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For X = Lr , and is a natural extension thereof in the other cases. Note that in the case where X = X(I) = Lq (I, Lr ) with q < ∞, the x dependence is tested in two steps for local regularity in Lr and behaviour at infinity in ∞ , with the time regularity tested in Lq in between the two. Those spaces are adequate for our purposes in subcritical situations, namely for r0 > nσ . In the critical cases r0 = nσ , however, we also need the spaces ∞ 0 (X) = {v : v ∈ ∞ (X) and χi v; X → 0 when |i| → ∞} .

And σ is at most Lr0 critical, namely nσ ≤ r0 , or at most H 1 critical, namely (n−2)σ ≤ 2, with the additional restriction that in the critical cases, u0 should tend to zero (however weakly) in local Lr0 norm or in local H 1 norm at infinity in Rn . This paper is organized as follows. In Sect. 2, as a preparation to the treatment in local spaces, we recall the treatment of the Cauchy problem in global spaces. This section is mostly a rewriting of the treatment given in [7] in a form suitable for later use.

This proves that v ∈ (2). vj (t) r ∞ r 0 (C(I, L )), ≤ε thereby completing the proof of Part In order to study the properties of the free evolution U (t) in local spaces, it is useful to develop a limited amount of abstract theory. We consider two (global) spaces X, X of the type considered above and an operator B from ∞ (X ) to L1loc of the appropriate variables (space or space time, depending on the nature of X), so that in particular ∞ (X) ⊂ L1loc . We assume that 58 (i) J. Ginibre, G. Velo for any bounded compactly supported ϕ and for any v ∈ → ϕ, B χR v ∞ (X ), when R → ∞ , ϕ, Bv where ·, · is the natural extension of the L2 scalar product in the appropriate variables (depending on X as above).