Symmetrization of some quasi-Banach function spaces Pawe÷ Kolwicz Institute of Mathematics Pozna´ n University of Technology POLAND PAWE× DOMA´ NSKI MEMORIAL CONFERENCE B ¾ edlewo 1.07-7.07.2018 Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 1 / 22
The outline of the talk Introduction. 1 The commutativity property of symmetrization with some known 2 constructions. Some factorization results. 3 The talk is supported by the Ministry of Science and Higher Education of Poland, grant number 04/43/DSPB/0094 and it is based on the papers: 1. Pawe÷ Kolwicz, Karol Le´ snik and Lech Maligranda, Pointwise products of some Banach function spaces and factorization, J. Funct. Anal. 266, 2, (2014), 616–659. 2. P. Kolwicz, K. Le´ snik and L. Maligranda, Symmetrization, factorization and arithmetic of quasi-Banach function spaces, submitted, avalaible on https://arxiv.org/pdf/1801.05799.pdf. Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 2 / 22
Introduction. Let ( I , Σ , m ) be a Lebesgue measure space with I = ( 0 , 1 ) or I = ( 0 , ∞ ) . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 3 / 22
Introduction. Let ( I , Σ , m ) be a Lebesgue measure space with I = ( 0 , 1 ) or I = ( 0 , ∞ ) . By L 0 = L 0 ( I ) we denote the set of all m -equivalence classes of real valued Lebesgue measurable functions de…ned on I . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 3 / 22
Introduction. Let ( I , Σ , m ) be a Lebesgue measure space with I = ( 0 , 1 ) or I = ( 0 , ∞ ) . By L 0 = L 0 ( I ) we denote the set of all m -equivalence classes of real valued Lebesgue measurable functions de…ned on I . Quasi-Banach ideal (function) space on I A quasi-Banach space E = ( E , k � k E ) is said to be a quasi-Banach ideal (function) space on I if E is a linear subspace of L 0 ( I ) and Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 3 / 22
Introduction. Let ( I , Σ , m ) be a Lebesgue measure space with I = ( 0 , 1 ) or I = ( 0 , ∞ ) . By L 0 = L 0 ( I ) we denote the set of all m -equivalence classes of real valued Lebesgue measurable functions de…ned on I . Quasi-Banach ideal (function) space on I A quasi-Banach space E = ( E , k � k E ) is said to be a quasi-Banach ideal (function) space on I if E is a linear subspace of L 0 ( I ) and if x 2 E , y 2 L 0 and j y j � j x j µ -a.e., then y 2 E and k y k E � k x k E ; 1 Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 3 / 22
Introduction. Let ( I , Σ , m ) be a Lebesgue measure space with I = ( 0 , 1 ) or I = ( 0 , ∞ ) . By L 0 = L 0 ( I ) we denote the set of all m -equivalence classes of real valued Lebesgue measurable functions de…ned on I . Quasi-Banach ideal (function) space on I A quasi-Banach space E = ( E , k � k E ) is said to be a quasi-Banach ideal (function) space on I if E is a linear subspace of L 0 ( I ) and if x 2 E , y 2 L 0 and j y j � j x j µ -a.e., then y 2 E and k y k E � k x k E ; 1 there exists a function x in E that is positive on the whole I . 2 Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 3 / 22
Introduction. Symmetric function space By a symmetric quasi-Banach function space on I , where I = ( 0 , 1 ) or I = ( 0 , ∞ ) with the Lebesgue measure m , we mean a quasi-Banach function space E = ( E , k � k E ) with the additional property that for any two equimeasurable functions x � y , x , y 2 L 0 ( I ) (that is, d x = d y , where d x ( λ ) = m ( f t 2 I : j x ( t ) j > λ g ) , λ � 0) and x 2 E we have y 2 E and k x k E = k y k E . In particular, k x k E = k x � k E , where x � ( t ) = inf f λ > 0 : d x ( λ ) � t g , t � 0 . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 4 / 22
The space of pointwise multipliers. Let ( E , k � k E ) and ( F , k � k F ) be quasi-Banach function spaces. The space of pointwise multipliers M ( E , F ) is de…ned by M ( E , F ) = f x 2 L 0 ( I ) : xy 2 F for all y 2 E g (1) and the functional on it k x k M ( E , F ) = sup fk xy k F , y 2 E , k y k E � 1 g (2) de…nes a quasi-norm. Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 5 / 22
The space of pointwise multipliers. Let ( E , k � k E ) and ( F , k � k F ) be quasi-Banach function spaces. The space of pointwise multipliers M ( E , F ) is de…ned by M ( E , F ) = f x 2 L 0 ( I ) : xy 2 F for all y 2 E g (1) and the functional on it k x k M ( E , F ) = sup fk xy k F , y 2 E , k y k E � 1 g (2) de…nes a quasi-norm. If F = L 1 we have M ( E , L 1 ) = E 0 , where E 0 is the classical associated space to E or the Köthe dual space of E . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 5 / 22
The space of pointwise multipliers. Let ( E , k � k E ) and ( F , k � k F ) be quasi-Banach function spaces. The space of pointwise multipliers M ( E , F ) is de…ned by M ( E , F ) = f x 2 L 0 ( I ) : xy 2 F for all y 2 E g (1) and the functional on it k x k M ( E , F ) = sup fk xy k F , y 2 E , k y k E � 1 g (2) de…nes a quasi-norm. If F = L 1 we have M ( E , L 1 ) = E 0 , where E 0 is the classical associated space to E or the Köthe dual space of E . Note that M ( E , F ) can be f 0 g . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 5 / 22
The space of pointwise multipliers. Let ( E , k � k E ) and ( F , k � k F ) be quasi-Banach function spaces. The space of pointwise multipliers M ( E , F ) is de…ned by M ( E , F ) = f x 2 L 0 ( I ) : xy 2 F for all y 2 E g (1) and the functional on it k x k M ( E , F ) = sup fk xy k F , y 2 E , k y k E � 1 g (2) de…nes a quasi-norm. If F = L 1 we have M ( E , L 1 ) = E 0 , where E 0 is the classical associated space to E or the Köthe dual space of E . Note that M ( E , F ) can be f 0 g . It is possible that supp M ( E , F ) is smaller than supp E \ supp F . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 5 / 22
Pointwise products. Given two quasi-Banach function spaces E and F de…ne the pointwise product space E � F as E � F = f x � y : x 2 E and y 2 F g . (3) with a functional k � k E � F de…ned by the formula k z k E � F = inf fk x k E k y k F : z = xy , x 2 E , y 2 F g . (4) Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 6 / 22
The Calderón-Lozanovski¼ ¬ space (construction). By P we denote the set of positively homogeneous and concave functions ρ : [ 0 , ∞ ) � [ 0 , ∞ ) ! [ 0 , ∞ ) . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 7 / 22
The Calderón-Lozanovski¼ ¬ space (construction). By P we denote the set of positively homogeneous and concave functions ρ : [ 0 , ∞ ) � [ 0 , ∞ ) ! [ 0 , ∞ ) . For two quasi-Banach function spaces E , F on I and ρ 2 P the Calderón-Lozanovski¼ ¬ space ( construction ) ρ ( E , F ) = f x 2 L 0 ( I ) : j x j � λ ρ ( j x 0 j , j x 1 j ) a.e. on I (5) for some x 0 2 E , x 1 2 F with k x 0 k E � 1 , k x 1 k F � 1 and for some λ > 0 g . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 7 / 22
The Calderón-Lozanovski¼ ¬ space (construction). By P we denote the set of positively homogeneous and concave functions ρ : [ 0 , ∞ ) � [ 0 , ∞ ) ! [ 0 , ∞ ) . For two quasi-Banach function spaces E , F on I and ρ 2 P the Calderón-Lozanovski¼ ¬ space ( construction ) ρ ( E , F ) = f x 2 L 0 ( I ) : j x j � λ ρ ( j x 0 j , j x 1 j ) a.e. on I (5) for some x 0 2 E , x 1 2 F with k x 0 k E � 1 , k x 1 k F � 1 and for some λ > 0 g . The quasi-norm k x k ρ = k x k ρ ( E , F ) = inf f λ > 0 g for which the above inequality holds. Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 7 / 22
The Calderón-Lozanovski¼ ¬ space (construction) Examples If ρ ( u , v ) = u θ v 1 � θ with 0 < θ < 1 we write E θ F 1 � θ instead of ρ ( E , F ) and these are Calderón spaces. Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 8 / 22
The Calderón-Lozanovski¼ ¬ space (construction) Examples If ρ ( u , v ) = u θ v 1 � θ with 0 < θ < 1 we write E θ F 1 � θ instead of ρ ( E , F ) and these are Calderón spaces. For 1 < p < ∞ , a p-convexi…cation E ( p ) of E is a special case of Calderón space E ( p ) = f x 2 L 0 : j x j p 2 E g E 1 / p ( L ∞ ) 1 � 1 / p = kj x j p k 1 / p and k x k E ( p ) = . E Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 8 / 22
Pointwise products. Useful characterization. Theorem (KLM 2014). Let E and F be a couple of quasi-Banach function spaces. Then E � F � ( E 1 / 2 F 1 / 2 ) ( 1 / 2 ) . (6) Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 9 / 22
Pointwise products. Useful characterization. Theorem (KLM 2014). Let E and F be a couple of quasi-Banach function spaces. Then E � F � ( E 1 / 2 F 1 / 2 ) ( 1 / 2 ) . (6) Corollary. Let E and F be a couple of quasi-Banach function spaces. Then E � F is a quasi-Banach function space and the triangle inequality is satis…ed with constant 2, i.e., k x + y k E � F � 2 ( k x k E � F + k y k E � F ) . Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 9 / 22
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