Decompositions of variable Lebesgue norms by ODE techniques

Transcription

1 Decompositions of variable Lebesgue norms by ODE techniques Septièmes journées Besançon-Neuchâtel d Analyse Fonctionnelle Jarno Talponen University of Eastern Finland talponen@iki.fi Besançon, June 217 Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

2 Abstract We study decompositions of Nakano type varying exponent Lebesgue norms and spaces. These function spaces are represented here in a natural way as tractable varying l p sums of projection bands. The main results involve embedding the varying Lebesgue spaces to such sums, as well as the corresponding isomorphism constants. The main tool applied here is an equivalent variable Lebesgue norm which is defined by a suitable ordinary differential equation introduced recently by the author. There is a preprint in the ArXiv with a similar title. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

3 Nakano spaces Variable Lebesgue spaces, or varying exponent L p spaces are often equipped with the following Musielak-Orlicz type norm: where ρ(g) = f p( ) = inf {λ > : ρ(f /λ) 1} g(t) dt, p, f, g L [, 1], p 1. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

4 Nakano spaces Variable Lebesgue spaces, or varying exponent L p spaces are often equipped with the following Musielak-Orlicz type norm: where ρ(g) = f p( ) = inf {λ > : ρ(f /λ) 1} g(t) dt, p, f, g L [, 1], p 1. There are some heuristic continuity considerations involving the modular ρ which suggest that an additional weight function should by applied in it. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

5 Nakano spaces Variable Lebesgue spaces, or varying exponent L p spaces are often equipped with the following Musielak-Orlicz type norm: where ρ(g) = f p( ) = inf {λ > : ρ(f /λ) 1} g(t) dt, p, f, g L [, 1], p 1. There are some heuristic continuity considerations involving the modular ρ which suggest that an additional weight function should by applied in it. In the paper we investigate Nakano L p( ) norms: f p( ) = inf { λ > : 1 f (t) λ (here the terminology varies in the literature). dt 1 } Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

6 Nakano spaces Variable Lebesgue spaces, or varying exponent L p spaces are often equipped with the following Musielak-Orlicz type norm: where ρ(g) = f p( ) = inf {λ > : ρ(f /λ) 1} g(t) dt, p, f, g L [, 1], p 1. There are some heuristic continuity considerations involving the modular ρ which suggest that an additional weight function should by applied in it. In the paper we investigate Nakano L p( ) norms: f p( ) = inf { λ > : 1 f (t) λ dt 1 (here the terminology varies in the literature). The above norms are derived from the Minkowski functional and are special cases of Musielak-Orlicz type L p( ) norms. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14 }

7 Decompositions As usual, we denote by X p Y the direct sum of Banach spaces X and Y with the norm given by (x, y) X py = x X p y Y, x X, y Y, 1 p < where a p b := (a p + b p ) 1 p, a, b. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

8 Decompositions As usual, we denote by X p Y the direct sum of Banach spaces X and Y with the norm given by (x, y) X py = x X p y Y, x X, y Y, 1 p < where a p b := (a p + b p ) 1 p, a, b. Recall that f L p = 1 f L p p 1 [,1]\ f L p for any f L p [, 1], p [1, ), and a measurable subset [, 1]. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

9 Decompositions As usual, we denote by X p Y the direct sum of Banach spaces X and Y with the norm given by (x, y) X py = x X p y Y, x X, y Y, 1 p < where Recall that a p b := (a p + b p ) 1 p, a, b. f L p = 1 f L p p 1 [,1]\ f L p for any f L p [, 1], p [1, ), and a measurable subset [, 1]. Now, if p( ) is a simple function with values p 1, p 2,..., p n then it seems reasonable to ask, how and with what isomorphism constants we can decompose L p( ) to the corresponding bands X i. For instance, L p( )? (... (X 1 p2 X 2 ) p3 X 3 ) p4... pn X n. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

10 Decompositions As usual, we denote by X p Y the direct sum of Banach spaces X and Y with the norm given by (x, y) X py = x X p y Y, x X, y Y, 1 p < where Recall that a p b := (a p + b p ) 1 p, a, b. f L p = 1 f L p p 1 [,1]\ f L p for any f L p [, 1], p [1, ), and a measurable subset [, 1]. Now, if p( ) is a simple function with values p 1, p 2,..., p n then it seems reasonable to ask, how and with what isomorphism constants we can decompose L p( ) to the corresponding bands X i. For instance, L p( )? (... (X 1 p2 X 2 ) p3 X 3 ) p4... pn X n. Preferably, the isomorphism constant should not depend on n. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

11 Motivation The definition of the Musielak-Orlicz norm is global. It is easy to write the definition. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

12 Motivation The definition of the Musielak-Orlicz norm is global. It is easy to write the definition. But is it easy to analyze these norms locally? Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

13 Motivation The definition of the Musielak-Orlicz norm is global. It is easy to write the definition. But is it easy to analyze these norms locally? In the analysis of classical integral operators different l p like auxiliary structures have turned out to be crucial. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

14 References on such applications Cruz-Uribe, D.; Fiorenza, A. Approximate identities in variable Lp spaces. Math. Nachr. 28 (27), no. 3, Cruz-Uribe, D.; Fiorenza, A.; Martell, J. M.; Pérez, C. The boundedness of classical operators on variable Lp spaces. Ann. Acad. Sci. Fenn. Math. 31 (26), no. 1, Cruz-Uribe, David; Wang, Li-An Daniel, Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Amer. Math. Soc. 369 (217), L. Diening, M. Ruzicka, Calderon-Zygmund operators on generalized Lebesgue spaces L p( ) and problems related to fluid dynamics. J. Reine Angew. Math. 563 (23), L. Diening, P. Harjulehto, P. Hästö, Function spaces of variable smoothness and integrability, Journal of functional Analysis, 256, (29), F. Maeda, Y. Sawano, T. Shimomura, Some norm inequalities in Musielak-Orlicz spaces. Ann. Acad. Sci. Fenn. Math. 41 (216), Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

15 Typical estimates in the paper Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

16 Typical estimates in the paper Theorem Let p L, p 1 and i, i = 1,..., n, be a measurable decomposition of [, 1], f L and r i = ess inf i p, s i = ess sup i p. Then and 1 2 ( 1 1 f p( ) s2 1 2 f p( ) ) s3... sn 1 n f p( ) f p( ) 2( 1 1 f p( ) r2 1 2 f p( ) ) r3... rn 1 n f p( ) ( 1 1 f r1 r2 1 2 f r2 ) r3... rn 1 n f rn f p( ) 12( 1 1 f s1 s2 1 2 f s2 ) s3... sn 1 n f sn. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

17 Norms defined by ODEs T. 216 introduced an ODE which defines L p( ) type norms. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

18 Norms defined by ODEs T. 216 introduced an ODE which defines L p( ) type norms. The general strategy: Let f be a candidate for an element in the L p( ) space sought after. The ODE is designed in such a way that its solution ϕ f : [, 1] R, satisfies ϕ f (t) =: 1 [,t] f. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

19 Norms defined by ODEs T. 216 introduced an ODE which defines L p( ) type norms. The general strategy: Let f be a candidate for an element in the L p( ) space sought after. The ODE is designed in such a way that its solution ϕ f : [, 1] R, satisfies ϕ f (t) =: 1 [,t] f. The absolutely continuous solution is defined by ϕ f () = +, ϕ f (t) = f (t) ϕ(t) 1 for a.e. t [, 1], (1) in the sense of Carathéodory. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

20 Norms defined by ODEs T. 216 introduced an ODE which defines L p( ) type norms. The general strategy: Let f be a candidate for an element in the L p( ) space sought after. The ODE is designed in such a way that its solution ϕ f : [, 1] R, satisfies ϕ f (t) =: 1 [,t] f. The absolutely continuous solution is defined by ϕ f () = +, ϕ f (t) = f (t) ϕ(t) 1 for a.e. t [, 1], (1) in the sense of Carathéodory. This produces a unique non-negative, non-decreasing solution for a suitable f L, e.g. f L. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

21 Function spaces defined by ODEs Given a measurable p : [, 1] [1, ), we eventually define L p( ) ODE := {f L [, 1]: ϕ f ϕ f (1) < } where ϕ f exists as Carathéodory s weak solution to and ϕ f () = +, ϕ f (t) = f (t) ϕ(t) 1 f p( ) L := ϕ f (1). ODE a.e. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

22 Function spaces defined by ODEs Given a measurable p : [, 1] [1, ), we eventually define L p( ) ODE := {f L [, 1]: ϕ f ϕ f (1) < } where ϕ f exists as Carathéodory s weak solution to ϕ f () = +, ϕ f (t) = f (t) ϕ(t) 1 a.e. and f p( ) L := ϕ f (1). ODE Of course, it is not a priori clear that this defines a Banach space. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

23 Function spaces defined by ODEs Given a measurable p : [, 1] [1, ), we eventually define L p( ) ODE := {f L [, 1]: ϕ f ϕ f (1) < } where ϕ f exists as Carathéodory s weak solution to ϕ f () = +, ϕ f (t) = f (t) ϕ(t) 1 and f p( ) L := ϕ f (1). ODE Of course, it is not a priori clear that this defines a Banach space. One has to be careful with the domains of definition of the solutions. Surprisingly, it turns out (see the paper) that linear structure fails before the -inequality. a.e. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

24 Function spaces defined by ODEs Given a measurable p : [, 1] [1, ), we eventually define L p( ) ODE := {f L [, 1]: ϕ f ϕ f (1) < } where ϕ f exists as Carathéodory s weak solution to ϕ f () = +, ϕ f (t) = f (t) ϕ(t) 1 and f p( ) L := ϕ f (1). ODE Of course, it is not a priori clear that this defines a Banach space. One has to be careful with the domains of definition of the solutions. Surprisingly, it turns out (see the paper) that linear structure fails before the -inequality. For instance, if p( ) is essentially bounded then L p( ) ODE becomes a Banach space. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14 a.e.

25 Illustration If p is a constant then the function ϕ f is clearly absolutely continuous and in fact the above ODE is separable. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

26 Illustration If p is a constant then the function ϕ f is clearly absolutely continuous and in fact the above ODE is separable. Let us solve it: ϕ (t) = f (t) p (ϕ(t)) 1 p p a.e. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

27 Illustration If p is a constant then the function ϕ f is clearly absolutely continuous and in fact the above ODE is separable. Let us solve it: ϕ (t) = f (t) p (ϕ(t)) 1 p p a.e. p(ϕ(t)) p 1 ϕ (t) = f (t) p a.e. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

28 Illustration If p is a constant then the function ϕ f is clearly absolutely continuous and in fact the above ODE is separable. Let us solve it: ϕ (t) = f (t) p (ϕ(t)) 1 p p a.e. p(ϕ(t)) p 1 ϕ (t) = f (t) p a.e. p ϕ p 1 ϕ dt = f (t) p dt, Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

29 Illustration If p is a constant then the function ϕ f is clearly absolutely continuous and in fact the above ODE is separable. Let us solve it: ϕ (t) = f (t) p (ϕ(t)) 1 p p a.e. p(ϕ(t)) p 1 ϕ (t) = f (t) p a.e. p ϕ p 1 ϕ dt = ϕ(1) p = f (t) p dt, f (t) p dt, Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

30 Illustration If p is a constant then the function ϕ f is clearly absolutely continuous and in fact the above ODE is separable. Let us solve it: ϕ (t) = f (t) p (ϕ(t)) 1 p p a.e. p(ϕ(t)) p 1 ϕ (t) = f (t) p a.e. p ϕ p 1 ϕ dt = ϕ(1) p = f (t) p dt, f (t) p dt, ( ϕ(1) = f (t) dt) 1 p p. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

31 Illustration II Thus for any p( ) p [1, ) L p( ) ODE = Lp. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

32 Illustration II Thus for any p( ) p [1, ) If then in the natural way isometrically. L p( ) ODE = Lp. p = p 1 1 [, 1 2 ] + p 21 [ 1 2,1] L p( ) ODE = Lp 1 [, 1/2] p2 L p 2 [1/2, 1] Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

33 Illustration II Thus for any p( ) p [1, ) If then in the natural way isometrically. L p( ) ODE = Lp. p = p 1 1 [, 1 2 ] + p 21 [ 1 2,1] L p( ) ODE = Lp 1 [, 1/2] p2 L p 2 [1/2, 1] This works similarly for successive l p summations. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

34 Proposition f p( ) f p( ) L 2 f p( ) ODE f L p( ) ODE. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

35 Proposition f p( ) f p( ) L 2 f p( ) ODE f L p( ) ODE. To check f p( ) L inf ODE { λ > : suppose that < ϕ f (1) = λ. 1 ( ) f (t) dt 1} λ Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

36 Proposition f p( ) f p( ) L 2 f p( ) ODE f L p( ) ODE. To check f p( ) L inf ODE { λ > : suppose that < ϕ f (1) = λ. Then so that λ = ϕ f (1) ϕ f (t) 1 ( ) f (t) dt 1} λ f (t) λ 1, f (t) λ 1 dt = λ 1 ( ) f (t) dt. λ Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

37 Proposition f p( ) f p( ) L 2 f p( ) ODE f L p( ) ODE. To check f p( ) L inf ODE { λ > : suppose that < ϕ f (1) = λ. Then so that λ = ϕ f (1) ϕ f (t) 1 ( ) f (t) dt 1} λ f (t) λ 1, f (t) λ 1 dt = λ 1 This is equivalent to ( ) 1 f (t) dt 1. λ ( ) f (t) dt. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14 λ

38 To check the latter inequality, we may restrict to the case f p( ) = 1 by the positive homogeneity of the norms. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

39 To check the latter inequality, we may restrict to the case f p( ) = 1 by the positive homogeneity of the norms. If f p( ) L 1 then we have the claim, so assume that < t < 1 is ODE such that ϕ f (t ) = 1, i.e. 1 [,t ]f p( ) L = 1. ODE Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

40 To check the latter inequality, we may restrict to the case f p( ) = 1 by the positive homogeneity of the norms. If f p( ) L 1 then we have the claim, so assume that < t < 1 is ODE such that ϕ f (t ) = 1, i.e. 1 [,t ]f p( ) L = 1. ODE Then ϕ f (t) f (t), for a.e. t [t, 1]. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

41 To check the latter inequality, we may restrict to the case f p( ) = 1 by the positive homogeneity of the norms. If f p( ) L 1 then we have the claim, so assume that < t < 1 is ODE such that ϕ f (t ) = 1, i.e. 1 [,t ]f p( ) L = 1. ODE Then Thus ϕ f (t) f (t), for a.e. t [t, 1]. ϕ f (1) 1+ t f (t) f (t) dt 1+ dt = 1+ f p( ) = 2. Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14

42 Merci! Jarno Talponen (UEF) Decompositions of norms by ODEs Besançon, June / 14