On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case

TL;DR

This paper investigates the optimal embeddings of Besov and Triebel-Lizorkin spaces into Lorentz spaces at critical smoothness levels, resolving several open problems in the field.

Contribution

It provides sharp embedding results for Besov and Triebel-Lizorkin spaces in the subcritical case, addressing open questions posed by Triebel and Haroske.

Findings

Established precise growth envelope characterizations.

Resolved open problems on embeddings at critical smoothness.

Connected embeddings to Lorentz space scale.

Abstract

We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B^s_{p,q}(\R^n)$ and $F^s_{p,q}(\R^n)$ for $s=\sigma_p=n\max(\frac 1p-1,0)$. These results may be also reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}(\R^n)$. We close several open problems outlined already by H. Triebel in [H. Triebel, The structure of functions, Birkh\"auser, Basel, 2001.] and explicitly formulated by D. D. Haroske in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall / CRC, Boca Raton, 2007.].