Abstract
The continuity envelope for the Besov and Triebel-Lizorkin spaces of generalized smoothness B (s,Ψ) pq (ℝn) and F (s,Ψ) pq (ℝn), respectively, are computed in the critical case s=n/p, provided that Ψ satisfies an appropriate critical condition. Surprisingly, in this critical situation, the corresponding optimal index is ∞, when compared with all the known results. Moreover, in the particular case of the classical spaces, we solve an open problem posed by Haroske in Envelopes and Sharp Embeddings of Function Spaces, Research Notes in Mathematics, vol. 437, Chapman & Hall, Boca Raton, 2007. As an immediate application of our results we give an upper estimate for approximation numbers of related embeddings.
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Moura, S.D., Neves, J.S. & Piotrowski, M. Continuity Envelopes of Spaces of Generalized Smoothness in the Critical Case. J Fourier Anal Appl 15, 775–795 (2009). https://doi.org/10.1007/s00041-009-9063-3
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DOI: https://doi.org/10.1007/s00041-009-9063-3