Let s ∈ R and p ≥ 1; the Sobolev space Lp,s(Rd) is the space of tempered distributions f such that (Id−∆)s/2 f ∈ Lp, where (Id−∆)s/2 is the Fourier multiplier by (1 + |ξ|2)s/2. If s > d/p, then Lp,s is composed of continuous functions; more precisely, the Sobolev embeddings state that Lp,s↩Cs−d/p, see [24, Chapter 11]. In order to state in which sense this embedding is sharp, we need to recall ...

S ome vaccines cause specific side effects, like the rare cases of poliomyelitis after oral poliovirus vaccination. But when people talk about rare side effects related to vaccination in general, two conditions are most often mentioned: anaphylaxis and GuillainBarré syndrome (pronounced “gheeYAN bah-RAY”). A previous column in this series reviewed the spectrum of severe allergic reactions known...