Non-strict singularity of optimal Sobolev embeddings

We investigate the operator-theoretic property of strict singularity for optimal Sobolev embeddings within the general framework of rearrangement-invariant function spaces (r.i. spaces). More specifically, we focus on studying the ``quality'' of non-compactness for optimal Sobolev embeddings Vᵐ₀X(Ω)→ YX(Ω), where X is a given r.i. space and YX is the corresponding optimal target r.i. space (i.e., the smallest among all r.i. spaces). For the class of sub-limiting norms (i.e., the norms whose fundamental function satisfies φYX(t)≈ t-m/nφX(t) as t→0⁺), we construct suitable spike-function sequences that establish a general framework for proving non-strict singularity of optimal (and thus non-compact) sublimiting Sobolev embeddings. As an application, we show that optimal sublimiting Sobolev embeddings are not strictly singular in a rather large subclass of r.i. spaces, namely weighted Lambda spaces X=Λqw, q∈[1, ∞). Except for the endpoint case X=Ln/m,1, our spike-function construction enables us to construct a subspace of Vᵐ₀X that is isomorphic to q, which we then leverage to prove the non-strict singularity of the corresponding optimal Sobolev embedding.