Page 98 - Textos de Matemática Vol. 34

88 JU´LIO S. NEVES
where |∇u| denotes the Euclidean norm of the gradient of u, Lemma 3.2 (with ν = −1 < 0) and the change of variables, we obtain
ω(u,t) t
t−1/r λr(t) r;(0,1)
t−1−1/r br (tn ) |∇u|∗ (τ n ) dτ
0 r;(0,1)
∥t−1+1/r′ br(tn) |∇u|∗(tn)∥r;(0,1)
≈ ∥t−1/rbr(t) |∇u|∗(t)∥r;(0,1).
Moreover, the estimate |∇u|∗(t) ≤ |∇u|∗∗(t) for every t ∈ (0,1), (2.1), Lemma 3.2 (with ν = −1 < 0) and (5.29) imply that
∥t−1/rbr(t) |∇u|∗(t)∥r;(0,1)
∥t−1/rbr(t) |∇u|∗∗(t)∥r;(0,1)
Consequently,
(5.31) t λ (t)
−1/r ω(u,t)
n−1/r t
∥u∥σ;n/(σ−1),q;b.
i=1 n−1/r
∂u∗∗ br(t)|∂x | (t)
i r;(0,1) ∂u∗
br(t)|∂x | (t)
i r;(0,1)
≈ t i=1
∥u∥σ;n/(σ−1),q;b.
By Proposition 5.4, we also have ∥u∥∞ ∥u∥σ;n/(σ−1),q;b. This and (5.31) yield
r r;(0,1)
∥u|Λλr(.)∥ ∥u∥ for all u ∈ S(Rn). ∞,r σ;n/(σ−1),q;b
References

[1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
[2] C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol.
129, Academic Press, New York, 1988.
[3] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation, Cambridge University Press,
Cambridge, 1987.
[4] H. Br´ezis and S. Wainger, A note on limiting cases of Sobolev embeddings, Comm. Partial
Differential Equations 5 (1980), 773-789.
[5] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der Mathematis-
chen Wissenschaften–A series of Comprehensive Studies in Mathematics, vol. 303, Springer-
Verlag, Berlin, 1993.
[6] D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability, Bessel potentials
and embedding theorems, Studia Math. 115 (1995), 151-181.
[7] , On embeddings of logarithmic Bessel potential spaces, J. Funct. Anal. 146 (1997),
no. 1, 116-150.