ABSTRACT
Therefore, Φ is now treated as a Riesz basis in L˜2ρ(IR N ) [40, 186]. We next
define a little Sobolev space h2mρ of functions v ∈ l2ρ such that ∑ |λβcβ |2 < ∞.
The scalar product and the induced norm in h2mρ are
(v, w)1 = (v, w)0 + (Bv,Bw)0,
and ‖v‖21 = (v, v)1 ≡ ∑(
1 + |λβ |2 )|cβ |2,
(216)
where our bounded operator B : h2mρ → l2ρ has the meaning B : {cβ} → {λβcβ}. This norm is equivalent to the graph norm induced by the positive operator (−B+ aI) with an a > 0. Then, h2mρ is the domain of B in l2ρ, and, by Sobolev’s embedding theorem,
h2mρ ⊂ l2ρ compactly, (217) which follows from the criterion of compactness in lp [242].
