Page 57 - Textos de Matemática Vol. 34

SUBATOMIC REPRESENTATION OF BESSEL POTENTIAL SPACES 47
(summability in S′) for some λ ≡ {λβνm}(β,ν,m)∈I ⊂ C satisfying |λ|ρ,s,p,q < ∞, where | · |ρ,s,p,q has the same meaning as in (4.3).
Moreover, an equivalent quasi-norm in HsLpq is given by inf |λ|ρ,s,p,q,
where, for each f ∈ HsLpq, the infimum is taken over all families λ≡{λβνm}(β,ν,m)∈I ⊂ C satisfying (4.6) and such that |λ|ρ,s,p,q is finite.
Further, given any f ∈ HsLpq, one can choose λ in an optimal way (in the sense that — besides (4.6) — it verifies
|λ|ρ,s,p,q ≤c∥f∥HsLpq, for some c > 0 independent of f), namely
[1] [2]
[3] [4] [5]
[6] [7]
[8] [9]
λ = λ(f) = {2−ρ|β| f,Ψβ,ρ} . νm (β,ν,m)∈I
References
J. Bergh and J. L¨ofstr¨om, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976.
M. Bricchi, Tailored function spaces and related h-sets, Ph.D. thesis, Friedrich-Schiller- Universita¨t Jena, 2001.
N. Dunford and J. T. Schwartz, Linear operators, part I, Interscience, New York, 1957.
S. Lang, Real and functional analysis, 3rd ed., Springer–Verlag, 1993.
S. Moura, Function spaces of generalised smoothness, Dissertationes Math. 398 (2001), 88 pp.
H. Triebel, Theory of function spaces, Birkh¨auser Verlag, Basel, 1983.
, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Am- brosius Barth, Heidelberg, 1995.
, Fractals and spectra, Birkh¨auser Verlag, Basel, 1997.
, The structure of functions, Birkh¨auser Verlag, Basel, 2001.
Anto´nio M. Caetano, Departamento de Matema´tica, Universidade de Aveiro, 3810-193 Aveiro, Portugal
E-mail address: acaetano@mat.ua.pt