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

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.].