Sharp lower bounds on the manifold widths of Sobolev and Besov spaces (Q6614421)

In the paper, the author proves a matching lower bound for the manifold widths of Sobolev and Besov unit \(L_q\)-balls with respect to the \(L_p\)-norm for all \(1\leq p,q\leq \infty\) for which a compact embedding holds. This solves an open problem for the case \(p<q\) and \(q>2\), where so far no mathching lower bound was known.\N\NIn order to bridge the gap, the author introduces a new notion of the ``sphere embedding widths'', that can decay slower than the Bernstein widths, which are used in the cases \(q\leq p\) and \(1\leq p\leq q\leq 2\) for getting the lower bound on the manifold widths, and which decay faster in the so-far open case \(p<q\) and \(q>2\).