Variable Lebesgue Spaces

Transcription

1 Variable Lebesgue Trinity College Summer School and Workshop Harmonic Analysis and Related Topics Lisbon, June 21-25, 2010

2 Joint work with: Alberto Fiorenza José María Martell Carlos Pérez

3 Special thanks to: Lars Diening Peter Hästö Aleš Nekvinda Stefan Samko

4 Lecture 1 Banach space properties of the variable Lebesgue spaces

5 Outline

6 Classical Lebesgue spaces L p (Ω), 1 p < : ( 1/p f L p (Ω) = f (x) dx) p < Ω L (Ω): f L p (Ω) = ess sup f (x) < x Ω Hereafter: Ω R n open set

7 Simple problem On R 1 consider f (x) = x 1/2 f L p (R), p, 1 p f L p ([ 1, 1]), 1 p < 2 f L p ([1, )), 2 < p Question: can we capture behavior w/o splitting domain?

8 More complicated problem Theorem (Calderón-Zygmund) Let Ω R n be bounded, and f L p (Ω). If u is a solution to u = f, then u W 2,p (Ω).

9 Intuition Replace constant exponent p by function p( ): : f (x) p(x) dx < Ω

10 Simple example p(x) = { 2 5 < x < x < 5. p = 2 p = 3

11 A simple example (continued) x + 1 1/3 L p( ) (( 5, 5)) x 1 1/3 L p( ) (( 5, 5)) p = 2 p = 3

12 Original motivations Generalized Orlicz spaces: replace Φ ( f (x) ) dx with Φ ( f (x), x ) dx Calculus of variations: minimize F(u, Ω) = f (u, Du) dx where z p(x) f (x, z) L(1 + z ) p(x) Electrorheological fluids: Energy = Du(x) p(x) dx Ω Ω

13 Exponent functions p( ) P(Ω) p( ) : Ω [1, ] Ω = {x Ω : p(x) = } For E Ω p (E) = ess inf{p(x) : x E} p + (E) = ess sup{p(x) : x E} Hereafter: p = p (Ω), p + = p + (Ω)

14 The modular & norm Given p( ) P(Ω) ρ p( ) (f ) = ρ(f ) = f (x) p(x) dx + f L (Ω ) Ω\Ω f = f p( ) = inf { λ > 0 : ρ p( ) (f /λ) 1 }

15 The space Theorem Given p( ) P(Ω), p( ) is a norm and = {f : f p( ) < } is a Banach function space.

16 A path not taken Given p( ) P(Ω), is: an Orlicz-Musielak/Nakano/modular space a Banach function space

17 The problem with direct proofs With apologies to Lewis Carroll

18 A meta-theorem If p + < or if p + (Ω \ Ω ) <, then L p( ) is GOOD and behaves like L p If p + =, then L p( ) is BAD and something interesting happens

19 Modular vs. norm convergence Theorem Given p( ) P(Ω) the equivalence f ρ p( ) (f ) < is true if and only if p + (Ω \ Ω ) < (or p = ).

20 Proof: p + (Ω \ Ω ) < ρ(f ) < f (x) p(x) dx + f L p( ) (Ω ) < Ω\Ω ( ) p(x) f (x) λ > 1 : dx 1/2 λ Ω\Ω λ 1 f L p( ) (Ω ) 1/2 f p( ) <

21 Proof: p + (Ω \ Ω ) <, continued f p( ) < λ > 1 : Ω\Ω ( ) p(x) f (x) dx + λ 1 f Lp( )(Ω ) 1 λ λ Ω\Ω p+(ω\ω ) f (x) p(x) dx + λ 1 f L p( ) (Ω ) 1 ρ(f ) <

22 Proof: p + (Ω \ Ω ) = Form sets E k Ω \ Ω : E k E k+1, E k \ E k+1 > 0 E k 0 p (E k ) > k ( f (x) = E k \ E k+1 1 χ Ek \E k+1 (x) k=1 ) 1/p(x) λ > 1 : ρ(f /λ) = λ p(x) dx E k \E k+1 k=1 k=1 λ k

23 Embedding in L p Theorem If Ω <, c 1 f p f p( ) c 2 f p+ Theorem f f = f 1 +f 2 : f 1 L p (Ω), f 2 L p+ (Ω)

24 Types of convergence Norm convergence: f f k p( ) 0 Modular convergence: β > 0: ρ(β f f k ) 0 in measure: ɛ, k: {x Ω : f (x) f k (x) ɛ} < ɛ

25 Norm Monotone convergence theorem: Fatou s lemma: f k f f k p( ) f p( ) f k f f p( ) lim inf f k p( ) k Dominated convergence theorem: Iff p + <, f k f, f k g f f k p( ) 0

26 Modular convergence Iff p + (Ω \ Ω ) < (or p = ) f k f in modular f k f in norm Key ingredient of proof: nested sets {E k } and functions ( f (x) = 2 k E k \ E k+1 1 χ Ek \E k+1 (x) k=1 f k (x) = f (x)χ Ek (x) ) 1/p(x)

27 in measure Theorem If p + <, TFAE: in norm in modular in measure and λ > 0 : ρ(λf k ) ρ(λf )

28 in measure Theorem If p + =, TFAE f k f in modular f k f in measure & λ > 0 : ρ(λf k ) ρ(λf ) {x : p(x) > N} 0 as N. Theorem If p + = & Ω = 0, f k f in measure & 0 < λ < 1 : ρ(λf ) < & ρ(λf k /3) ρ(λf /3) f k f in modular

29 Density Theorem TFAE: p + < Bounded functions of compact support are dense in

30 Simple example Let Ω = R, p(x) = x + 1 f (x) 1 L p( ) (R) : ρ(f /2) = 2 (1+ x ) dx < supp(g) [ N, N] f (x) g(x) 1+ x dx = x >N f g p( ) 1. R

31 Theorem TFAE: p + < is separable: has countable dense subset Key ingredients of proof: nested sets {E k }, function f and associate norm

32 Hölder s inequality Ω f (x) g(x) dx K p( ) f p( ) g p ( ) K p( ) = 1 p 1 p + + χ Ω1 + χ Ω + χ Ω 1 p(x) + 1 p (x) = 1

33 Associate norm f p( ) = sup f (x)g(x) dx g p ( ) 1 Ω Theorem k p( ) f p( ) f p( ) K p( ) f p( ) k 1 p( ) = χ Ω 1 + χ Ω + χ Ω

34 Linear functionals Given g L p ( ) (Ω), Φ g : R, Φ g (f ) = f (x)g(x) dx is a bounded linear functional: g. Ω

35 Theorem TFAE: p + < = L p ( ) (Ω) Key ingredient in proof: nested sets {E k } and f defined above

36 The dual space Characterize if p + =. Conjecture: Depends on whether or not L (Ω)

37 I W. Orlicz. Über konjugierte Exponentenfolgen. Stud. Math., 3: , O. Kováčik and J. Rákosník. On spaces L p(x) and W k,p(x). Czechoslovak Math. J., 41(116)(4): , X. Fan and D. Zhao. On the spaces L p(x) (Ω) and W m,p(x) (Ω). J. Math. Anal. Appl., 263(2): , 2001.

38 II L. Diening, P. Hästö, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings (Drabek and Rakosnik (eds.); Milovy, Czech Republic, pages Academy of Sciences of the Czech Republic, Prague, S. Samko. On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct., 16(5-6): , 2005.

39 III DCU and A. Fiorenza. in variable Lebesgue spaces. Publ. Mat., 54 (2) (2010),

40 IV L. Diening, P. Harjulehto, P. Hästö, and M. Růžička. Lebesgue and Sobolev spaces with variable exponent. Forthcoming. DCU and A. Fiorenza. Harmonic Analysis on variable Lebesgue spaces. Forthcoming.