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 $B^s_{p,q}(\mathbb{R}^n)$ and $F^s_{p,q} (\mathbb{R}^n)$ in the limiting case $s=\sigma_p:=n\max(\frac 1p-1,0)$. These results may also be reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}(\mathbb{R}^n)$. We close several open problems outlined already in [H. Triebel, The structure of functions, Birkhäuser, Basel, 2001] and explicitly stated in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall/CRC, Boca Raton, FL, 2007].