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

Abstract

We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B_{p,q}^s({\mathbb{R}}^n)$ and $F_{p,q}^s({\mathbb{R}}^n)$ for $s={\sigma }_p=n\max (\frac{1}{p}-1,0)$. These results may be also reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}({\mathbb{R}}^n)$. We close several open problems outlined already by H. Triebel in [H. Triebel, The structure of functions, Birkhäuser, 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.].