A composition property in supercritical Besov spaces
by
Gérard Bourdaud
Univ. Paris VII, France

The full characterization of functions f:R®R which act on the Sobolev space - in the sense that f°g Î H^s(\R) for all g Î H^s(\R) - is an open problem for noninteger real numbers s such that s > 1. The existence of the so-called "triviality area" 3/2 £ s < n/2 - for which the only acting functions are the affine ones - leads to restrict the problem to bounded functions. In view of the known necessary acting conditions, and of the case when s is integer, we can formulate the following:

Conjecture. A function f acts on H^sÇL^¥ (\R) if and only if it satisfies the following properties:

• f(0)=0 ;

• f is locally Lipschitz continuous;

• f belongs locally to H^s(\re).
  

We establish the conjecture in case n=1 et s > 3/2, by proving a more precise result:

Theorem 1. Let assume s > 3/2. For all function f such that f(0)=0 and f¢ Î H^s-1(\re), the composition operator T_f: g® f°g takes the space H^s(\re) into itself. Moreover there exists a constant c=c(s) > 0 such that

||f°g||Hs(\re) £ c  ||f¢|| Hs-1(\re) ( 1+ ||g||Hs(\re))s-(1/2) .

Theorem 1 has a remarkable corollary concerning functions on the unit circle S¹ of the complex plane:

Corollary. Let assume s > 3/2. If the functions f₁ and f₂ belong to H^s(S¹, S¹), the same is true for f₁°f₂.

Here we denote by H^s(S¹, S¹) the set of functions f:S¹® S¹ such that x®f(e^ix) belongs to the periodic Sobolev space H^s(T).

We know how to extent Theorem 1 to higher dimension, but with a "microscopic" loss of regularity:

Theorem 2. Let assume s¢ > s > 3/2. For all function f such that f(0)=0 and f¢ Î H^s¢-1(\re), the composition operator T_f: g® f°g takes the space H^sÇL^¥(\R) into itself.

The preceding results are obtained, up to certain restrictions on parameters, in the more general framework of Besov spaces \besovnspq, the critical exponent being s=1+(1/p).

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Date received: June 8, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carf-08.