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

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