Symmetrization of some quasi-Banach function spaces Pawe Kolwicz - - PowerPoint PPT Presentation

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

1

Introduction.

2

The commutativity property of symmetrization with some known constructions.

3

Some factorization results.

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 L0 = L0(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 L0 = L0(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 kE ) is said to be a quasi-Banach ideal (function) space on I if E is a linear subspace of L0(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 L0 = L0(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 kE ) is said to be a quasi-Banach ideal (function) space on I if E is a linear subspace of L0(I) and

1

if x 2 E, y 2 L0 and jyj jxj µ-a.e., then y 2 E and kykE kxkE ;

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 L0 = L0(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 kE ) is said to be a quasi-Banach ideal (function) space on I if E is a linear subspace of L0(I) and

1

if x 2 E, y 2 L0 and jyj jxj µ-a.e., then y 2 E and kykE kxkE ;

2

there exists a function x in E that is positive on the whole I.

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)

• r I = (0, ∞) with the Lebesgue measure m, we mean a

quasi-Banach function space E = (E, k kE ) with the additional property that for any two equimeasurable functions x y, x, y 2 L0(I) (that is, dx = dy, where dx(λ) = m(ft 2 I : jx(t)j > λg), λ 0) and x 2 E we have y 2 E and kxkE = kykE . In particular, kxkE = kxkE , where x(t) = inffλ > 0: dx(λ) tg, 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 kE ) and (F, k kF ) be quasi-Banach function spaces. The space of pointwise multipliers M(E, F) is de…ned by M(E, F) = fx 2 L0(I) : xy 2 F for all y 2 Eg (1) and the functional on it kxkM(E ,F ) = supfkxykF , y 2 E, kykE 1g (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 kE ) and (F, k kF ) be quasi-Banach function spaces. The space of pointwise multipliers M(E, F) is de…ned by M(E, F) = fx 2 L0(I) : xy 2 F for all y 2 Eg (1) and the functional on it kxkM(E ,F ) = supfkxykF , y 2 E, kykE 1g (2) de…nes a quasi-norm. If F = L1 we have M(E, L1) = 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 kE ) and (F, k kF ) be quasi-Banach function spaces. The space of pointwise multipliers M(E, F) is de…ned by M(E, F) = fx 2 L0(I) : xy 2 F for all y 2 Eg (1) and the functional on it kxkM(E ,F ) = supfkxykF , y 2 E, kykE 1g (2) de…nes a quasi-norm. If F = L1 we have M(E, L1) = 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 f0g.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 5 / 22

The space of pointwise multipliers.

Let (E, k kE ) and (F, k kF ) be quasi-Banach function spaces. The space of pointwise multipliers M(E, F) is de…ned by M(E, F) = fx 2 L0(I) : xy 2 F for all y 2 Eg (1) and the functional on it kxkM(E ,F ) = supfkxykF , y 2 E, kykE 1g (2) de…nes a quasi-norm. If F = L1 we have M(E, L1) = 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 f0g. 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 = fx y : x 2 E and y 2 Fg . (3) with a functional k kE F de…ned by the formula kzkE F = inf fkxkE kykF : z = xy, x 2 E, y 2 Fg . (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) = fx 2 L0(I) : jxj λ ρ(jx0j, jx1j) a.e. on I (5) for some x0 2 E, x1 2 F with kx0kE 1, kx1kF 1 and for some λ > 0g.

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) = fx 2 L0(I) : jxj λ ρ(jx0j, jx1j) a.e. on I (5) for some x0 2 E, x1 2 F with kx0kE 1, kx1kF 1 and for some λ > 0g. The quasi-norm kxkρ = kxkρ(E ,F ) = inf fλ > 0g 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) = uθ v1θ with 0 < θ < 1 we write E θF 1θ 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) = uθ v1θ with 0 < θ < 1 we write E θF 1θ 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 1/p(L∞)11/p = E (p) = fx 2 L0 : jxjp 2 Eg and kxkE (p) = kjxjpk1/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 1/2F 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 1/2F 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., kx + ykE F 2 (kxkE F + kykE F ) .

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 9 / 22

The symmetrizations.

For a quasi-Banach function space E on I = (0, 1) or I = (0, ∞) de…ne a space E () (symmetrization of E) as E () = fx 2 L0(I) : x 2 Eg, with the functional kxkE () = kxkE .

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 10 / 22

The symmetrizations.

For a quasi-Banach function space E on I = (0, 1) or I = (0, ∞) de…ne a space E () (symmetrization of E) as E () = fx 2 L0(I) : x 2 Eg, with the functional kxkE () = kxkE . Examples.

• 1. The Marcinkiewicz spaces M()

w

= (L∞ (w))() .

• 2. The Lorentz spaces Λp,w p = (Lp (w))() .
• 3. The Lorentz spaces Lp,q = (Lq (w))() with w (t) = t1/p1/q.
• 4. The generalized Orlicz-Lorentz spaces Λϕ = (Lϕ)() .

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 10 / 22

The symmetrizations.

For a quasi-Banach function space E on I = (0, 1) or I = (0, ∞) de…ne a space E () (symmetrization of E) as E () = fx 2 L0(I) : x 2 Eg, with the functional kxkE () = kxkE . Examples.

• 1. The Marcinkiewicz spaces M()

w

= (L∞ (w))() .

• 2. The Lorentz spaces Λp,w p = (Lp (w))() .
• 3. The Lorentz spaces Lp,q = (Lq (w))() with w (t) = t1/p1/q.
• 4. The generalized Orlicz-Lorentz spaces Λϕ = (Lϕ)() .

The dilation operator Ds, s > 0, Dsx(t) = x(t/s)χI (t/s), t 2 I, is bounded in any symmetric space E on I and kDskE !E max(1, s).

• A. Kami´

nska and Y. Raynaud showed that k kE () is a quasi-norm if and only if there is a constant C > 0 such that kD2xkE C kxkE for all x 2 E, (7)

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 10 / 22

The Hardy operators.

Consider the Hardy operator H and its formal Köthe dual H de…ned for x 2 L0(I) by Hx(t) = 1 t

Z t

0 x(s) ds, Hx(t) =

Z l

t

x(s) s ds (8) with l = m(I), t 2 I.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 11 / 22

The Hardy operators.

Consider the Hardy operator H and its formal Köthe dual H de…ned for x 2 L0(I) by Hx(t) = 1 t

Z t

0 x(s) ds, Hx(t) =

Z l

t

x(s) s ds (8) with l = m(I), t 2 I. Note that if 0 < p < 1, then neither H nor H are bounded on Lp(w) spaces for any weight w, therefore we need to consider their “r-convexi…cations" for 0 < r < ∞, which are de…ned by Hrx = [H(jxjr)]1/r and H

r x = [H(jxjr)]1/r,

(9) provided the corresponding integrals are …nite.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 11 / 22

The symmetrizations

Denote by E # – the cone of nonnegative and nonincreasing elements x = x 2 E.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 12 / 22

The symmetrizations

Denote by E # – the cone of nonnegative and nonincreasing elements x = x 2 E.

Theorem

(A. Kami´ nska and Y. Raynaud 2009, KLM 2018) Let E be a quasi-normed ideal space on I. The following statements are equivalent: (i) E () is a linear space. (ii) For each x 2 E # we have D2x 2 E #. (iii) There is a constant 1 A < ∞ such that kD2xkE A kxkE for all x 2 E #. (iv) (E (), k kE ()) is a quasi-normed space.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 12 / 22

The symmetrizations

Denote by E # – the cone of nonnegative and nonincreasing elements x = x 2 E.

Theorem

(A. Kami´ nska and Y. Raynaud 2009, KLM 2018) Let E be a quasi-normed ideal space on I. The following statements are equivalent: (i) E () is a linear space. (ii) For each x 2 E # we have D2x 2 E #. (iii) There is a constant 1 A < ∞ such that kD2xkE A kxkE for all x 2 E #. (iv) (E (), k kE ()) is a quasi-normed space.

• Remark. Let E be a quasi-Banach function space on I and

E () 6= f0g . If kHrxkE C kxkE for all x 2 E #, then kD2xkE 21/rC kxkE for all x 2 E #.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 12 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018). Let E and F be quasi-Banach ideal spaces such that E () 6= f0g, F () 6= f0g.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 13 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018). Let E and F be quasi-Banach ideal spaces such that E () 6= f0g, F () 6= f0g. (i) If the operator D2 is bounded both on E # and F # with the constants AE , AF , then Calderón–Lozanovski¼ ¬ construction ρ(E (), F ()) 6= f0g, ρ(E (), F ()) ρ(E, F)() and kxkρ(E ,F )() C1 kxkρ(E (),F ()) for all x 2 ρ(E (), F ()) (10) with C1 max(AE , AF ).

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 13 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018). Let E and F be quasi-Banach ideal spaces such that E () 6= f0g, F () 6= f0g. (i) If the operator D2 is bounded both on E # and F # with the constants AE , AF , then Calderón–Lozanovski¼ ¬ construction ρ(E (), F ()) 6= f0g, ρ(E (), F ()) ρ(E, F)() and kxkρ(E ,F )() C1 kxkρ(E (),F ()) for all x 2 ρ(E (), F ()) (10) with C1 max(AE , AF ). (ii) If, additionally, the operator H

r is bounded on the spaces E, F for

some r > 0, then ρ(E, F)() ρ(E (), F ()) and kxkρ(E (),F ()) C2 kxkρ(E ,F )() for all x 2 ρ(E, F)() (11) with C2 21/r max(1, 21/r1) max (AE kH

r kE !E , AF kH r kF !F ) .

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 13 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018).

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 14 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018). (iii) In particular, the inequalities (10) and (11) imply that the functional kkρ(E ,F )() (12) is a quasi-norm on the space ρ(E, F)() and ρ(E, F)() = ρ(E (), F ()). (13)

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 14 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018). (iii) In particular, the inequalities (10) and (11) imply that the functional kkρ(E ,F )() (12) is a quasi-norm on the space ρ(E, F)() and ρ(E, F)() = ρ(E (), F ()). (13) kkρ(E ,F )() is a quasi-norm, D2 is bounded on ρ(E, F)#.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 14 / 22

The symmetrization commutates with the Calderón-Lozanovski¼ ¬ construction.

Theorem (KLM 2018). (iii) In particular, the inequalities (10) and (11) imply that the functional kkρ(E ,F )() (12) is a quasi-norm on the space ρ(E, F)() and ρ(E, F)() = ρ(E (), F ()). (13) kkρ(E ,F )() is a quasi-norm, D2 is bounded on ρ(E, F)#.

• Remark. The assumption of theorem (ii) is essential.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 14 / 22

The symmetrization commutates with the pointwise product.

Corollary (KLM 2018). Let E and F be quasi-Banach ideal spaces such that E () 6= f0g and F () 6= f0g.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 15 / 22

The symmetrization commutates with the pointwise product.

Corollary (KLM 2018). Let E and F be quasi-Banach ideal spaces such that E () 6= f0g and F () 6= f0g. (i) If the operator D2 is bounded both on E # and F #, then E () F () 6= f0g, E () F () (E F)() and kxk(E F )() (C1)2 kxkE ()F () for all x 2 E () F (), (14) where C1 is the constant from previous theorem with the function ρ(s, t) = s1/2t1/2.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 15 / 22

The symmetrization commutates with the pointwise product.

Corollary (KLM 2018). Let E and F be quasi-Banach ideal spaces such that E () 6= f0g and F () 6= f0g. (i) If the operator D2 is bounded both on E # and F #, then E () F () 6= f0g, E () F () (E F)() and kxk(E F )() (C1)2 kxkE ()F () for all x 2 E () F (), (14) where C1 is the constant from previous theorem with the function ρ(s, t) = s1/2t1/2. (ii) If, additionally, the operator H

r is bounded on the spaces E, F for

some r > 0, then (E F)() E () F () and kxkE ()F () (C2)2 kxk(E F )() for all x 2 (E F)(), (15) where C2 is the constant from the previous theorem with the function ρ(s, t) = s1/2t1/2.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 15 / 22

The symmetrization commutates with the pointwise product.

Corollary (KLM 2018). (iii) In particular, the inequalities (14) and (15) imply that the functional kk(E F )() is a quasi-norm on the space (E F)() and (E F)() = E () F ().

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 16 / 22

The symmetrization commutates with the Köthe dual.

Theorem (A. Kami´ nska and M. Masty÷o 2007, KLM 2018).

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 17 / 22

The symmetrization commutates with the Köthe dual.

Theorem (A. Kami´ nska and M. Masty÷o 2007, KLM 2018). Suppose E is a quasi-Banach ideal space such that E () 6= f0g. If the

• perator D2 is bounded on E # and (E 0)() 6= f0g, then

(E 0)() (E ())0 and kxk(E ())0 kxk(E 0)() for all x 2 (E 0)(). (16)

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 17 / 22

The symmetrization commutates with the Köthe dual.

Theorem (A. Kami´ nska and M. Masty÷o 2007, KLM 2018). Suppose E is a quasi-Banach ideal space such that E () 6= f0g. If the

• perator D2 is bounded on E # and (E 0)() 6= f0g, then

(E 0)() (E ())0 and kxk(E ())0 kxk(E 0)() for all x 2 (E 0)(). (16) If, additionally, the operator H is bounded on the space E, then (E ())0 (E 0)() and kxk(E 0)() kHkE !E kxk(E ())0 for all x 2 (E ())0. (17)

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 17 / 22

The symmetrization commutates with the Köthe dual.

Theorem (A. Kami´ nska and M. Masty÷o 2007, KLM 2018). Suppose E is a quasi-Banach ideal space such that E () 6= f0g. If the

• perator D2 is bounded on E # and (E 0)() 6= f0g, then

(E 0)() (E ())0 and kxk(E ())0 kxk(E 0)() for all x 2 (E 0)(). (16) If, additionally, the operator H is bounded on the space E, then (E ())0 (E 0)() and kxk(E 0)() kHkE !E kxk(E ())0 for all x 2 (E ())0. (17) In particular, the inequalities (16) and (17) imply that the functional kk(E 0)() is a quasi-norm on the space (E 0)() and (E 0)() = (E ())0.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 17 / 22

The symmetrization commutates with the Köthe dual.

Theorem (A. Kami´ nska and M. Masty÷o 2007, KLM 2018). Suppose E is a quasi-Banach ideal space such that E () 6= f0g. If the

• perator D2 is bounded on E # and (E 0)() 6= f0g, then

(E 0)() (E ())0 and kxk(E ())0 kxk(E 0)() for all x 2 (E 0)(). (16) If, additionally, the operator H is bounded on the space E, then (E ())0 (E 0)() and kxk(E 0)() kHkE !E kxk(E ())0 for all x 2 (E ())0. (17) In particular, the inequalities (16) and (17) imply that the functional kk(E 0)() is a quasi-norm on the space (E 0)() and (E 0)() = (E ())0.

• Remark. The assumption about the operator H is essential.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 17 / 22

The symmetrization commutates with the pointwise multipliers.

Theorem (KLM 2018) Let E, F be Banach ideal spaces on I such that F has the Fatou property, E () 6= f0g, F () 6= f0g are normable spaces, the operator D2 is bounded on F #, (F 0)() 6= f0g and

• (E F 0)0()

6= f0g. Assume that the following conditions hold: (i) The operator H is bounded on the spaces F and E F 0. (ii) For some r > 0, the operator H

r is bounded on E, F 0,

kHrxkE CE kxkE for all x 2 E and kHrxkF 0 CF 0 kxkF 0 for all x 2 F 0. Then M(E (), F ()) = M(E, F)().

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 18 / 22

The symmetrization commutates with the pointwise multipliers.

Theorem (KLM 2018) Let E, F be Banach ideal spaces on I such that F has the Fatou property, E () 6= f0g, F () 6= f0g are normable spaces, the operator D2 is bounded on F #, (F 0)() 6= f0g and

• (E F 0)0()

6= f0g. Assume that the following conditions hold: (i) The operator H is bounded on the spaces F and E F 0. (ii) For some r > 0, the operator H

r is bounded on E, F 0,

kHrxkE CE kxkE for all x 2 E and kHrxkF 0 CF 0 kxkF 0 for all x 2 F 0. Then M(E (), F ()) = M(E, F)(). Apllications for weighted spaces E = Lp (ta) , F = Lq tb with a, b 2 R and 1 p, q ∞.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 18 / 22

Multipliers between Lorentz spaces.

For 0 < p, q ∞ consider the classical Lorentz function spaces Lp,q = Lp,q(I) = (Lq (w))() with w (t) = t1/p1/q on I = (0, 1) or I = (0, ∞) de…ned by the quasi-norms kxkp,q = 8 > > > < > > > : m(I ) R [t1/px(t)]q dt

t

1/q , for 0 < p ∞, 0 < q < ∞, sup

0<t<m(I )

t1/px(t), for 0 < p ∞, q = ∞.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 19 / 22

Multipliers between Lorentz spaces.

For 0 < p, q ∞ consider the classical Lorentz function spaces Lp,q = Lp,q(I) = (Lq (w))() with w (t) = t1/p1/q on I = (0, 1) or I = (0, ∞) de…ned by the quasi-norms kxkp,q = 8 > > > < > > > : m(I ) R [t1/px(t)]q dt

t

1/q , for 0 < p ∞, 0 < q < ∞, sup

0<t<m(I )

t1/px(t), for 0 < p ∞, q = ∞. Lp,p Lp for 0 < p ∞ and L∞,q = f0g for 0 < q < ∞.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 19 / 22

Multipliers between Lorentz spaces.

Theorem (KLM 2018). Let 0 < p1, p2 < ∞, 0 < q1, q2 ∞ and I = (0, 1) or I = (0, ∞). (i) If either p1 < p2 or p1 = p2 and q1 > q2, then M(Lp1,q1, Lp2,q2) = f0g. (ii) If either p1 > p2 or p1 = p2 and q1 q2, then M(Lp1,q1, Lp2,q2) = Lp3,q3, where 1 p3 = 1 p2 1 p1 and 1 q3 =

• 1

q2 1 q1

if q1 > q2, if q1 q2 (18)

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 20 / 22

Factorization.

The Lozanovski¼ ¬ theorem: L1 = E E

0. Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 21 / 22

Factorization.

The Lozanovski¼ ¬ theorem: L1 = E E

0.

The generalization: F = E M(E, F).

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 21 / 22

Factorization.

The Lozanovski¼ ¬ theorem: L1 = E E

0.

The generalization: F = E M(E, F). F = E M(E, F) = ) F (w1) = E (w0) M(E (w0) , F (w1)).

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 21 / 22

Factorization.

The Lozanovski¼ ¬ theorem: L1 = E E

0.

The generalization: F = E M(E, F). F = E M(E, F) = ) F (w1) = E (w0) M(E (w0) , F (w1)). Corollary (KLM 2018). Suppose the following assumptions are satis…ed:

• from theorem M (X, Y )() = M
• X (), Y ()

, for the spaces E, F,

• from the corollary: (X Y )() = X () Y (), for the spaces

E, M (E, F) . If F factorizes through E, i.e.,F = E M(E, F), then the symmetrization F () factorizes through the symmetrization E (), that is, F () = E () M(E (), F ()). (19)

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 21 / 22

Factorization.

The Lozanovski¼ ¬ theorem: L1 = E E

0.

The generalization: F = E M(E, F). F = E M(E, F) = ) F (w1) = E (w0) M(E (w0) , F (w1)). Corollary (KLM 2018). Suppose the following assumptions are satis…ed:

• from theorem M (X, Y )() = M
• X (), Y ()

, for the spaces E, F,

• from the corollary: (X Y )() = X () Y (), for the spaces

E, M (E, F) . If F factorizes through E, i.e.,F = E M(E, F), then the symmetrization F () factorizes through the symmetrization E (), that is, F () = E () M(E (), F ()). (19) Corollary (KLM 2018). The factorization of Lp spaces !The factorization of Lp (w) spaces !The factorization of Lorentz and Marcinkiewicz spaces.

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 21 / 22

Thank You very much for Your attention

Pawe÷ Kolwicz Pozna´ n () Symmetrization B ¾ edlewo 1.07-7.07.2018 22 / 22