Generalized pointwise Hölder spaces

Transcription

1 Generalized pointwise Hölder spaces D. Kreit & S. Nicolay Nord-Pas de Calais/Belgium congress of Mathematics October

2 The idea A function f L loc (Rd ) belongs to Λ s (x 0 ) if there exists a polynomial of degree at most s s.t. for sufficiently large. sup f (x 0 + h) P(h) C2 s, h 2 Is it possible to be sharper and replace the sequence (2 s ) with a more general sequence σ = (σ ) : f L loc (Rd ) belongs to Λ σ,m (x 0 ) if there exists a polynomial of degree at most M s.t. for sufficiently large. sup f (x 0 + h) P(h) Cσ, h 2

3 Generalized Besov spaces B s p,q(r d ) generalization Λ s (R d ) = B s, (R d ) generalization generalization Bp,q 1/σ (R d ), where σ is some sequence generalization Λ σ (R d ) = B 1/σ, Λ σ (R d ) pointwise version Λ σ (x 0 )

4 Admissible sequence A sequence of real positive numbers is called admissible if is bounded. For such a sequence, we set σ +1 σ and s(σ) = lim s(σ) = lim log 2 (inf k N log 2 (sup k N σ +k σ ) σ +k σ ).

5 Notations the open unit ball centered at the origin is denoted B, the set of polynomials of degree at most n is denoted P[n], [s] = sup{n Z : n s}, if f is defined on R d, 1 hf (x) = f (x + h) f (x) and for any x, h R d n+1 h f (x) = 1 h n hf (x),

6 Definition of the generalized global Hölder spaces Definition Les s > 0 and σ an admissible sequence ; a function f L (R d ) belongs to Λ σ,m (R d ) iff there exists C > 0 s.t. Proposition sup [M]+1 h f Cσ h 2 Les s > 0 and σ an admissible sequence ; a function f L (R d ) belongs to Λ σ,s (R d ) iff there exists C > 0 s.t. for any x 0 R d and any N. inf f P L P P (2 B+x 0 ) Cσ, [M]

7 The pointwise version Definition A function f L loc (Rd ) belongs to Λ σ,m (x 0 ) iff there exists C > 0 and J N s.t. for any J. Definition inf f P L P P[M] (2 B+x 0 ) Cσ, A function f L loc (Rd ) belongs to Λ σ,m (x 0 ) iff there exists C > 0 and J N s.t. for any J, there exists P P[M] for which sup h <2 f (x 0 + h) P (x 0 + h) Cσ.

8 What about the classical case A function f L loc (Rd ) belongs to Λ s (x 0 ) (s R) iff there exists C > 0, a polynomial P of degree less than s and J N s.t. for any J, sup h <2 f (x 0 + h) P(x 0 + h) C2 s. There is one polynomial, independant from the scale.

9 Two lemmata Lemma If M < s(σ 1 ), the sequence of polynomials occuring in the definition of Λ σ,m (x 0 ) satisfies D β P k D β P L (x 0 +2 k B) C2 β σ, for any multi-index β s.t. β M and k J. In particular, (D β P(x 0 )) is a Cauchy sequence.

10 Two lemmata Lemma If M < s(σ 1 ), and (P ) is a sequence of polynomials in the definition of Λ σ,m (x 0 ), for any multi-index β s.t. β M, the limit f β (x 0 ) = lim D β P (x 0 ) is independant of the chosen sequence (P ). f β (x 0 ) is the Peano derivative of order β of f at x 0.

11 There can be only one Theorem If M < s(σ 1 ), then f Λ σ,m (x 0 ) iff there exist C > 0 and a polynomial P P[M] s.t. f P L (x 0 +2 B) Cσ, for sufficiently large. The polynomial is unique. One has P(x) = β M f β (x 0 ) (x x 0) β. β!

12 The classical case For s (0, ), let σ = 2 s M = [s(σ 1 )] = [s] if s N M = s 1 if s N We have Λ s (x 0 ) = Λ σ,m (x 0 ). Corollary If M < s(σ 1 ), one has Λ σ,m (x 0 ) Λ M (x 0 ).

13 Finite differences Let B M h (x 0, ) = {x : [x, x + (M + 1)h] x B}. Proposition Let f L loc (Rd ) ; one has f Λ σ,m (x 0 ) iff there exist C, J > 0 s.t. for any J. sup M+1 h f L (Bh M h B (x 0,)) Cσ,

14 Convolutions Let ρ a radial function s.t. ρ C c (B), ρ(b) [0, 1] and ρ 1 = 1. One sets, for any N 0, ρ = 2 d ρ( /2 ). Lemma Let N N 0 ; if f L 1 loc (Rd ) satisfies sup f ρ k f L (x 0 +2 B) Cσ, k for J, then, for any multi-index β s.t. β N, one has for any J. D β (f ρ f ρ 1 ) L (x 0 +2 B) C2 N σ,

15 Convolutions Proposition If f Λ σ,m (x 0 ), then there exists Φ C c (R d ) s.t. sup f f Φ k L (x 0 +2 B) Cσ, k for sufficiently large. Conversely, if σ 0, f Λ ɛ (R d ) for some ɛ > 0 and f satisfies the previous relation for some function Φ Cc (R d ), then f Λ σ,m (x 0 ) for any M s.t. M + 1 > s(σ 1 ).

16 Definitions Under some general conditions, there exist a function φ and 2 d 1 functions ψ (i) called wavelets s.t. {φ( k) : k Z d } {ψ (i) (2 k) : k Z d, N 0 } forms an orthogonal basis of L 2 (R d ). Any function f L 2 (R d ) can be decomposed as follows, f (x) = k Z d C k φ(x k) + 0,k Z d,1 i<2 d c (i),k ψ(i) (2 x k), with C k = f (x)φ(x k) dx, c (i),k = 2d f (x)ψ (i) (2 x k) dx.

17 Definitions We assume φ, ψ (i) C n (R d ) with n > M, D β φ, D β ψ (i) ( β n) have fast decay, supp(ψ (i) ) 2 0 B for some 0.

18 Definitions We set λ = λ(i,, k) = k 2 + i [0, 1 2 c λ = c (i),k ψ λ = ψ (i) (2 k). +1 )d

19 Definitions The wavelet leaders are defined by d λ = sup c λ λ λ If 3λ denotes the 3 d dyadic cubes adacent to λ and λ (x 0 ) the dyadic cube of length 2 containing x 0, one sets d (x 0 ) = sup d λ λ 3λ (x 0 )

20 Definitions

21 The caracterization Theorem If f Λ σ,m (x 0 ), then there exists C > 0 s.t. d (x 0 ) Cσ, for sufficiently large. Conversely, if σ 0, f Λ ɛ (R d ) for some ɛ > 0 and f satisfies the previous relation, then f Λ τ,m (x 0 ), where τ is the sequence defined by τ = σ log 2 σ, M is any number satisfying M + 1 > s(σ 1 ).

22 Definitions If, for any s > 0, σ (s) is an admissible sequence, the application σ ( ) : s > 0 σ (s) is called a family of admissible sequences. A family of admissible sequences is decreasing for x 0 if s < t Λ σ(t),[t] (x 0 ) Λ σ(s),[s] (x 0 ). Let σ ( ) a family of decreasing sequences for x 0 and f L loc (Rd ) ; the Hölder exponent of f at x 0 for σ ( ) is h σ( ) f (x 0 ) = sup{s > 0 : f Λ σ(s),[s] (x 0 )}.

23 How to check if a family of admissible sequences is decreasing? Let Θ (m) σ (m) = sup k N k+1 σ (m) k, Θ (m) σ (m) = inf k N k+1 σ (m) k, Proposition A family of admissible sequences is decreasing for x 0 if it satisfies the following conditions : if m s < t < m + 1 with m N 0, σ (t) Cσ (s) for sufficiently large for any m N, at least one of the following conditions is satisfied : there exists ɛ 0 > 0 s.t. for any ɛ (0, ɛ 0 ), σ (m) Cσ (m ɛ) if 1 < 2 m Θ (m) : (Θ (m) ) Cσ (m ɛ) if 1 > 2 m Θ (m) : 2 m Cσ (m ɛ) if 1 = 2 m Θ (m) : 2 m Cσ (m ɛ) 2 m Cσ (m ɛ) if 1 < 2 m Θ (m) : σ (m) if 1 > 2 m Θ (m) : σ (m) if 1 = 2 m Θ (m) : σ (m) Cσ (m ɛ) (2 m Θ (m) ) Cσ (m ɛ) Cσ (m ɛ)