Generalized Lorentz spaces and Köthe duality Anna Kaminska´ and Yves Raynaud University of Memphis, Paris University VI Conference on Non Linear Functional Analysis, Universitat Politecnica de Valencia, Spain, 17-20 October 2017 Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality µ measure on the the measure space (Ω; A; µ) L0(Ω) = L0(Ω; A; µ) , µ-measurable real valued functions on Ω 0 0 L+(Ω) non-negative functions from L (Ω). L1 = L1(Ω), kf k1, L1 = L1(Ω), kf k1 L1 + L1(Ω), kf kL1+L1(Ω) = inffkgk1 + khk1 : f = g + hg < 1 L1 \ L1(Ω), kf kL1\L1(Ω) = maxfkf k1; kf k1g < 1. A Banach function space E over (Ω; A), is a complete vector space 0 E ⊂ L (Ω) equipped with a norm k · kE such that if 0 ≤ f ≤ g, where 0 g 2 E and f 2 L (Ω), then f 2 E and kf kE ≤ kgkE . The space E satisfies the Fatou property whenever for any 0 f 2 L (Ω), fn 2 E such that fn " f a.e. and sup kfnkE < 1 it follows that f 2 E and kfnkE " kf kE . µ distribution of f with respect to µ, df (s) = µfjf j > sg, s ≥ 0, and its ∗,µ µ decreasing rearrangement f (t) = inffs > 0 : dg (s) ≤ tg, 0 < t < µ(Ω). f ; g are equimeasurable (with respect to the measures µ and ν) if µ ν ∗,µ ∗,ν df (s) = dg (s), s ≥ 0; equivalently, f = g . Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality A Banach function space E is symmetric space (with respect to µ) whenever kf kE = kgkE for every µ-equimeasurable functions f ; g 2 E. fundamental function of a symmetric space E is φE (t) = kχAkE , µ(A) = t. support of the symmetric space E is the entire set Ω whenever χA 2 E for any A 2 A with µ(A) < 1. R x ∗ R x ∗ Hardy-Littlewood order f ≺ g for f ; g 2 L1 + L1(Ω), 0 f ≤ 0 g for every x 2 (0; µ(Ω)); (f + g) ≺ f ∗ + g∗. E fully symmetric if E is symmetric and if for any f 2 L0(Ω) and g 2 E with f ≺ g we have that f 2 E and kf kE ≤ kgkE . Köthe dual space E 0 of E is the set of f 2 L0(Ω) such that Z kf kE 0 = sup jfgj dµ : kgkE ≤ 1 < 1: Ω 0 (E ; k · kE 0 ) is Banach function space with the Fatou property. If E is symmetric then E 0 is fully symmetric, and (Z µ(Ω) ) ∗ ∗ kf kE 0 = sup f g dm : kgkE ≤ 1 : 0 Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality I = (0; a), 0 < a ≤ 1, L0 = L0(I) m ∗ ∗;m If Ω = I and µ = m then df = df , f = f . weight function w : I ! (0; 1) decreasing positive R measure on I, ! = wdm, !(A) = A w for A ⊂ I w ∗;w df and f , distribution and decreasing rearrangement of f w. r. to the measure ! = wdm Z t Z 1 W (t) = w dm; t 2 I; W (1) = w dm if I = (0; 1) 0 0 w is regular if W (t) ≤ Ctw(t) for some C ≥ 1 and all t 2 I Let b = !(I) = W (a) 2 (0; +1] and J = (0; b) If a = 1 then I = (0; 1) = J J is equipped with the Lebesgue measure m symbol E will always stand for a fully symmetric Banach function space E ⊂ L0(J), support of E is equal to J Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality 0 0 weighted E space, Ew ⊂ L = L (I), 0 ∗;w ∗;w Ew = ff 2 L : f 2 Eg; kf kEw = kf kE ; f 2 Ew : 0 ∗;w 0 If f 2 L (I) then f 2 L (J). 1 ≤ p < 1, then Ew = (Lp)w is a weighted Lp space on (I;!). f 2 Ew = (Lp)w , Z Z ∗;w p 1=p p ∗;w 1=p kf k(Lp)w = ( (f ) dm) = ( (jf j ) dm) J J Z Z = ( jf jpd!)1=p = ( jf jpwdm)1=p: I I Analogously for the Orlicz space E = L'(J), Ew = (L')w is a weighted Orlicz space associated with the Orlicz modular Z Z '(f ∗;w ) dm = '(jf j) wdm: J I 0 Ew is a Banach function space in L (I), not symmetric w. r. to m, isometrically order isomorphic to E on (J; m). Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality Lemma Assume that W < 1 on I. Then (i) Every f 2 L0 is equimeasurable with respect to !, to f ◦ W −1 with respect to m. Consequently, (f ◦ W −1)∗ = f ∗;w : 0 −1 (ii) f 2 L belongs to Ew if and only if f ◦ W belongs to E, and then −1 kf kEw = kf ◦ W kE : 0 −1 Consequently, Ew is an order ideal in L and the map f 7! f ◦ W induces an order isometry from Ew onto E. Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality LORENTZ SPACES generalized Lorentz space ΛE;w , the symmetrization of Ew , 0 ∗ ∗ ΛE;w = ff 2 L : f 2 Ew g; kf kΛE;w = kf kEw : If W (t) = 1 for t > 0, then ΛE;w = f0g or ΛE;w = L1(I). Assume that W < 1 on I. If E = L'(J) is Orlicz space then ΛE;w = Λ';w is the Orlicz-Lorentz ∗ space; kf kΛ';w = kf kL';w . p If '(t) = t , 1 ≤ p < 1, E = Lp(J) then ΛE;w = Λp;w is the Lorentz R ∗p 1=p space; kf kΛp;w = ( I f w) . If E = L1(J) then Ew = L1(I) = ΛE;w . Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality Theorem Let W < 1 on I. (i) The support of ΛE;w is I. (ii) For all f 2 ΛE;w , ∗ −1 kf kΛE;w = kf ◦ W kE : (iii) The functional k · kΛE;w is a norm, and the Lorentz space ΛE;w is a fully symmetric Banach space. If E has the Fatou property then ΛE;w has also this property. Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality Proof. −1 −1 ∗ −1 (iii) D2W ≤ (1=2)W , D2f (t) = f (t=2). f 2 ΛE;w ) f ◦ W 2 E, and D2f 2 ΛE;w since D2 is a bounded operator in E, and ∗ −1 ∗ −1 ∗ −1 (D2f ) ◦ W = (D2f ) ◦ W = f ◦ ((1=2)W ) ∗ −1 ∗ −1 ≤ f ◦ (D2W ) = D2(f ◦ W ) 2 E: ∗ −1 ∗ −1 f ; g 2 ΛE;w ) (f + g) ◦ W ≤ (D2(f + g)) ◦ W ≤ ∗ −1 ∗ −1 D2(f ◦ W ) + D2(g ◦ W ) 2 E ) f + g 2 ΛE;w . 0 Let f 2 L , ; g 2 ΛE;w with f ≺ g, and x 2 J.Then Z x Z W −1(x) Z W −1(x) Z x f ∗ ◦ W −1 = f ∗w ≤ g∗w = g∗ ◦ W −1: 0 0 0 0 Hence f ∗ ◦ W −1 ≺ g∗ ◦ W −1. By full symmetry of E, f ∗ ◦ W −1 2 E, ∗ −1 ∗ −1 kf kΛE;w = kf ◦ W kE ≤ kg ◦ W kE = kgkΛE;w ; so k · kΛE;w is fully symmetric. Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality Proof. ∗ ∗ ∗ We also have kf + gkΛE;w ≤ kf kΛE;w + kgkΛE;w by (f + g) ≺ f + g . The normed function space ΛE;w is complete since it is a symmetrization of the complete space Ew . 0 Suppose now that E has the Fatou property. Take fn; f 2 L (J), fn " f ∗ −1 ∗ −1 a.e., and sup kfnkΛE;w < 1. Then fn ◦ W " f ◦ W a.e., and by (ii) ∗ −1 sup kfn ◦ W kE = sup kfnkΛE;w < 1. Now by the Fatou property of E, f ∗ ◦ W −1 2 E and ∗ −1 ∗ −1 kfnkΛE;w = kfn ◦ W kE " kf ◦ W kE = kf kΛE;w . Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality Classes ME;w 0 0 Define a class ME;w contained in L = L (I) which will be used for finding the Köthe dual of the Lorentz space ΛE;w . Let the class ME;w and the gauge on ME;w be defined by ∗ ∗ ∗ ∗;w 0 f f f ME;w = f 2 L : 2 Ew and kf kM = = : w E;w w w Ew E Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality Theorem 0 (i) The class ME;w is a solid symmetric subset of L , that is ∗ 0 kf kME;w = kf kME;w and if f 2 L , g 2 ME;w and jf j ≤ jgj a.e. then f 2 ME;w and kf kME;w ≤ kgkME;w . (ii) For all x 2 I, χ(0;x) 2 ME;w . Consequently the support of ME;w is equal to the entire interval I. (iii) The fundamental function φME;w (x) = kχ(0;x)kME;w , x 2 I, verifies 1 φ (x) ≤ 2φ (1 ^ b) x + : ME;w E w(x) (iv) If W < 1 on I, then ∗ ∗ f −1 f −1 f 2 ME;w () ◦ W 2 E and kf kME;w = ◦ W : w w E (v) If E has the Fatou property then the class ME;w has this 0 property, that is for every f 2 L , 0 ≤ fn 2 ME;w with fn " f a.e.