Page 48 - Textos de Matemática Vol. 34

38 ANTO´NIO M. CAETANO
Bessel potential spaces modelled on Lorentz spaces, something which, in itself, is a new result.
Our departure point as long as subatomic representations are concerned is what is known in this respect for the usual Bessel potential spaces Hs(Rn) = Fs (Rn),
p p,2 s>0,1<p<∞. Thiscanbeseenin[9,Section2]orbyspecializingour
Section 4 below for this setting (taking q = p in Propositions 4.1, 4.2 and Theorem 4.3). We need to explain here what are the functions Ψβ,ρ in the Schwartz class
νm
S(Rn) which are considered there: they are the same as in [9, Corol. 2.12] and
its construction depends directly only on the consideration of the dyadic resolution of unity (φk)k∈N0 in Rn introduced in [9, 2.8] and, obviously, on the ρ, β, ν, m which show up in our Section 4 below. In particular, they are independent of s, p, q considered either in [9, Section 2] or in the present paper.
The Bessel potential spaces HsLpq(Rn), s > 0, 1 < p < ∞, 1 ≤ q ≤ ∞, for which we get subatomic representations are defined by
sn ′n 2s/2ˆ
H Lpq(R ):={f ∈S (R ):∥((1+|x| ) f)ˇ∥Lpq(Rn) <∞},
where Lpq(Rn) are Lorentz spaces and ˆ, ˇ stand, respectively, for Fourier and inverse Fourier transformation. The subatomic results we can prove for them are collected in Section 4.
A word or two about notation: the symbol → shall mean that the space on the left-hand side of the symbol is continuously embedded in the space on its right- hand side; an equality sign between spaces shall mean that each one is continuously embedded in the other; the use of the same letter c in adjacent formulae does not necessarily mean they assume the same value.
2. Vector-valued Lorentz spaces
We shall consider Rn the usual Euclidean space endowed with Lebesgue measure.
We recall that, given a (real or complex) Banach space (E,| · |E), a function f : Rn → E is called strongly measurable if it is a.e. the limit of step functions, i.e., of functions of the form Nj=1 aj χAj , where aj can be any elements of E, N can be any natural number and Aj can be any measurable subsets of Rn with finite measure. It is well-known (see [4, p. 124]) that, in this context, strong measurability of f is equivalent to measurability together with the hypothesis that there exists a null set Z such that f(Zc) is separable. In particular, in the case when E equals R or C the two notions of measurability coincide.
As usual (see [3, pp. 101-106]), the vector space M(Rn,E) of equivalence classes of strongly measurable functions from Rn into E (two functions being in the same class if they are equal a.e., i.e., if they differ on a null set only) is endowed with the metric of convergence in measure, becoming then a topological vector space which, in particular, is Hausdorff. Also as usual, we shall sometimes refer to the