded open set in R N.