Function spaces with variable exponents

Transcription

1 Function spaces with variable exponents Henning Kempka September 22nd 2014 September 22nd 2014 Henning Kempka 1 / 50

2 Outline 1. Introduction & Motivation First motivation Second motivation 2. Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Hölders inequality 3. The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator Boundedness on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces 4. Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Wavelet characterization of L p( ) (Ω) September 22nd 2014 Henning Kempka 2 / 50

3 References W. Orlicz: Über konjugierte Exponentenfolgen. Studia Math. 3, (1931), O. Kováčik, J. Rákosník: On spaces L p(x) and W 1,p(x). Czechoslovak Math. J. 41(116)(1991), L. Diening: Maximal function on generalized Lebesgue spaces L p( ). Mathematical Inequalities and Applications 7 no.2, (2004), L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics 2017,Springer, Heidelberg (2011). D. V. Cruz-Uribe, A. Fiorenza: Variable Lebesgue spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg (2013). September 22nd 2014 Henning Kempka 3 / 50

4 Introduction & Motivation Table of Contents 1. Introduction & Motivation First motivation Second motivation 2. Variable exponent Lebesgue spaces 3. The Hardy-Littlewood maximal operator on L p( ) (Ω) 4. Rubio de Francia extrapolation September 22nd 2014 Henning Kempka 4 / 50

5 Introduction & Motivation First motivation Consider on R the function f(x) = x 1/3 f is well behaving, but f / L p (R) for every 1 p it either grows to quickly at the origin or it decays to slow at infinity But: f L 2 ([ 2, 2]) and f L 4 (R \ [ 2, 2]) September 22nd 2014 Henning Kempka 5 / 50

6 Introduction & Motivation First motivation Consider on R the function f(x) = x 1/3 f is well behaving, but f / L p (R) for every 1 p it either grows to quickly at the origin or it decays to slow at infinity But: f L 2 ([ 2, 2]) and f L 4 (R \ [ 2, 2]) September 22nd 2014 Henning Kempka 5 / 50

7 Introduction & Motivation First motivation Consider on R the function f(x) = x 1/3 f is well behaving, but f / L p (R) for every 1 p it either grows to quickly at the origin or it decays to slow at infinity But: f L 2 ([ 2, 2]) and f L 4 (R \ [ 2, 2]) September 22nd 2014 Henning Kempka 5 / 50

8 Introduction & Motivation First motivation Consider now g(x) = x 1/3 + x 1 1/4 : Then g L p ([ 2, 2]) for p < 3 and g L q (R \ [ 2, 2]) for q > 4 But: We lost informatin on the local behaviour at the singularity x = 1! First solution: We can subdivide R even more: g L 2 ([ 1, 1/2]), g L 3 ([1/2, 2]) and g L 9 (R \ [ 1, 2]) 2 September 22nd 2014 Henning Kempka 6 / 50

9 Introduction & Motivation First motivation Consider now g(x) = x 1/3 + x 1 1/4 : Then g L p ([ 2, 2]) for p < 3 and g L q (R \ [ 2, 2]) for q > 4 But: We lost informatin on the local behaviour at the singularity x = 1! First solution: We can subdivide R even more: g L 2 ([ 1, 1/2]), g L 3 ([1/2, 2]) and g L 9 (R \ [ 1, 2]) 2 September 22nd 2014 Henning Kempka 6 / 50

10 Introduction & Motivation First motivation Variable solution Introduce the exponent: p(x) = 9 2 5/2 2 x +1, then and p(0) = 2, p(1) = 11 3 and p(x) 9 2 for x. We have f(x) p(x) dx < and g(x) p(x) dx <. R R September 22nd 2014 Henning Kempka 7 / 50

11 Introduction & Motivation First motivation Variable solution Introduce the exponent: p(x) = 9 2 5/2 2 x +1, then and p(0) = 2, p(1) = 11 3 and p(x) 9 2 for x. We have f(x) p(x) dx < and g(x) p(x) dx <. R R September 22nd 2014 Henning Kempka 7 / 50

12 Introduction & Motivation First motivation Variable solution II Furthermore, for the exponent: q(x) = x +1 we have q(0) = 2, p(1) = 10 3 and p(x) 4 for x. Then we have f(x) q(x) dx < R and R g(x) q(x) dx =. September 22nd 2014 Henning Kempka 8 / 50

13 Introduction & Motivation First motivation Variable solution II Furthermore, for the exponent: q(x) = x +1 we have q(0) = 2, p(1) = 10 3 and p(x) 4 for x. Then we have f(x) q(x) dx < R and R g(x) q(x) dx =. September 22nd 2014 Henning Kempka 8 / 50

14 Introduction & Motivation Second motivation Electrorheological fluids non-newtonian fluids change their viscosity dramatically (factor 1000) in reaction to an electrical field First observed in 1949 by Winslow many applications: fast acting hydraulic valves & clutches ER brakes and shock absorbers Electrorheological fluid, with and without electrical field September 22nd 2014 Henning Kempka 9 / 50

15 Introduction & Motivation Second motivation Electrorheological fluids II The model of these electrorheological fluids is connected via some nasty PDEs (with non standard growth conditions) to the Dirichlet energy integral: where: Du... symmetricpart of the gradient Ω Du(x) p(x) dx p(x) = p(e(y, t))... is a function of the electrical field Questions: What is Ω f(x) p(x) dx <? To which type of space does f belong? Comparison to L p (Ω)? Further properties (inequalities, embeddings,... ) September 22nd 2014 Henning Kempka 10 / 50

16 Introduction & Motivation Second motivation Electrorheological fluids II The model of these electrorheological fluids is connected via some nasty PDEs (with non standard growth conditions) to the Dirichlet energy integral: where: Du... symmetricpart of the gradient Ω Du(x) p(x) dx p(x) = p(e(y, t))... is a function of the electrical field Questions: What is Ω f(x) p(x) dx <? To which type of space does f belong? Comparison to L p (Ω)? Further properties (inequalities, embeddings,... ) September 22nd 2014 Henning Kempka 10 / 50

17 Variable exponent Lebesgue spaces Table of Contents 1. Introduction & Motivation 2. Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Hölders inequality 3. The Hardy-Littlewood maximal operator on L p( ) (Ω) 4. Rubio de Francia extrapolation September 22nd 2014 Henning Kempka 11 / 50

18 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) For the rest Ω R n is an arbitrary but fixed open set. Definition 1 The class of variable exponents is P(Ω) = {p : Ω [1, ] measurable }. For p P(Ω) and U Ω we introduce p U = ess inf x U p(x) p + U = ess sup x U p(x) p = p Ω p + = p + Ω Ω = {x Ω : p(x) = } Ω 1 = {x Ω : p(x) = 1} Ω = {x Ω : 1 < p(x) < } September 22nd 2014 Henning Kempka 12 / 50

19 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) For the rest Ω R n is an arbitrary but fixed open set. Definition 1 The class of variable exponents is P(Ω) = {p : Ω [1, ] measurable }. For p P(Ω) and U Ω we introduce p U = ess inf x U p(x) p + U = ess sup x U p(x) p = p Ω p + = p + Ω Ω = {x Ω : p(x) = } Ω 1 = {x Ω : p(x) = 1} Ω = {x Ω : 1 < p(x) < } September 22nd 2014 Henning Kempka 12 / 50

20 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Definition 2 For p P(Ω) we define the convex modular ϱ p( ) (f) = f(x) p(x) dx + ess sup f(x) Ω\Ω x Ω and the variable exponent Lebesgue space L p( ) (Ω) by L p( ) (Ω) = {f : Ω C : ϱ p( ) (f/λ) < for some λ > 0}. September 22nd 2014 Henning Kempka 13 / 50

21 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Lemma 1 (Properties of the modular) Let p P(Ω), then: (M1) For all f, we have ϱ(f) 0 and ϱ( f ) = ϱ(f); (M2) ϱ(f) = 0 f(x) = 0 almost everywhere; (M3) If ϱ(f) <, then f(x) < for a.e. x Ω; (M4) ϱ is convex, i.e. ϱ(αf + βg) αϱ(f) + βϱ(g) for all α, β [0, 1] with α + β = 1; (M5) If f(x) g(x) a.e, then ϱ(f) ϱ(g); (M6) Continuity property If ϱ(f/λ) < for some Λ > 0, then λ ϱ(f/λ) is continuous and decreasing on [Λ, ) Further lim λ ϱ(f/λ) = 0. September 22nd 2014 Henning Kempka 14 / 50

22 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Remark 1 From the convexity (M4) of the modular it follows ϱ(αf) αϱ(f) for 0 < α < 1 ϱ(αf) αϱ(f) for α > 1 Example 1 Let Ω = (1, ), p(x) = x and f(x) = 1, then ϱ(f) = but for all λ > 1 ϱ(f/λ) = 1 λ x dx = 1 λ log λ <. September 22nd 2014 Henning Kempka 15 / 50

23 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Theorem 1 For given p P(Ω), the space L p( ) (Ω) is a normed vector space with the Luxemburg norm f p( ) = inf{λ > 0 : ϱ p( ) (f/λ) 1}. Proof. L p( ) (Ω) is a vector space (N1) f p( ) 0 and f p( ) = 0 f = 0 (N2) αf p( ) = α f p for all α R (N3) f + g p( ) f p( ) + g p( ) September 22nd 2014 Henning Kempka 16 / 50

24 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Theorem 1 For given p P(Ω), the space L p( ) (Ω) is a normed vector space with the Luxemburg norm f p( ) = inf{λ > 0 : ϱ p( ) (f/λ) 1}. Proof. L p( ) (Ω) is a vector space (N1) f p( ) 0 and f p( ) = 0 f = 0 (N2) αf p( ) = α f p for all α R (N3) f + g p( ) f p( ) + g p( ) September 22nd 2014 Henning Kempka 16 / 50

25 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Lemma 2 Fix p P(Ω) then: 1. If f p( ) 1, then ϱ p( ) (f) f p( ) ; 2. If f p( ) > 1, then ϱ p( ) (f) f p( ). Proof. Remember: ϱ p( ) (f) = Ω\Ω f(x) p(x) dx + f L (Ω ) f p( ) = inf{λ > 0 : ϱ p( ) (f/λ) 1} ϱ p( ) (αf) αϱ p( ) (f) for 0 < α 1 September 22nd 2014 Henning Kempka 17 / 50

26 Variable exponent Lebesgue spaces Definition of L p( ) (Ω) Lemma 2 Fix p P(Ω) then: 1. If f p( ) 1, then ϱ p( ) (f) f p( ) ; 2. If f p( ) > 1, then ϱ p( ) (f) f p( ). Proof. Remember: ϱ p( ) (f) = Ω\Ω f(x) p(x) dx + f L (Ω ) f p( ) = inf{λ > 0 : ϱ p( ) (f/λ) 1} ϱ p( ) (αf) αϱ p( ) (f) for 0 < α 1 September 22nd 2014 Henning Kempka 17 / 50

27 Variable exponent Lebesgue spaces Hölders inequality Theorem 2 (Hölders inequality) Fix p P(Ω) and let p P(Ω) be defined by 1 p(x) + 1 p (x) = 1 pointwise. If f L p( ) (Ω) and g L p ( )(Ω), then f g L 1 (Ω) and f(x)g(x) dx K p( ) f p( ) g p ( ), where Ω ( 1 K p( ) = p 1 ) p χ Ω + χ Ω + χ Ω1. September 22nd 2014 Henning Kempka 18 / 50

28 Variable exponent Lebesgue spaces Hölders inequality Proof. 1. If f p( ) = 0 or g p ( ) = 0, then f g = 0. Therefore: f p( ) > 0 and g p ( ) > Norm is homogenuous: We assume f p( ) = 1 and g p ( ) = We consider the integral fg dx on the sets Ω, Ω 1 and Ω x Ω : p(x) = and p (x) = 1 x Ω 1 : p(x) = 1 and p (x) = x Ω : 1 < p(x) < Youngs inequality: a b ap(x) p(x) + (x) bp p (x) September 22nd 2014 Henning Kempka 19 / 50

29 Variable exponent Lebesgue spaces Hölders inequality Proof. 1. If f p( ) = 0 or g p ( ) = 0, then f g = 0. Therefore: f p( ) > 0 and g p ( ) > Norm is homogenuous: We assume f p( ) = 1 and g p ( ) = We consider the integral fg dx on the sets Ω, Ω 1 and Ω x Ω : p(x) = and p (x) = 1 x Ω 1 : p(x) = 1 and p (x) = x Ω : 1 < p(x) < Youngs inequality: a b ap(x) p(x) + (x) bp p (x) September 22nd 2014 Henning Kempka 19 / 50

30 Variable exponent Lebesgue spaces Hölders inequality Proof. 1. If f p( ) = 0 or g p ( ) = 0, then f g = 0. Therefore: f p( ) > 0 and g p ( ) > Norm is homogenuous: We assume f p( ) = 1 and g p ( ) = We consider the integral fg dx on the sets Ω, Ω 1 and Ω x Ω : p(x) = and p (x) = 1 x Ω 1 : p(x) = 1 and p (x) = x Ω : 1 < p(x) < Youngs inequality: a b ap(x) p(x) + (x) bp p (x) September 22nd 2014 Henning Kempka 19 / 50

31 Variable exponent Lebesgue spaces Hölders inequality Further properties of L p( ) (Ω) Banach spaces L p( ) (Ω) is complete Banach spaces Equivalent norm Introduce f p( ) = sup f(x)g(x)dx, then ϱ p ( ) (g) f p( ) f p( ) 4 f p( ) Dual spaces ( L p( ) (Ω) ) = Lp ( ) p L (Ω) For 1 < p p + < the spaces L p( ) (Ω) are reflexive Seperability L p( ) (Ω) is seperable p L (Ω) Ω September 22nd 2014 Henning Kempka 20 / 50

32 Variable exponent Lebesgue spaces Hölders inequality Further properties of L p( ) (Ω) Banach spaces L p( ) (Ω) is complete Banach spaces Equivalent norm Introduce f p( ) = sup f(x)g(x)dx, then ϱ p ( ) (g) f p( ) f p( ) 4 f p( ) Dual spaces ( L p( ) (Ω) ) = Lp ( ) p L (Ω) For 1 < p p + < the spaces L p( ) (Ω) are reflexive Seperability L p( ) (Ω) is seperable p L (Ω) Ω September 22nd 2014 Henning Kempka 20 / 50

33 Variable exponent Lebesgue spaces Hölders inequality Differences to L p (Ω) Translation invariance If p( ) is not constant, then there exists an f L p( ) (Ω) and an h R n such that f( + h) / L p( ) (Ω). Example 2 Let Ω = ( 1, 1), 1 r < s < and define { x 1/s, for 0 < x < 1 f(x) = p(x) = 0, for 1 < x 0 { r, for 0 < x < 1 s, for 1 < x 0. Then f L p( ) ( 1, 1) but f( + h) / L p( ) ( 1, 1) for every h (0, 1). Youngs convolution inequality If there exists a constant c > 0 such that for all f L p( ) (Ω) and all g L 1 (Ω) f g p( ) c f p( ) g 1, then p( ) = const almost everywhere. September 22nd 2014 Henning Kempka 21 / 50

34 Variable exponent Lebesgue spaces Hölders inequality Differences to L p (Ω) Translation invariance If p( ) is not constant, then there exists an f L p( ) (Ω) and an h R n such that f( + h) / L p( ) (Ω). Example 2 Let Ω = ( 1, 1), 1 r < s < and define { x 1/s, for 0 < x < 1 f(x) = p(x) = 0, for 1 < x 0 { r, for 0 < x < 1 s, for 1 < x 0. Then f L p( ) ( 1, 1) but f( + h) / L p( ) ( 1, 1) for every h (0, 1). Youngs convolution inequality If there exists a constant c > 0 such that for all f L p( ) (Ω) and all g L 1 (Ω) f g p( ) c f p( ) g 1, then p( ) = const almost everywhere. September 22nd 2014 Henning Kempka 21 / 50

35 Variable exponent Lebesgue spaces Hölders inequality Generalizations of L p( ) (Ω) Semimodular spaces On the vector space X there is defined a semimodular ϱ : X [0, ] with some properties and X ϱ = {f X : ϱ(f/λ) 1 for some λ > 0} is then a Semimodular space. Musielak-Orlicz spaces The space X(Ω) consists of all functions f such that there exists a λ > 0 with Ω ( Φ x, f(x) ) dx λ is finite. The function Φ : Ω [0, ) [0, ) is for almost every x Ω a Young function Φ(x, ). The function ϱ Φ (f) = Ω Φ(x, f(x) )dx defines a semimodular on the space of measurable functions. If Φ(x, t) = χ {p(x)< } (x)t p(x) + χ {p(x)= } χ (1, ) (t), then X(Ω) = L p( ) (Ω). September 22nd 2014 Henning Kempka 22 / 50

36 Variable exponent Lebesgue spaces Hölders inequality Generalizations of L p( ) (Ω) Semimodular spaces On the vector space X there is defined a semimodular ϱ : X [0, ] with some properties and X ϱ = {f X : ϱ(f/λ) 1 for some λ > 0} is then a Semimodular space. Musielak-Orlicz spaces The space X(Ω) consists of all functions f such that there exists a λ > 0 with Ω ( Φ x, f(x) ) dx λ is finite. The function Φ : Ω [0, ) [0, ) is for almost every x Ω a Young function Φ(x, ). The function ϱ Φ (f) = Ω Φ(x, f(x) )dx defines a semimodular on the space of measurable functions. If Φ(x, t) = χ {p(x)< } (x)t p(x) + χ {p(x)= } χ (1, ) (t), then X(Ω) = L p( ) (Ω). September 22nd 2014 Henning Kempka 22 / 50

37 Variable exponent Lebesgue spaces Hölders inequality Generalizations of L p( ) (Ω) Semimodular spaces On the vector space X there is defined a semimodular ϱ : X [0, ] with some properties and X ϱ = {f X : ϱ(f/λ) 1 for some λ > 0} is then a Semimodular space. Musielak-Orlicz spaces The space X(Ω) consists of all functions f such that there exists a λ > 0 with Ω ( Φ x, f(x) ) dx λ is finite. The function Φ : Ω [0, ) [0, ) is for almost every x Ω a Young function Φ(x, ). The function ϱ Φ (f) = Ω Φ(x, f(x) )dx defines a semimodular on the space of measurable functions. If Φ(x, t) = χ {p(x)< } (x)t p(x) + χ {p(x)= } χ (1, ) (t), then X(Ω) = L p( ) (Ω). September 22nd 2014 Henning Kempka 22 / 50

38 Variable exponent Lebesgue spaces Hölders inequality History of L p( ) (Ω) 1. Early period 1931 Orlicz (he only wrote one paper about them and then studied Orlicz spaces) 1950 Nakano introduced modular spaces 1961 Tsenov 1979 Sharapudinov, introduced the local log-hölder condition 1986 Zhikov applied Lp( ) (Ω) to problems in calculus of variations 2. Modern period 1991 Kováčik & Rákosník, good overview of properties of L p( ) (Ω) 1990 A lot of papers on PDEs with non standard growth and the p( ) Laplacian (Fan, Zhao, Harjulehto, Mingione,... ) 2000 Modelling of electrorheological fluids with Lp( ) (Ω) by Růžička 2004 Boundedness of the Hardy-Littlewood maximal operator (Diening) September 22nd 2014 Henning Kempka 23 / 50

39 Variable exponent Lebesgue spaces Hölders inequality History of L p( ) (Ω) 1. Early period 1931 Orlicz (he only wrote one paper about them and then studied Orlicz spaces) 1950 Nakano introduced modular spaces 1961 Tsenov 1979 Sharapudinov, introduced the local log-hölder condition 1986 Zhikov applied Lp( ) (Ω) to problems in calculus of variations 2. Modern period 1991 Kováčik & Rákosník, good overview of properties of L p( ) (Ω) 1990 A lot of papers on PDEs with non standard growth and the p( ) Laplacian (Fan, Zhao, Harjulehto, Mingione,... ) 2000 Modelling of electrorheological fluids with Lp( ) (Ω) by Růžička 2004 Boundedness of the Hardy-Littlewood maximal operator (Diening) September 22nd 2014 Henning Kempka 23 / 50

40 The Hardy-Littlewood maximal operator on L p( ) (Ω) Table of Contents 1. Introduction & Motivation 2. Variable exponent Lebesgue spaces 3. The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator Boundedness on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces 4. Rubio de Francia extrapolation September 22nd 2014 Henning Kempka 24 / 50

41 The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator Definition 3 Let f L loc 1 (Rn ) then the Hardy-Littlewood maximal operator Mf is defined for every x R n by 1 Mf(x) = sup f(y) dy, Q x Q Q where the supremum is taken over all cubes Q R n containing x and sides parallel to the axes. Remark 2 The cubes can be open or closed, they can be replaced by balls B or everything can be centered in x. All these definitions are up to constants the same. September 22nd 2014 Henning Kempka 25 / 50

42 The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator Definition 3 Let f L loc 1 (Rn ) then the Hardy-Littlewood maximal operator Mf is defined for every x R n by 1 Mf(x) = sup f(y) dy, Q x Q Q where the supremum is taken over all cubes Q R n containing x and sides parallel to the axes. Remark 2 The cubes can be open or closed, they can be replaced by balls B or everything can be centered in x. All these definitions are up to constants the same. September 22nd 2014 Henning Kempka 25 / 50

43 ext. Marcinkiewicz interpolation September 22nd 2014 Henning Kempka 26 / 50 Theorem 3 The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator If 1 < p the M : L p (R n ) L p (R n ) is bounded, i.e. Mf p c f p for all f L p (R n ). Proof. easy M is strong (, ), i.e. hard M is weak (1, 1), i.e. Mf f. {x R n f 1 : Mf(x) > t} c 1. t

44 ext. Marcinkiewicz interpolation September 22nd 2014 Henning Kempka 26 / 50 Theorem 3 The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator If 1 < p the M : L p (R n ) L p (R n ) is bounded, i.e. Mf p c f p for all f L p (R n ). Proof. easy M is strong (, ), i.e. hard M is weak (1, 1), i.e. Mf f. {x R n f 1 : Mf(x) > t} c 1. t

45 ext. Marcinkiewicz interpolation September 22nd 2014 Henning Kempka 26 / 50 Theorem 3 The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator If 1 < p the M : L p (R n ) L p (R n ) is bounded, i.e. Mf p c f p for all f L p (R n ). Proof. easy M is strong (, ), i.e. hard M is weak (1, 1), i.e. Mf f. {x R n f 1 : Mf(x) > t} c 1. t

46 ext. Marcinkiewicz interpolation September 22nd 2014 Henning Kempka 26 / 50 Theorem 3 The Hardy-Littlewood maximal operator on L p( ) (Ω) Basics about the maximal operator If 1 < p the M : L p (R n ) L p (R n ) is bounded, i.e. Mf p c f p for all f L p (R n ). Proof. easy M is strong (, ), i.e. hard M is weak (1, 1), i.e. Mf f. {x R n f 1 : Mf(x) > t} c 1. t

47 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Conditions on p( ) Definition 4 Let p P(Ω). (i) p is said to be locally log Hölder continuous, p Clog loc (Ω), if there exists a constant c 0 such that p(x) p(y) c 0 log( x y ) for all x, y Ω with x y < 1 2. (ii) p is log Hölder continuous at infinity, p Clog (Ω), if there exist constants c and p such that p(x) p c log(e + x ) for all x Ω. We write p P log (Ω) if 1/p Clog (Ω) Cloc log (Ω). September 22nd 2014 Henning Kempka 27 / 50

48 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Conditions on p( ) Definition 4 Let p P(Ω). (i) p is said to be locally log Hölder continuous, p Clog loc (Ω), if there exists a constant c 0 such that p(x) p(y) c 0 log( x y ) for all x, y Ω with x y < 1 2. (ii) p is log Hölder continuous at infinity, p Clog (Ω), if there exist constants c and p such that p(x) p c log(e + x ) for all x Ω. We write p P log (Ω) if 1/p C log (Ω) Cloc log (Ω). September 22nd 2014 Henning Kempka 27 / 50

49 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Lemma 3 (i) If p P log (Ω), then p can be extented to p P log (R n ) with the same constants c 0, p, p and p +. (ii) If p P(R n ) with p + <, then t.f.a.e. (a) p Clog loc(rn ) (b) There exists a constant c = c(n) such that for every cube Q (or ball B) with x Q: Q p(x) p+ Q c and Q p Q p(x) c. September 22nd 2014 Henning Kempka 28 / 50

50 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Lemma 3 (i) If p P log (Ω), then p can be extented to p P log (R n ) with the same constants c 0, p, p and p +. (ii) If p P(R n ) with p + <, then t.f.a.e. (a) p Clog loc(rn ) (b) There exists a constant c = c(n) such that for every cube Q (or ball B) with x Q: Q p(x) p+ Q c and Q p Q p(x) c. September 22nd 2014 Henning Kempka 28 / 50

51 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Theorem 4 Let Ω R n and p P log (Ω) with 1 p p +, then {tχ {x Ω:Mf(x)>t} } p( ) c f p( ) and if p > 1, then Mf p( ) c f p( ). September 22nd 2014 Henning Kempka 29 / 50

52 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) History of the theorem 2000 Michael Růžička conjectured in his book Electrorheological fluids: modeling and mathematical theory: September 22nd 2014 Henning Kempka 30 / 50

53 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) History of the theorem II 2001 Lubos Pick and Michael Růžička provided a counterexample to the boundedness, if p / C loc log (Ω) 2002 His PhD student, Lars Diening, gave the first proof of the boundedness, for p + < and p is constant outside a large ball 2004 generalizations of the conditions used (p + and Clog (Ω) condition) 2005 Andrei Lerner p(x) = 2 a(1 + sin(log log(e + x + 1/ x ))) for small a > 0 Then p( ) is not continuous in x = 0 = p / C loc log (Ω) But M : L p( ) (R n ) L p( ) (R n ) is bounded September 22nd 2014 Henning Kempka 31 / 50

54 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) History of the theorem II 2001 Lubos Pick and Michael Růžička provided a counterexample to the boundedness, if p / C loc log (Ω) 2002 His PhD student, Lars Diening, gave the first proof of the boundedness, for p + < and p is constant outside a large ball 2004 generalizations of the conditions used (p + and Clog (Ω) condition) 2005 Andrei Lerner p(x) = 2 a(1 + sin(log log(e + x + 1/ x ))) for small a > 0 Then p( ) is not continuous in x = 0 = p / C loc log (Ω) But M : L p( ) (R n ) L p( ) (R n ) is bounded September 22nd 2014 Henning Kempka 31 / 50

55 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) To prove Theorem 4 We prove Theorem 4 only in the special case 1 < p p + <, p C loc log (Ω) and Ω <. We need the following variable version of Jensens inequality: Lemma 4 Let p P(Ω) with p + < and p Clog loc (Ω). Then there exists a constant c(p( )) > 0 such that for all f p 1 for all x Ω. (Mf(x)) p(x) p [( ) ] c(p( )) M f( ) p( ) p (x) + 1 September 22nd 2014 Henning Kempka 32 / 50

56 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof. 1. Jensen inequality If q [1, ] and f L q (Ω), then for all Q Ω ( ) 1 q f(y) dy 1 f(y) q dy since t t q is convex. Q Q Q Q 2. Set q( ) = p( ), then q + < and q C loc p log (Ω) 3. Let Q r = Q Ω with x Q and l(q) = r ( ) q(x) 1 r 1 Q Q f(y) dy c(p( )) 0 < r < 1 ( 1 Q Q f(y) dy ) q(x) c ( 1 Q 4. Take supremum over all cubes Q with x Q. ) Q f(y) q(y) dy + 2 September 22nd 2014 Henning Kempka 33 / 50

57 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof. 1. Jensen inequality If q [1, ] and f L q (Ω), then for all Q Ω ( ) 1 q f(y) dy 1 f(y) q dy since t t q is convex. Q Q Q Q 2. Set q( ) = p( ), then q + < and q C loc p log (Ω) 3. Let Q r = Q Ω with x Q and l(q) = r ( ) q(x) 1 r 1 Q Q f(y) dy c(p( )) 0 < r < 1 ( 1 Q Q f(y) dy ) q(x) c ( 1 Q 4. Take supremum over all cubes Q with x Q. ) Q f(y) q(y) dy + 2 September 22nd 2014 Henning Kempka 33 / 50

58 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof. 1. Jensen inequality If q [1, ] and f L q (Ω), then for all Q Ω ( ) 1 q f(y) dy 1 f(y) q dy since t t q is convex. Q Q Q Q 2. Set q( ) = p( ), then q + < and q C loc p log (Ω) 3. Let Q r = Q Ω with x Q and l(q) = r ( ) q(x) 1 r 1 Q Q f(y) dy c(p( )) 0 < r < 1 ( 1 Q Q f(y) dy ) q(x) c ( 1 Q 4. Take supremum over all cubes Q with x Q. ) Q f(y) q(y) dy + 2 September 22nd 2014 Henning Kempka 33 / 50

59 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof. 1. Jensen inequality If q [1, ] and f L q (Ω), then for all Q Ω ( ) 1 q f(y) dy 1 f(y) q dy since t t q is convex. Q Q Q Q 2. Set q( ) = p( ), then q + < and q C loc p log (Ω) 3. Let Q r = Q Ω with x Q and l(q) = r ( ) q(x) 1 r 1 Q Q f(y) dy c(p( )) 0 < r < 1 ( 1 Q Q f(y) dy ) q(x) c ( 1 Q 4. Take supremum over all cubes Q with x Q. ) Q f(y) q(y) dy + 2 September 22nd 2014 Henning Kempka 33 / 50

60 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof of Theorem additional assumptions: p C loc log (Ω) 1 < p p + < Ω < 2. Since p + < and M is homogenuous it is enough to show: 3. Set q( ) = p( ) p ϱ p (Mf) c(p( )) for all f L p( ) (Ω) with f p( ) 1. and use (Mf) q(x) (x) c(p( ))((M f( ) q( ) )(x) + 1) M : Lp L p is bounded September 22nd 2014 Henning Kempka 34 / 50

61 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof of Theorem additional assumptions: p C loc log (Ω) 1 < p p + < Ω < 2. Since p + < and M is homogenuous it is enough to show: 3. Set q( ) = p( ) p ϱ p (Mf) c(p( )) for all f L p( ) (Ω) with f p( ) 1. and use (Mf) q(x) (x) c(p( ))((M f( ) q( ) )(x) + 1) M : Lp L p is bounded September 22nd 2014 Henning Kempka 34 / 50

62 The Hardy-Littlewood maximal operator on L p( ) (Ω) Boundedness on L p( ) (Ω) Proof of Theorem additional assumptions: p C loc log (Ω) 1 < p p + < Ω < 2. Since p + < and M is homogenuous it is enough to show: 3. Set q( ) = p( ) p ϱ p (Mf) c(p( )) for all f L p( ) (Ω) with f p( ) 1. and use (Mf) q(x) (x) c(p( ))((M f( ) q( ) )(x) + 1) M : Lp L p is bounded September 22nd 2014 Henning Kempka 34 / 50

63 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces A positive interpolation result Theorem 5 (Complex interpolation) Let p 0, p 1 P(Ω), Θ (0, 1) and define Then 1 p Θ (x) = 1 Θ p 0 (x) + Θ p 1 (x). [ Lp0 ( )(Ω), L p1 ( )(Ω) ] Θ = L p Θ ( )(Ω). September 22nd 2014 Henning Kempka 35 / 50

64 Open question The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces This question for a Marcinkiewicz theorem in variable exponent Lebesgue spaces was posed in 2004 by Diening, Hästö and Nekvinda. Question: Let T be a sublinear operator which is of weak type (π 0 ( ), π 0 ( )) and of weak type (π 1 ( ), π 1 ( )). Is T then bounded from L πθ ( ) to L πθ ( ) with 1 π Θ ( ) = 1 Θ π 0 ( ) + Θ π 1 ( )? weak type (π( ), π( )) means, there exist a constant such that Lπ( ) c f Lπ( ), tχ{x R n : T f(x) >t} i.e. the operator T is bounded from L π( ) into L π( ),. September 22nd 2014 Henning Kempka 36 / 50

65 Open question The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces This question for a Marcinkiewicz theorem in variable exponent Lebesgue spaces was posed in 2004 by Diening, Hästö and Nekvinda. Question: Let T be a sublinear operator which is of weak type (π 0 ( ), π 0 ( )) and of weak type (π 1 ( ), π 1 ( )). Is T then bounded from L πθ ( ) to L πθ ( ) with 1 π Θ ( ) = 1 Θ π 0 ( ) + Θ π 1 ( )? weak type (π( ), π( )) means, there exist a constant such that Lπ( ) c f Lπ( ), tχ{x R n : T f(x) >t} i.e. the operator T is bounded from L π( ) into L π( ),. September 22nd 2014 Henning Kempka 36 / 50

66 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Negative Result In general Marcinkiewicz Interpolation does not hold in the variable exponent setting, ie. T... sublinear operator T : L π0 ( ) L π0 ( ), T : L π1 ( ) L π1 ( ), Then in general it does not hold: T : L πθ ( ) L πθ ( ) with 1/π θ ( ) = (1 θ)/π 0 ( ) + θ/π 1 ( ) Idea for counterexample: Use usual Marcinkiewicz interpolation H. Kempka, J. Vybíral: Lorentz spaces with variable exponents, Math. Nachr. 287 no. 8-9 (2014), September 22nd 2014 Henning Kempka 37 / 50

67 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Negative Result In general Marcinkiewicz Interpolation does not hold in the variable exponent setting, ie. T... sublinear operator T : L π0 ( ) L π0 ( ), T : L π1 ( ) L π1 ( ), Then in general it does not hold: T : L πθ ( ) L πθ ( ) with 1/π θ ( ) = (1 θ)/π 0 ( ) + θ/π 1 ( ) Idea for counterexample: Use usual Marcinkiewicz interpolation H. Kempka, J. Vybíral: Lorentz spaces with variable exponents, Math. Nachr. 287 no. 8-9 (2014), September 22nd 2014 Henning Kempka 37 / 50

68 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Theorem 6 (Marcinkiewicz interpolation) Let T be a sublinear operator which is bounded from L p0 into L q0, and from L p1 into L q1,, where 0 < p 0 p 1 and 0 < q 0 q 1. Let 0 < Θ < 1 and put 1 p = 1 Θ + Θ, p 0 p 1 1 q = 1 Θ + Θ. q 0 q 1 If p q, then T is also bounded from L p into L q. There exist a sublinear operator T and 0 < p 0 p 1, 0 < q 0 q 1 and 0 < θ < 1 such that T : L p0 ([0, 1]) L q0, ([0, 1]) T : L p1 ([0, 1]) L q1, ([0, 1]) and p > q and T is not bounded from L p ([0, 1]) to L q ([0, 1]). September 22nd 2014 Henning Kempka 38 / 50

69 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Theorem 6 (Marcinkiewicz interpolation) Let T be a sublinear operator which is bounded from L p0 into L q0, and from L p1 into L q1,, where 0 < p 0 p 1 and 0 < q 0 q 1. Let 0 < Θ < 1 and put 1 p = 1 Θ + Θ, p 0 p 1 1 q = 1 Θ + Θ. q 0 q 1 If p q, then T is also bounded from L p into L q. There exist a sublinear operator T and 0 < p 0 p 1, 0 < q 0 q 1 and 0 < θ < 1 such that T : L p0 ([0, 1]) L q0, ([0, 1]) T : L p1 ([0, 1]) L q1, ([0, 1]) and p > q and T is not bounded from L p ([0, 1]) to L q ([0, 1]). September 22nd 2014 Henning Kempka 38 / 50

70 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Theorem 6 (Marcinkiewicz interpolation) Let T be a sublinear operator which is bounded from L p0 into L q0, and from L p1 into L q1,, where 0 < p 0 p 1 and 0 < q 0 q 1. Let 0 < Θ < 1 and put 1 p = 1 Θ + Θ, p 0 p 1 1 q = 1 Θ + Θ. q 0 q 1 If p q, then T is also bounded from L p into L q. There exist a sublinear operator T and 0 < p 0 p 1, 0 < q 0 q 1 and 0 < θ < 1 such that T : L p0 ([0, 1]) L q0, ([0, 1]) T : L p1 ([0, 1]) L q1, ([0, 1]) and p > q and T is not bounded from L p ([0, 1]) to L q ([0, 1]). September 22nd 2014 Henning Kempka 38 / 50

71 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Counterexample Use the counterexample T to usual Marcinkiewicz from above and define T by T f(x) := { T (χ [0,1] f)(x 1), if x [1, 2] 0 if x [0, 1). Put π 0 (x) := { p 0, x [0, 1) q 0, x [1, 2] and π 1 (x) := { p 1, x [0, 1) q 1, x [1, 2] then T is weak type (π0 ( ), π 0 ( )) T is weak type (π1 ( ), π 1 ( )) but not strong type (π θ ( ), π θ ( )) September 22nd 2014 Henning Kempka 39 / 50

72 The Hardy-Littlewood maximal operator on L p( ) (Ω) Interpolation in variable exponent Lebesgue spaces Counterexample Use the counterexample T to usual Marcinkiewicz from above and define T by T f(x) := { T (χ [0,1] f)(x 1), if x [1, 2] 0 if x [0, 1). Put π 0 (x) := { p 0, x [0, 1) q 0, x [1, 2] and π 1 (x) := { p 1, x [0, 1) q 1, x [1, 2] then T is weak type (π0 ( ), π 0 ( )) T is weak type (π1 ( ), π 1 ( )) but not strong type (π θ ( ), π θ ( )) September 22nd 2014 Henning Kempka 39 / 50

73 Rubio de Francia extrapolation Table of Contents 1. Introduction & Motivation 2. Variable exponent Lebesgue spaces 3. The Hardy-Littlewood maximal operator on L p( ) (Ω) 4. Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Wavelet characterization of L p( ) (Ω) September 22nd 2014 Henning Kempka 40 / 50

74 Definition 5 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces A weight w : R n (0, ) belongs to the Muckenhoupt class A p (1 < p < ), if ( 1 1 p 1 w(x)dx w(x) dx) 1 p C (1) Q Q Q Q for all cubes Q R n. w is an A 1 weight if Mw(x) Cw(x) for a.e. x R n and A = 1 p< A p. The smallest constant in (1) is denoted by A p (w). Remark 3 A p A q for p < q M : L p (w) L p (w) if, and only if, w A p. September 22nd 2014 Henning Kempka 41 / 50

75 Definition 5 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces A weight w : R n (0, ) belongs to the Muckenhoupt class A p (1 < p < ), if ( 1 1 p 1 w(x)dx w(x) dx) 1 p C (1) Q Q Q Q for all cubes Q R n. w is an A 1 weight if Mw(x) Cw(x) for a.e. x R n and A = 1 p< A p. The smallest constant in (1) is denoted by A p (w). Remark 3 A p A q for p < q M : L p (w) L p (w) if, and only if, w A p. September 22nd 2014 Henning Kempka 41 / 50

76 Definition 5 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces A weight w : R n (0, ) belongs to the Muckenhoupt class A p (1 < p < ), if ( 1 1 p 1 w(x)dx w(x) dx) 1 p C (1) Q Q Q Q for all cubes Q R n. w is an A 1 weight if Mw(x) Cw(x) for a.e. x R n and A = 1 p< A p. The smallest constant in (1) is denoted by A p (w). Remark 3 A p A q for p < q M : L p (w) L p (w) if, and only if, w A p. September 22nd 2014 Henning Kempka 41 / 50

77 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Theorem 7 Given Ω R n and a family of measureable functions F = {f, g} such that for a p 0 1 and all w A 1 and all (f, g) F f(x) p 0 w(x)dx c 1 g(x) p 0 w(x)dx, Ω where c 1 only depends on A 1 (w). Let p P log (Ω) with p 0 p p + <, then f p( ) c(p( )) g p( ) for all (f, g) F. Ω September 22nd 2014 Henning Kempka 42 / 50

78 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Theorem 7 Given Ω R n and a family of measureable functions F = {f, g} such that for a p 0 1 and all w A 1 and all (f, g) F f(x) p 0 w(x)dx c 1 g(x) p 0 w(x)dx, Ω where c 1 only depends on A 1 (w). Let p P log (Ω) with p 0 p p + <, then f p( ) c(p( )) g p( ) for all (f, g) F. Ω September 22nd 2014 Henning Kempka 42 / 50

79 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Proof. Set s( ) = p( ) p 0, then 1 s s + < and s P log (Ω). We use 1. f p0 p( ) = f p0 s( ) 2. f s( ) sup h s ( ) 1 Ω f(x)h(x)dx 3. Mh s ( ) A h s ( ) Define the Rubio de Francia operator Rh(x) = k=0 M k h(x) (2A) k, where Mk = M M and M 0 = id. September 22nd 2014 Henning Kempka 43 / 50

80 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Proof. Set s( ) = p( ) p 0, then 1 s s + < and s P log (Ω). We use 1. f p0 p( ) = f p0 s( ) 2. f s( ) sup h s ( ) 1 Ω f(x)h(x)dx 3. Mh s ( ) A h s ( ) Define the Rubio de Francia operator Rh(x) = k=0 M k h(x) (2A) k, where Mk = M M and M 0 = id. September 22nd 2014 Henning Kempka 43 / 50

81 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Proof. Set s( ) = p( ) p 0, then 1 s s + < and s P log (Ω). We use 1. f p0 p( ) = f p0 s( ) 2. f s( ) sup h s ( ) 1 Ω f(x)h(x)dx 3. Mh s ( ) A h s ( ) Define the Rubio de Francia operator Rh(x) = k=0 M k h(x) (2A) k, where Mk = M M and M 0 = id. September 22nd 2014 Henning Kempka 43 / 50

82 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Proof. The Rubio de Francia operator Rh(x) = k=0 M k h(x) (2A) k has the properties: (a) h Rh (b) Rh s ( ) 2 h s ( ) (c) M(Rh) 2ARh and therfore Rh is an A 1 weight. September 22nd 2014 Henning Kempka 44 / 50

83 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Remark 4 If one uses Rubio de Francia interpolation in the constant exponent case and A 1 A p we get the following version: If there exists a p 0 1 such that for all w A p0 f(x) p 0 w(x)dx c 1 g(x) p 0 w(x)dx, Ω where c 1 only depends on A p0 (w). Let p P log (Ω) with 1 < p p + <, then f p( ) c(p( )) g p( ) for all (f, g) F. Ω September 22nd 2014 Henning Kempka 45 / 50

84 Rubio de Francia extrapolation Extrapolation in variable Lebesgue spaces Remark 4 If one uses Rubio de Francia interpolation in the constant exponent case and A 1 A p we get the following version: If there exists a p 0 1 such that for all w A p0 f(x) p 0 w(x)dx c 1 g(x) p 0 w(x)dx, Ω where c 1 only depends on A p0 (w). Let p P log (Ω) with 1 < p p + <, then f p( ) c(p( )) g p( ) for all (f, g) F. Ω September 22nd 2014 Henning Kempka 45 / 50

85 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) Wavelet bases in R n Let D = {Q jk = 2 j ([0, 1) n + k) : j Z and k Z n } be dyadic cubes. A finite set of functions Ψ = {ψ 1,..., ψ L } L 2 (R n ) is called an orthonormal Waveletbasis, if {Ψ l Q } = {ψl Q jk (x) = 2 j n 2 ψ l (2 j x k) : j Z, k Z n, 1 l L} is an ONB of L 2 (R n ). Define: 1/2 L W Ψ f = f, ψq l 2 Q 1 χ Q = l=1 Q D j= k Z n l=1 L f, ψq l 2 jk 2 jn χ jk 1/2 September 22nd 2014 Henning Kempka 46 / 50

86 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) Wavelet bases in R n Let D = {Q jk = 2 j ([0, 1) n + k) : j Z and k Z n } be dyadic cubes. A finite set of functions Ψ = {ψ 1,..., ψ L } L 2 (R n ) is called an orthonormal Waveletbasis, if {Ψ l Q } = {ψl Q jk (x) = 2 j n 2 ψ l (2 j x k) : j Z, k Z n, 1 l L} is an ONB of L 2 (R n ). Define: 1/2 L W Ψ f = f, ψq l 2 Q 1 χ Q = l=1 Q D j= k Z n l=1 L f, ψq l 2 jk 2 jn χ jk 1/2 September 22nd 2014 Henning Kempka 46 / 50

87 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) A wavelet system is called admissible, if there exists an p 0 (1, ) such that for all w A p0. Remark 5 Cw 1 f L p0 (w) W Ψ f L p0 (w) C w f L p0 (w) Admissible wavelet systems are: Haar wavelets Spline wavelets Daubechies wavelets Wavelets out of Multi resolution analysis September 22nd 2014 Henning Kempka 47 / 50

88 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) A wavelet system is called admissible, if there exists an p 0 (1, ) such that for all w A p0. Remark 5 Cw 1 f L p0 (w) W Ψ f L p0 (w) C w f L p0 (w) Admissible wavelet systems are: Haar wavelets Spline wavelets Daubechies wavelets Wavelets out of Multi resolution analysis September 22nd 2014 Henning Kempka 47 / 50

89 Further use: L suppf compact is dense in L p( ) (R n ). September 22nd 2014 Henning Kempka 48 / 50 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) Theorem 8 Let p P log (R n ) with 1 < p p + < and Ψ be an admissible wavelet system. Then for all f L p( ) (R n ) we have 1/2 L f p( ) f, ψq l 2 Q 1 χ Q. l=1 Q D p( ) Proof. Apply Theorem 7 and Remark 4 on F 1 = {( f, W Ψ f) : f L and suppf compact} and F 2 = {(W Ψ f, f ) : f L and suppf compact}.

90 Further use: L suppf compact is dense in L p( ) (R n ). September 22nd 2014 Henning Kempka 48 / 50 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) Theorem 8 Let p P log (R n ) with 1 < p p + < and Ψ be an admissible wavelet system. Then for all f L p( ) (R n ) we have 1/2 L f p( ) f, ψq l 2 Q 1 χ Q. l=1 Q D p( ) Proof. Apply Theorem 7 and Remark 4 on F 1 = {( f, W Ψ f) : f L and suppf compact} and F 2 = {(W Ψ f, f ) : f L and suppf compact}.

91 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) Further applications of Rubio de Francia extrapolation Boundedness of the fractional maximal and the Riesz potential operator on L p( ) (R n ) Boundedness of Calderon-Zygmund integral operators on L p( ) (R n ) Vector-valued maximal inequality, i.e. the maximal operator is bounded on L p( ) (l q ). September 22nd 2014 Henning Kempka 49 / 50

92 Rubio de Francia extrapolation Wavelet characterization of L p( ) (Ω) Thank you for your attention! September 22nd 2014 Henning Kempka 50 / 50