Continuity Envelopes of Spaces of Generalized Smoothness in the Critical Case

Abstract

The continuity envelope for the Besov and Triebel-Lizorkin spaces of generalized smoothness B (s, ) pq (R n ) and F (s, ) pq (R n ), respectively, are computed in the critical case s = n/p, provided that 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.

Optimal Embeddings of Spaces of Generalized Smoothness in the Critical Case

“…A corresponding proof for (ii) can be found in [24,Lemma 2.6]. The proof in the case u = ∞ is analogous.…”

“…We refer to the end of the introduction for an example. Under this circuntances, the continuous envelope of B ( n p , ) p,q (R n ) was obtained in [24,Theorem 3.5]. Furthermore, if ≡ 1, condition (3.8) implies the violation of condition (2.10).…”

“…This assertion is surprising in comparison with all the previously known results, where the optimal index was always q for the Besov spaces and p for the Triebel-Lizorkin spaces. In particular, in [24] the former result of Haroske in [20,Theorem 9.10] was improved and the open question posed by her in [20, Remark 9.11] was answered. However, it turns out now that in this critical situation the continuity envelopes of the spaces mentioned above do not yield optimal embedding results.…”

Lipschitz continuity in Muckenhoupt A1weighted function spaces

“…Remark 5.5. In the same spirit one can prove criteria for limiting embeddings between spaces of type A s p,q (w); for instance, we have shown in [15] with similar arguments that for w ∈ A 1 with (20), 0 < p 0 < p < p 1 < ∞, s 1 < s < s 0 which satisfy (21), and 0…”

“…where we have, in addition, to assume that 0 < p ≤ 1 in case of F n/p p,q (w) and 0 < q ≤ 1 in case of B n/p p,q (w). However, the celebrated result in [21] for the unweighted case is rather tricky to prove, so its weighted counterpart is postponed. Remark 4.7.…”