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 () YX (), 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 ₘₗ (t) t^-m/nX (t) as t0^+), 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=qw, q[1, ). Except for the endpoint case X=L^n/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.
Lang et al. (Mon,) studied this question.