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so that,
In a similar way, it holds
This in (6.21) completes the proof.
SUSANA D. MOURA
Ψ♯ (β−1 ) ≤ cΨ♯ (β−1). d,n j+1 d,n
Ψ♯ (β−1) ≤ c Ψ♯
d,n
(β−1 ). j+1
d,n
As a direct consequence of Theorem 6.3 and Corollary 6.2 (with b = 1 ∈ L∞(Γ)) we can state the following:
Corollary 6.4. Under the same assumptions of Theorem 6.3 and assuming further that Ψ♯d, the Bruijn conjugate of Ψ(r1/d)−1, satisfy
Ψ♯d(ra) ∼ Ψ♯d(r)
for any a > 0, then there are positive numbers c1, c2 and β0 such that
d −1
βd+1 Ψβ−1
2−n+d
Ψ(r1/(2−n+d))−1, respectively, according to Proposition 2.1(iv).
Remark 6.3. Having into consideration what we have mentioned in Remark 6.2, the above corollary improves [24, Theorem 4.2.5]. We should remark that the condition (6.14) is satisfied for all admissible functions (see e.g. [9, Lemma 2.3(ii)]) but not for all slowly varying functions. Moreover, there is no easy method for calculating Bruijn conjugates; we refer to [1, Appendix 5] in this connection.
d+1,
n=1,
d ♯ −1 n−2
c1 β2−n+d Ψd,n(β )2−n+d ≤ Nβ ≤ c2
β, β d
n = 2, 2−n+d, n>2,
n−2
for all β > β0, where Ψ♯d and Ψ♯d,n denote the Bruijn conjugates of Ψ(r1/d)−1 and
n−2 (1/Ψ♯)♯ β− d
[1] [2] [3] [4]
[5] [6]
References
N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, Cambridge University Press, 1987.
M. Bricchi and S. D. Moura, Complements on growth envelopes of spaces with generalised smoothness in the sub-critical case, Z. Anal. Anwendungen 22 (2003), no. 2, 383-398.
M. Bricchi, On the relationship between Besov spaces B(s,Ψ)(Rn) and Lp-spaces defined on pq
a (d, Ψ)-set, Preprint Math/Inf/00/13, Universit¨at Jena, 2000.
, Tailored function spaces and related h-sets, Ph.D. thesis, Friedrich-Schiller-
Universita¨t Jena, 2001.
, Existence and properties of h-sets, Georgian Math. J. 9 (2002), 13-32.
, Complements and results on h-sets, Function Spaces, Differential Operators and
Nonlinear Analysis. The Hans Triebel Anniversary Volume (Basel) (D. Haroske, T. Runst, and H.-J. Schmeisser, eds.), 219-229, Birkh¨auser, BAsel, 2003.
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