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result, but the authors don’t seem to be aware of this earlier work.
    No, the Problem hasn’t been cracked yet. Besides, it would have been a real disappointment if the solution had turned out to be so simple! An article of thirty pages or so that doesn’t resolve any major difficulty is unlikely to do the trick, however good it may be otherwise. Deep down I am convinced that the solution will require completely new tools, which will allow us to look at the problem in a new way.
    I need a new norm.
    A norm, in mathematical jargon, is a special sort of ruler, or measuring stick, designed for the purpose of estimating the size of some quantity one wishes to investigate. If we want to compare the pluviometry of Brest with that of Bordeaux, for example, should we compare the maximum rainfall for a single day in each place or integrate over the whole year? Comparing maximum quantities involves the L ∞ norm, usually called the supremum (or sup) norm; comparing integrated quantities involves another norm with an equally lovely name, L 1 . There are many, many others.
    To qualify as a true norm in the mathematical sense, certain conditions must be satisfied. The norm of a sum of two terms, for example, must be less than or equal to the sum of the norms of these terms taken separately. But that still leaves a vast number of norms to choose from.
    I need the right norm.
    The concept of a norm was formalized more than a century ago. Since then, mathematicians have not stopped inventing new ones. The second-year course I teach at ENS-Lyon is full of norms. Not only the Lebesque norm but also Sobolev, Hilbert, and Lorentz norms, Besov and Hölder norms, Marcinkiewicz and Lizorkin norms, L p , W s,p , H s , L p,q , B s,p,q , M p , and F s,p,q norms—and who knows how many more!
    But this time none of the norms I’m familiar with seems to be up to the job. I’ll just have to come up with a new one myself—pull it out of a great mathematical hat somehow.
    The norm of my dreams would be fairly stable under composition with elements close to identity, and capable of accommodating the filamentation typically associated with the Vlasov equation in large time. Gott im Himmel! Could such a thing really exist? I tried taking a weighted sup; perhaps I’ve got to introduce a delay.… Clément was saying we need to preserve the memory of elapsed time, in order to permit comparison with the solution of the free transport equation. That’s fine with me—but which one is supposed to be taken as the basis for comparison??
    While I was rereading the book by Alinhac and Gérard this fall, one exercise in particular caught my eye. Show that a certain norm W is an algebraic norm. In other words, show that the norm W of the product of two terms is at most equal to the product of the norms W of these terms taken separately. I’ve known about this exercise for a long time, but looking at it again I suspected that it might be useful in wrestling with the Problem.
    Maybe so—but even if I’m right, we’ll still have to modify the evaluation at 0 by inserting a sup, or otherwise an integral. But then that’s not going to work very well in the position variable, so we’ll have to use another algebraic norm … perhaps with Fourier? Or else with …
    One fruitless attempt after another. Until yesterday. Finally, I think I’ve found the norm I need. I’ve been scribbling away for weeks now, evening after evening, page after page, sending the results to Clément as I go along. The machine is cranked up. Cédurak go!
    *   *   *
     
    Let D be the unit disk in , and W ( D ) the space of holomorphic functions f on D satisfying

     
    Show that if and if g is holomorphic near the values taken by f on then (Remark that and that W ( D ) is an algebra; then write

     
    where N is chosen sufficiently large that the series is well defined and converges in W ( D ) .)
[Serge Alinhac and Patrick Gérard, Pseudo-Differential Operators and the Nash–Moser