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By M. Aizenman (Chief Editor)

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The second part of the theorem is obvious. 5. If CDL(ψ) holds, then the numbers ak are fast decaying: for any s > 0, one can reorder {ak } so that ak ≤ C(s)k −s . 3, because they do not depend on the system {ek }, and inevitably, one loses some information about the moments dk (p). If dk (p) grow very fast as k → ∞, it is even possible that k akν < +∞ for all ν > 0, but k ak dk (p) = +∞ for all p > 0. Finally, in this section we shall give necessary conditions for DL(ψ, p) in terms of projections of ψ on Hλk .

Central charges and their solitonic carriers have been much discussed in the recent literature because it is hoped that they may provide a window on M-Theory and its non-perturbative degrees of freedom. More specifically, it has been argued in [5] that a proper description of the non-perturbative M-Theory degrees of freedom might require supplementing ordinary space-time coordinates by central charge coordinates. Solitonic charges also play an important role in the microscopic description of black hole entropy: for maximally extended N = 8 supergravity, the latter is conjectured to be given by an E7(7) invariant formula [20, 8].

7) directly from the definition of R(n, α). 8) 40 S. Tcheremchantsev where the summation is carried only over k such that ak > 0, so dk (r) < +∞. Let s = 2/(2 − δ), s = 2/δ. Applying the Hölder inequality, and using the fact that 2 k |ek (n)| ≤ 1, one obtains: |ek (n)| S≡ k δ s −δ/2 dk (α)dk (r) 2/(2−δ) ≤ k dk −δ/(2−δ) (α)dk (r). 9) Let w < q. Applying the Hölder inequality with s = q/w, s = q/(q − w), one can estimate: |ek (n)|2/s (|n| + 1)w |ek (n)|2/s dk (w) = n 1/s = (dk (q))w/q . 9) imply dk (r)(2α/r−δ)/(2−δ) .

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