Convexity and concavity constants in Lorentz and Marcinkiewicz spaces

Transcription

1 J Mah Anal Appl ) wwwelseviercom/locae/jmaa Convexiy and concaviy consans in Lorenz and Marciniewicz spaces Anna Kamińsa, Anca M Parrish Deparmen of Mahemaical Sciences, The Universiy of Memphis, Memphis, TN 3852, USA Received 23 July 27 Available online 9 January 28 Submied by K Jarosz Absrac We provide here he formulas for he q-convexiy and q-concaviy consans for funcion and sequence Lorenz spaces associaed o eiher decreasing or increasing weighs I yields also he formula for he q-convexiy consans in funcion and sequence Marciniewicz spaces In his paper we exen and enhance he resuls from [GJO Jameson, The q-concaviy consans of Lorenz sequence spaces and relaed inequaliies, Mah Z ) 29 42] and [A Kamińsa, AM Parrish, The q-concaviy and q-convexiy consans in Lorenz spaces, in: Banach Spaces and Their Applicaions in Analysis, Conference in Honor of Nigel Kalon, May 26, Waler de Gruyer, Berlin, 27, pp ] Published by Elsevier Inc Keywords: q-convexiy concaviy) consans; Lorenz spaces; Marciniewicz spaces Given <q< and a quasi-normed laice E, E ), we define he following consans: he q-convexiy consan M q) E) is he leas consan M such ha for all f,,f n E, /q f i q M f i E) E q, he q-concaviy consan M q) E) is he leas consan K such ha for all f,,f n E, /q f i E) q K f i q E We hen say ha E, E ) is q-convex, respecively q-concave,ifm q) E), M q) E) < These noions are closely relaed o he noions of ype and coype, and hey play an essenial role in sudies of he local geomery of Banach * Corresponding auhor addresses: aminsa@memphisedu A Kamińsa), abuican@memphisedu AM Parrish) X/$ see fron maer Published by Elsevier Inc doi:6/jjmaa2834

2 338 A Kamińsa, AM Parrish / J Mah Anal Appl ) spaces/laices [4,8,7] I is of special ineres o deermine he exac values of he convexiy and concaviy consans for paricular classes of laices equipped wih heir original quasi-)norms [5 7] For insance i is well nown ha every q-convex Banach laice can be given an equivalen laice norm ha is q-convex wih consan one and he same holds for q-concaviy) [7] Such a renorming is hen a saring poin for invesigaion of several geomeric properies In [6], GJO Jameson found he q-concaviy and q-convexiy consans in Lorenz sequence spaces dw,p) for p< and w decreasing In ha paper he saed he problem of finding a direc approach in proving his resul [6, Theorem 3], where he formula for q-concaviy consan in dw,p) is esablished In his proof he applied general relaionships beween convexiy and concaviy consans in Banach spaces and heir duals, as well as he well-nown represenaion of he dual space o he Lorenz space In his paper we presen a direc and simpler way in proving ha heorem Theorem 6) The mehod we use here has also an addiional advanage, namely, i can be applied afer some modificaions for calculaion of he convexiy and concaviy consans in dw,p) for increasing weighs as well Noice ha if w is increasing, hen Lorenz spaces are no Banach spaces, and for a large class of weighs hey are no even normable [] Thus he dualiy mehod is no applicable in his class of spaces Marciniewicz and Lorenz spaces play an imporan role in he heory of Banach spaces, in paricular hey are ey objecs in he inerpolaion heory of linear operaors The origins of he Marciniewicz spaces go bac o he heorem on wea ype operaors [7, h 2b5], which was originally due o K Marciniewicz in he 93 s The Lorenz spaces inroduced by GG Lorenz in 95 have appeared in a naural way as inerpolaion spaces beween suiable Lebesgue spaces by classical resul of Lions and Peere [7, h 2g8] This heory has been exensively developed and along wih hese invesigaions he heory of Lorenz and Marciniewicz spaces including he sudies of heir local geomeric srucure has evolved independenly eg [2,3,9,,2,6,7,9]) In our paper we provide he formulas for q-concaviy and q-convexiy consans in Lorenz funcion and sequence spaces, Λ p,w and dw,p), for all <p<, and any decreasing or increasing weigh w Consequenly, we obain he appropriae formulas for hese consans in Marciniewicz funcion or sequence spaces ha are duals o Lorenz spaces Λ,w or dw,) Here we complemen and improve he Jameson s resuls in [6] as well as he resuls of our earlier paper [3] The firs secion is devoed o funcion spaces, he second one o sequence spaces, and he hird one o specific sequence spaces associaed o he power weigh sequences u n = n α wih α> As a consequence of he esablished formulas, we ge among ohers ha hese consans are equal o one if and only if he spaces are isomeric o he Lebesgue spaces L p or l p Le R, R + and N sand for he ses of real numbers, posiive real numbers and naural numbers, respecively Le L be he se of all real-valued -measurable funcions defined on R + or N, where is he Lebesgue measure on R or he couning discree) measure on N Thedisribuion funcion d f of a funcion f L is given by d f λ) = {>: f) >λ}, for all λ We say ha wo funcions f,g L are equimeasurable and we denoe i by f g if d f λ) = d g λ), for all λ For f L we define is decreasing rearrangemen as f ) = inf{s >: d f s) }, > In he case of discree measure, he elemens of L coincide wih real-valued sequences x = xn)), and hen x = x n)) is a decreasing rearrangemen of x defined equivalenly as x n) = inf{s >: d x s) n },forn N By w : R + R + or w : N R + we denoe he weigh funcion w), >, or he weigh sequence w n ) Leing W):= w, >, or Wj)= W j = j w i, j N, we shall assume ha W)< for all >, and W ) = w = w i = WealsoassumehaW saisfies he Δ 2 -condiion, ha is W2s) KWs) for some K>and all s>ors N Given <p< and a weigh funcion w, helorenz space Λ p,w is a subse of L such ha f = f p,w := f p w) /p = f p )w) d) /p < Recall also ha he Marciniewicz space M W is he space of all funcions f L saisfying f MW = sup > f s) ds < W)

3 A Kamińsa, AM Parrish / J Mah Anal Appl ) In he case of discree measure, we denoe by dw,p) he Lorenz sequence space associaed o a weigh sequence w = w n ) The space dw,p) consiss of all real sequences x = xn)) such ha ) /p x = x p,w := x j) p w j < Similarly, he Marciniewicz sequence space m W is he space of all sequences x = xn)) saisfying x j) x mw = sup < W Under he assumpion of Δ 2 -condiion on W, p,w is a quasi-norm, and Λ p,w, p,w ) or dw, p), p,w ) is a quasi-banach space [,4] If w is decreasing, hen p,w is a norm [,5] and /p f p,w = sup f v) p or x p,w = sup xj) ) /p p v j ) v w v w In his case, Λ,w or dw,) is a separable Banach space and is dual is he Marciniewicz space M W [5, Theorem 52], or m W [, Theorem 44], respecively In he following, we will consider boh funcion and sequence spaces in wo cases, when w is decreasing or increasing To prove he resuls for increasing weigh, we need o recall he definiion of he increasing rearrangemen and some of is properies [3] Similarly o he disribuion funcion d f, define for f L and for all λ>he funcion γ f λ) = {s supp f : fs) <λ} We say ha wo funcions f and g are equivalen and denoe i by f γ g,ifγ f λ) = γ g λ) for all λ> Then he increasing rearrangemen of f is he funcion f defined as { sup{λ : γf λ) }, if [, supp f ); f ) =, if supp f Analogously, for a sequence x = xn)), define for s>, γ x s) = { i supp x: xi) <s }, and he increasing rearrangemen x of x by { sup{s : γx s) j }, if j, supp x ]; x j) =, if j> supp x Recall [3, Theorem 25] ha for any f,g L such ha g> ae and γ f λ) < for every λ>, we have ha fg f g By Theorem 27 in [3], if w is increasing and lim w) =, hen for any bounded funcion f wih supp f <, { /p } f p,w = inf f v) p : v γ w, v > ae 3) The similar fac can be proved analogously in sequence spaces, and hus for any increasing weigh w = w n ) wih lim n w n =, and for any x wih finie suppor, we have { ) /p } x p,w = inf xj) p vj : v γ w, v > 4) 2)

4 34 A Kamińsa, AM Parrish / J Mah Anal Appl ) Recall ha given <p<, hep-convexificaion of a quasi-banach laice E, E ) is he space E p) ={f : f p E} wih he quasi-norm f E p) = f p /p E I is well nown and easy o show ha M q) E p) ) = M q/p) E) and M q) E p)) = M q/p) E) 5) The spaces Λ p,w or dw,p) are p-convexificaions of Λ,w or dw,), respecively This allows us o assume ha p = in he process of compuing he convexiy and concaviy consans for hese spaces Funcion spaces The firs resul presened here was proved in [3, Theorem 32], where we have used Jameson s dualiy mehod [6, Theorem 3] accommodaed o funcion spaces Below we provide a direc proof Theorem Le q>pand w be a decreasing weigh funcion Then M q) Λ p,w ) = sup wr ) /r > w, where p q + r = Proof Assuming p =, we shall prove ha M q) Λ,w ) sup wr ) /r > w := B Leing f i ) n Λ,w, here exiss a sequence λ i ) n of non-negaive numbers such ha λ r i = and λ i f i = f i q By ), for ε> and for i =,,nhere exis h i, h i w such ha f i =sup v w f i v f i h i + ε nλ i Applying hen he Hölder inequaliy we obain ) f i q λ i f i h i + ε nλ i = Le now g = n λ r i hr i )/r Thus for all >, ) g r = λ r i hr i λ r i h r ) i = Hence g = g w r ) /r /q = By Hardy s lemma and by 7), we have { } { } f i q g B f i q w λ r i λ i f i h i + ε h r i = w r ) /r f i q λ r i hr i + ε ) r ) /r w r = wr ) /r w w B 6) w 7)

5 A Kamińsa, AM Parrish / J Mah Anal Appl ) Thus by 6) and by he Hardy Lilewood inequaliy, f i q f i q g + ε B and he proof is compleed { } f i q g + ε { } f i q w + ε = B f i q + ε, We obain he formula for he consan in Marciniewicz space M W by dualiy o Λ,w Corollary 2 For <p< and decreasing weigh funcion w, M p) M W ) = sup wp ) /p > w The space M W is no p-concave for any <p< 8) Proof I is well nown [7, Proposiion d4] ha if E is a Banach laice, hen for q, M q) E) = M q )E ) and M q) E) = M q ) E ), where /q + /q = Given ha for w decreasing, M W is a dual space o Λ,w, we obain 8) Using Theorem 73 in [8], if he norm of a Banach laice is no order coninuous, hen he Banach laice conains an order isomorphic copy of l, and since l is no p-concave for any <p<, neiher is he Banach laice So all we need is o show ha MW is no order coninuous For his, consider he funcions f n = wχ,/n) So f n, f n w, and since for all >, wχ,/n) = W) {, if </n; W n ) W), if /n, and /W) is increasing, we obain ha f n MW = for all n N Thus he norm is no order coninuous In he nex heorem we sae he formula for he q-convexiy consan in Lorenz space Λ p,w for he weigh w increasing I is an improvemen of he previous resul given in [3, Theorem 35, Corollary 36], where we have found only some esimaes of he consan Theorem 3 If <q<pand w is an increasing weigh saisfying lim w) =, hen M q) Λ p,w ) = sup > where p q + r = w wr ) /r, Proof In view of 5) and he obvious fac ha he p-convexificaion of Λ,w is Λ p,w, we assume ha p = Since he inequaliy w M q) Λ,w ) sup > wr ) /r was proved in [3, Theorem 35], we shall show only he reverse one Le f i ) n Λ,w We can assume ha each f i is a simple funcion wih bounded suppor, since such funcions are dense in Λ,w By he reverse Hölder inequaliy for <q<, here exiss a sequence a i ) n of posiive numbers such ha ai r = and a i f i = f i q

6 342 A Kamińsa, AM Parrish / J Mah Anal Appl ) Le ε> Since by 3), { } f i =inf f i v: v γ w, v > ae for all i =, 2,,n, here exis h i γ w, h i >, ae such ha f i f i h i ε na i By Hölder s inequaliy for <q<, f i q a i f i h i ε na i ) = a i f i h i ε ) /r f i q ai r hr i ε 9) Le g = n ai rhr i )/r > and le > Noice ha r< and ha f r ) = f ) r ae [3, Proposiion 234)] Hence by n ai r = and he subaddiiviy of he operaor f f, [ ) /r ) ] r ) g ) r = ai r hr i = ai r hr i a r i h r ) i = ai r ) h r i = a r i [ hi ) ] r = a r i w r = w r, since [h i ) ] r are equimeasurable o w r Byr<wege g ) r ) /r wr ) /r Thus if we denoe by B := sup > w we ge ha for all >, g = wr ) /r, g g ) r ) /r /q w r ) /r So by Hardy s lemma [, Proposiion 36] we ge { } { } f i q g B f i q w Now by 9) and 2), f i q f i q g ε ) r = wr ) /r )/r w w B w { } f i q g ε { } B f i q w ε = B f i q ε Therefore f i q B f i q and so M q) Λ,w ) B

7 A Kamińsa, AM Parrish / J Mah Anal Appl ) Corollary 4 If w is decreasing respecively increasing), hen for all q>p, M q) Λ p,w ) = respecively for all <q<p, M q) Λ p,w ) = ) if and only if Λ p,w is isomeric o L p Proof Assume ha p = and w is decreasing Then q>and by Theorem, for every >, w r w On he oher hand by Hölder s inequaliy w w r ) /r /r = w r ) /r Then for every >, wr = w, and by he equaliy condiion in Hölder s inequaliy, w) = C, for all >, and some C> Hence f p,w = C /p f L p For increasing w, he proof is analogous Recall ha for <p,q<, he classical Lorenz spaces L q,p are obained from Λ p,w by seing w) = p/q see [,7]) The following resul is an improvemen and complemen of Corollary 37 in [3] Corollary 5 ) If p q, hen a) M s) L q,p ) = for s p and he space is no s-convex for s>p; b) for s>q, p M s) L q,p ) = q[ p, q )r + ]/r where r + p s = Fors q, he space is no s-concave 2) If p>q, hen a) M s) L q,p ) = for s p and he space is no s-concave for s<p; b) for <s<q, q[ p q )r + ]/r p where r + p s = Fors q he space is no s-convex 2 Sequence spaces [ ) p /r M s) L q,p ) q r + ], We sar wih presening a direc proof of he formula for q-concaviy consan of dw,p) in case when w is decreasing, originally proved by GJO Jameson in [6, Theorem 3] and by his answering his quesion posed here Theorem 6 Le q>pand w be a decreasing weigh sequence Then ) M q) dw,p) = sup where p q + r = wj r )/r, w j Proof Assume p = In view of Theorem 3 in [6], we wish o prove only he inequaliy M q) dw,) ) sup wj r )/r w j

8 344 A Kamińsa, AM Parrish / J Mah Anal Appl ) We noice firs ha for all, w r ) /r j B := sup )/r /q wj r ) w j W Le x i ) n dw,) Then here exiss a non-negaive sequence a i) n such ha ai r = and a i x i = x i q Le ε> Proceeding in he same way as in Theorem, here exis h i w, h i and such ha for all i =,,n, x i xi j) hi j) + ε na i Then by Hölder s inequaliy [ x i q x i j) ) /r ] q ai r h ij) r + ε ) Leing g be a sequence defined by gj) = n a r i h ij) r ) /r,wehave gj) ) r wj r, and from ), we also have ha g j) B w j for all So by Hardy s lemma, { x i j) } { q g j) B x i j) } q w j Then from ) and he Hardy Lilewood inequaliy { x i q B x i j) } q w j + ε = B x i j) q + ε, which shows he heorem Corollary 7 For <p< and w = w n ) a decreasing weigh sequence, M p) m W ) = sup w p j )/p w j The space m W is no p-concave for any <p< 2) Proof Using he same reasoning as in Corollary 2 and ha m W is he dual space of dw,), we obain he p-convexiy consan for m W Similarly, we show ha he space m W is no p-concave for any <p< by using ha m W conains an order isomorphic copy of l if he norm is no order coninuous To show ha he norm is no order coninuous, consider

9 A Kamińsa, AM Parrish / J Mah Anal Appl ) he sequence x n ) n= defined by x n = w n χ En + wχ N\En, where E n ={,,n} Then clearly <x n w, x n, and elemenary calculaions show ha lim n x n mw > In he nex resul we presen he convexiy and concaviy consans for he weigh w increasing Theorem 8 Le w be an increasing weigh sequence wih lim n w n = If <q<p, hen dw,p) is no q-concave and M q) dw,p) ) = sup w j wj r, )/r where p q + r = If q p, hen dw,p) is no q-convex and M q) dw, p)) = Proof Since dw,p) conains an order isomorphic copy of l p see [6, Proposiion 4e3] for w decreasing, and [4, Theorem 3] for arbirary w), i is no q-concave for q<pand no q-convex for q>p Le now q = p and assume ha p = Since w is increasing, Wn)/n is also increasing Moreover, by he assumpion lim n w n =, we have lim n Wn)/n= Indeed, denoing by s he leas ineger bigger han or equal o s, for any n N, Wn) n n i= n/2 w i 2 ) w n/2 n Assume for a conrary ha dw,) is -convex Le <<l, n = l/, and E i ={i ) +,,i} for i =,,n Define x i = χ Ei Then x i =W) and n x i = χ {,,i} =Wn)Byl n and n l/ +, and by -convexiy of he space, for some C>, Wl) Wn)= x i C x i =CnW) Cl/ + )W ) 2Cl/)W) Hence Wl)/l 2CW )/ for every <<lbuwn)/n, a conradicion So dw,) is no -convex If q p we ge ha M q) dw, p)) = by applying formula 4) Le now <q<p In view of 5) we also suppose ha p = In order o show ha M q) dw,) ) sup w j wj r, 3) )/r we will follow he mehod of he proof of Theorem 3 in [6] Indeed, le and le x = w α,wα 2,,wα,,,), where we define α by αq = α + = r noice ha α<) Define x 2,,x by all differen cyclic permuaions of he firs coordinaes of x Then for all i =,,, ) x i = w α+ j and On he oher hand, x q i = W w αq j x i q)/q = /q w α+ j

10 346 A Kamińsa, AM Parrish / J Mah Anal Appl ) Hence, denoing M = M q) dw, )), we ge ) W wj r M /q wj r, and so 3) is proved To show he reverse inequaliy o 3), le w j W B := sup wj r = sup )/r /q wj r )/r Le x i ) n dw,) be elemens wih finie suppors Then analogously o he proof of Theorem 3, here exiss a sequence of posiive numbers a i ) n such ha ai r = and a i x i = x i q Le ε> Following he similar seps as in Theorem 3, in view of 4) here exiss h i γ w, h i >, such ha x i x i j) h i j) ε na i By Hölder s inequaliy for <q<, [ x i q x i j) ) /r ] q ai r h ij) r ε 4) Le g be a sequence defined by gj) = n a r i h ij) r ) /r Then for all, ) r gj) wj r, and herefore g j) B w j By Hardy s lemma, { x i j) } q { g j) B x i j) } q w j, and so from 4), we have q)/q x i B Since ε> is arbirary, we finish he proof xi j) q ε The proof of he nex resul is analogical o he one of Corollary 4 Corollary 9 If w is decreasing respecively increasing), hen for all q>p, M q) dw, p)) = respecively for all <q<p, M q) dw, p)) = ) if and only if dw,p) is isomeric o L p

11 A Kamińsa, AM Parrish / J Mah Anal Appl ) Power weigh sequences In his chaper we shall consider he class of weigh sequences given by power funcions, ha is u = u n ), where u n = n α and α> The case of u n = n α,<α, was sudied in [6] We are ineresed in finding similar resuls when he weighs are increasing For his, define also a weigh sequence v = v n ) given by v n = n n + α) α d I is clear ha v n = n +α n ) +α and V = v j = +α By sandard comparison wih he inegral of f)= α on [,] and [,+ ] we obain ha +α + α U + )+α + α In he following, for simpliciy, we will denoe by A = u j u r and B = j )/r By 5) we immediaely ge he following resul: v j vj r )/r Proposiion Le <q<pand r = p q Then, for <α< r, lim u j + αr)/r u r = j )/r + α 5) Theorem Le r<, v defined as above and <α< r Then sup v j + αr)/r vj r )/r + α Proof Noice firs ha since V = +α, B r = +α)r r vj r For r<, he funcion ϕu) = u r is convex, so by Jensen s inequaliy we have j r + α) d) j α + α) r αr d j Thus by he definiion of v j, v r j + α)r I follows ha for all, B r +α)r r and since r<, sup B j αr d = + α)r + αr +αr + αr + αr + α) r = +αr + α) r, + αr)/r + α 6)

12 348 A Kamińsa, AM Parrish / J Mah Anal Appl ) The nex lemma can be proved following he same reasoning as in [6, Lemma 7] Lemma 2 Wih u, v defined as above, v j u j ) is increasing Lemma 3 Le r<, x j ), y j ) be increasing and such ha x j y j ) is also increasing Then n x n j y j n xj r )/r n yj r )/r Proof For x = x,,x n ) R n, we will denoe by X = x + +x n By [6, Lemma 6], if x j ), y j ) are finie or infinie) sequences of posiive numbers, and x j y j ) is decreasing or increasing), hen so is X j Y j ) n x j Le n = C, ha is X y n = CY n Since x j j y j ) is increasing, X j Y j ) is also increasing, so X j X n = C for all j n Y j Y n Since for r<, he funcion f)= r is convex, and x j ), y j ) are increasing non-negaive elemens of R n such ha X Y, for all n, and X n = Y n, using he discree version of Karamaa s inequaliy for decreasing sequences [6, Lemma 8] and rearranging he erms, i can be shown ha fx j ) fy j ) Now by r<, ) /r ) /r xj r C yj r, which proves our claim Theorem 4 Le r< and v be defined as above Then for <α< r, v j + αr)/r sup vj r = )/r + α If r = q, hen his is he exac value of M q) dv, )) Proof By Lemma 2, v j u j ) is increasing, and hen by Lemma 3, u j v j u r j )/r vj r, )/r ha is A B Thus from Theorem, A B sup B Therefore using Proposiion, lim B + αr)/r =, + α so he claim is proved + αr)/r + α

13 A Kamińsa, AM Parrish / J Mah Anal Appl ) Theorem 5 Le r< and u be defined as above Then for <α< r, u j + αr)/r sup u r = j )/r + α If r = q, hen his is he exac value of M q) du, )) Proof For all, as noed in he previous proof, A B +αr)/r +α, and so by Proposiion, + αr)/r sup A = lim + α A, and we are done Theorem 6 Le <q<pand r = p q Then du,p) and dv,p) are q-convex for <α< r for α r and no q-convex Proof By Theorems 8, 4 and 5, if <α< r, M q) du,p) ) = M q) dv,p) ) + αr)/r =, + α so he q-convexiy consans are bounded and he spaces are q-convex For he space du,p), in he case when α = r = r, A = /r = +/ r j / r / r ) j ) = / r /r j r + r + ) ln + ) ) r = r ) ln + ) r r + ) / r j /r / r d The righ side is unbounded as, so in view of Theorem 8, he space is no q-convex Consider now he space dv,p) Since by definiion v n = n n + α)α d, + α)n ) α v n + α)n α Then by r<, + α) r j αr vj r + α)r j αr j= Since α = r, we have ha V = +α and + α) r j vj r + α)r j Therefore B = /r /r vj r )/r + α) j ) = /r + α) j ) / r ) / r = + α j + α j= + ) / r d = ) / r ln + ) + α ) + d ) / r 7)

14 35 A Kamińsa, AM Parrish / J Mah Anal Appl ) Again, he righ side is unbounded as, so he space is no q-convex If α> r, hen by Eq 5), U +α /r /r + α = α+/r + α Since α + /r >, he lef side is unbounded as,soa is unbounded Similar calculaions apply o dv,p) For <p,q<, we define he classical Lorenz sequence spaces l q,p analogously o L q,p, ha is as he space dw,p) wih w = w n ) such ha w n = n p/q In he sequence case we canno expec o obain he consans as easy as for funcion spaces compare Corollary 5 here and Corollary 37 in [3]) However applying resuls from [6] and he previous heorem, we ge he concaviy and convexiy consans for he space l q, Corollary 7 ) If q, hen for s>q, M s) l q, ) = q[ q, )r + ]/r where r + s = Fors q, he space is no s-concave 2) If <q<, hen for s<q, [ ) /r M s) l q, ) = q q r + ], where r + s = Fors q, he space is no s-convex Proof ) Applying Theorem 6 in [6] for α = q and r = s, we obain he s-concaviy consan By Proposiion 4 in [6], he space is no s-concave for s q 2) Applying Theorem 6 for α = q and r = s, he claim is rue References [] C Benne, R Sharpley, Inerpolaion of Operaors, Pure Appl Mah Ser, vol 29, Academic Press Inc, 988 [2] NL Carohers, SJ Dilworh, Geomery of Lorenz spaces via inerpolaion, in: Texas Funcional Analysis Seminar , Ausin, TX, , in: Longhorn Noes, Univ Texas, Ausin, TX, 986, pp 7 33 [3] NL Carohers, SJ Dilworh, Equidisribued random variables in L p,q, J Func Anal 84 ) 989) [4] J Diesel, H Jarchow, A Tonge, Absoluely Summing Operaors, Cambridge Sud Adv Mah, vol 43, Cambridge Universiy Press, Cambridge, 995 [5]CHao,AKamińsa, N Tomcza-Jaegermann, Orlicz spaces wih convexiy or concaviy consan one, J Mah Anal Appl 32 ) 26) [6] GJO Jameson, The q-concaviy consans of Lorenz sequence spaces and relaed inequaliies, Mah Z ) [7] GJO Jameson, 2-convexiy and 2-concaviy in Schaen ideals, Mah Proc Cambridge Philos Soc 2 996) [8] NJ Kalon, Convexiy, ype and he hree space problem, Sudia Mah 69 98) [9] NJ Kalon, A Kamińsa, Type and order convexiy of Marciniewicz and Lorenz spaces and applicaions, Glasg Mah J 47 ) 25) [] A Kamińsa, HJ Lee, M-ideal properies in Marciniewicz spaces, Commen Mah Prace Ma 24, Tomus specialis in Honorem Juliani Musiela, pp [] A Kamińsa, L Maligranda, Order convexiy and concaviy of Lorenz spaces, Sudia Mah 6 3) 24) [2] A Kamińsa, L Maligranda, LE Persson, Convexiy, concaviy, ype and coype of Lorenz spaces, Indag Mah 9 998) [3] A Kamińsa, AM Parrish, The q-concaviy and q-convexiy consans in Lorenz spaces, in: Banach Spaces and Their Applicaions in Analysis, Conference in Honor of Nigel Kalon, May 26, Waler de Gruyer, Berlin, 27, pp [4] A Kamińsa, Y Raynaud, Copies of l p and c in general quasi-normed Orlicz Lorenz sequence spaces, in: Funcion Spaces, in: Conemp Mah, vol 435, 27, pp [5] SG Kreĭn, YuĪ Peunīn, EM Semënov, Inerpolaion of Linear Operaors, Transl Mah Monogr, vol 54, AMS, 982 [6] J Lindensrauss, L Tzafriri, Classical Banach Spaces I, Springer-Verlag, 977

15 A Kamińsa, AM Parrish / J Mah Anal Appl ) [7] J Lindensrauss, L Tzafriri, Classical Banach Spaces II, Springer-Verlag, Berlin, 979 [8] WAJ Luxemburg, AC Zaanen, Riesz Spaces II, Norh-Holland, Amserdam, 983 [9] Y Raynaud, On Lorenz Sharpley spaces, in: Inerpolaion Spaces and Relaed Topics, Haifa, 99, in: Israel Mah Conf Proc, vol 5, Bar-Ilan Univ, Rama Gan, 992, pp