An explicit expression for the Kr, functionals of interpolation between Lp spaces

Publicacions Ma emá iques, Vol 35 (1991, 97-117 . AN EXPLICIT EXPRESSION FOR THE IV, FUNCTIONALS OF INTERPOLATION BETWEEN LP SPACES JESÚS BASTERO * , YVES RAYNAUD AND M . LUISA REZOLA ** In oduc ion and No a ion When dealing wi h in e pola ion spaces by eal me hods one is lead o com- pu e (o a leas o es ima e) he K- unc ional associa ed o he couple o in e pola ion spaces . This concep was i s in oduced by J .Pee e (see [8], [9]) and some e o s ha e been done o ind explici exp essions o i o he case o Lebesgue spaces . I is well known ha o he couple consis ing o Ll and L°° on [0, oo) K is gi en by K( ; , L 1 , L°°) = 0 * whe e * deno es he non inc easing ea angemen o he unc ion . In [7] Nilsson and Pee e compu ed he K- unc ional also be ween spaces LP and L9 when 1 _< p < q < oo . Mo e ecen ly he wo i s named au ho s ob ained an explici exp ession o a sui able modi ica ion o he K- unc ional o he case (LP, LM) whe e L M s ands o an O licz space (see [1] ) . The aim o his pape is o answe a ques ion aised by J . Pee e o he au ho s and o ex end he esul s in [1] and [7] o he mo e gene al case o he K, .- unc ionals be ween LP spaces . The no ion o K, .- unc ional was in oduced in [4] by Holms ed and Pee e ob aining also some es ima es o hose unc ionals be ween gene al compa ible couples o in e pola ion spaces . We shall w i e LP o he Lebesgue space LP([0, oo» . Fo E LP + L9,1 <_ p < q < oc, > 0 and 1 < < oo we de ine he K - unc ional by K,( ; ) = in (¡¡gil' + 11 hil ) l/ whe e he in imum uns o e all possible decomposi ions = g + h wi h g E LP and h E L9 . (Ob iously K,,, will mean K, > , > ( ; ) =in max{ligJ I P , i jhJ l .}) No e ha = 1 co esponda o he classical de ini ion o K- unc ional . The eade is e e ed o [2], [3] and [9] o backg ound on in e pola ion spaces . *Resea ch pa ially suppo ed by CAICYT 0804-84 and DGICYT PS87-0338 **Resea ch pa ially suppo ed by CAICYT, PB 85-0338 98  J . BASTERO, Y . RAINAUD, M .L . REZOLA Clea ly he K, .- unc ional de ines a ea angemen in a ian no m on LP -F L9 equi alen o he na u al one . Hence K,( ; ) = K,( ; *) and his allows us o es ic ou sel es o non nega i e and non inc easing unc ions E Lp - - L9 . The pape is di ided in o h ee sec ions and one appendix . Sec ion Iis de o ed o show he exis ence o ex emal unc ions which minimize he'K - unc ionals . In sec ion II we gi e a p ocedu e o ge such ex emal unc ions o he cases 1 < < oo and inally he case = oo is conside ed in sec ion III . In he appendix we in oduce and compu e he unc ionals K, , ,, and K 9 . We apply hese esul s o he isome ic p oblem o in e pola ion be ween he couple o spaces (L P ,L 9) . Le us men ion ha he me hod we use in sec ions II and III is ac ually a simpli ied e sion o he calculus o a ia ions, bu we shall include some p oo s o he sake o comple eness . As i happens in [1] and [7] he solu ions o p = 1 o p > 1 a e essen ially di e en . In he case p = 1 he ex emal decomposi ion o is achie ed by a ho izon al slicing o he unc ion . I . Exis ence o ex emal solu ions In he sequel 1 _< p < q < oo and will deno e a non nega i e and non inc easing ixed unc ion on [0, oo) . I is e y easy o see ha whe e he in imum is aken o e all unc ions g E LP wi h 0 < g < . Le g be a non nega i e measu able unc ion on [0, oo) such ha 0 <_ We de ine he unc ional <P(g ; ; ; ) o simply ~¿(g) by max{11911P, il -911 9 }, I is clea ha 0 < (D(g) < oo and K ( ; ) = in (ligli + il - 9,,y)lI K ( ; . ) = in {¿(9) 1/ ;0 < 9 < . ,9 E L P } The main esul o his sec ion is he ollowing heo em ¡ 1< <oo ; i = oo . 9 < . I .1 Theo em . Le 1 _< <_ oo and E LP + L9 .  The e exis s a non inc easing unc ion g E LP, 0 <_ g <_ , such ha K ( ; ) = $( g) l / .  I 1 < < oo , his unc ion g is unique and - g is also non inc easing . In o de o . p o e his heo em we need se e al lemmas . We begin by es ab- lishing he exis ence o ex emal solu ions . THE K FUNCTIONALS OF INTERPOLATION  99 1 .2 Lemma .  The e exis s a unc ion g E LP, 0 < g < , such ha K,( ; ) _ q>( g )1/ . P oo . . Le a be de ined by a = in {,¿(g) ;0 _< g <_ , g E LP} .  We can choose a sequence (gn)n in LP such ha 0 < 9n < and a= lim~-,, 4>(g n ) . As ( - g),, is a bounded sequence in L9 we may suppose, by passing o subsequence i necessa y, ha ( -g n ) n con e ges weakly o a unc ion h E L q . Then (g n ) n is a weakly Cauchy sequence in LP and so w-limg n = -h E LP . Hence i g = - h we ha e ~¿(g) < limin 119n11p + limin 1I -9n1Iq = li,~¿(gn) = a . The same a gumen s can be modi ied o = oo . Rema k . Le us poin ou ha he same ideas used in p eceding lemma may be applied o K,- unc ionals be ween a la ge class o in e pola ion spaces . Ac ually, i (A 0 , A l ) is a compa ible couple o Banach spaces le us de ine K on A o +A 1 by K,( ; ) = in (11911Á', + ilhIIA,)1/ whe e he in imum uns all possible decomposi ions = g + h wi h g E A o and h E A l . I we suppose ha A o is weakly sequen ially comple e, A 1 is e lexi e and Aá n Ai is dense in A*, hen he e exis g E A o and hE A 1 such ha =9+h and K,( ; ) = (11911Á o + 1Ih1i ,) 1/ 1 .3 De ini ion . A non inc easing unc ion g in LP is an ex ema] solu ion o he unc ional K,( ; ) i 0 <_ g < and K ( ; ) = ¿(g)1/ (K ( ; ) = (p(9) i = oo) . Nex we will s udy he uniqueness o he ex ema] solu ions . 1 .4 Lemma . Le¡ 1 < < oo .  I K ( ; ) = ¿( 91 )1/ = <P( 92 )1/ , hen 91 = 92- P oo . . I is e y easy o check ha he unc ion g = 91 2 g2 also e i ies K ( ; ) = ~¿(g)1/ o all 1 <_ < oo . Indeed, by using Minkowski's inequali y we ha e , ¿(9) 1/ _< 1 [(119111,+119211,) +( ll -911Iq+ ll -92jjq) [ 1/ < 1 [x(91 _ ) 1/ +4 >(92) 1/ ] = K ( ; ) and hen 1191 + 9211, = 119111, + 119211  11( - 91) + ( . - 92)1I9 = II - 9111 9 + Il -9211q- I 1 < < oo he ec o s in R2 (1191 11,, I1 - 91 I1 q ) and (11921I,, 11 - 9211 q ) a e also colinea . Since Lq is s ic 1y con ex (q > 1) we ob ain ha , o ins ance, 100  J . BASTERO, Y . RAINAUD, M .L . REZOLA - 91 = A( - 92) o some A > 0 . Thus, 11911¡p = X119211, and oD(g1) = A (P(92) which implies ha A = 1 and consequen ly g1 = 92- Suppose now ha = oo .  We ealize ha K,,~( ; ) = 4>(g) implies ha J l9IJp = J ~ - 9 JJq . Indeed, i i we e JigJip > J I -9Jj9, we choose some posi i a a such ha he se A = {x E [0, oo) ; g(x) > a} has m(A) > 0 . Take 0 < S small enough e i ying il - 9liq < il - 9+áXAliy < 119 -6 XA11p < liglip Hence $(9 - 6XA) = 119 - 6 XA1 ¡p < (P(g) and consequen ly K,,~(i ; ) < ~¿(g) . (An analogous a gumen wo ks i we suppose liglip < il - glw) . Le 91,92 be wo unc ions in Lp such ha K~( ; ) = <P(91) = <P(92) . We may epea he a gumen a used o 1 < < oo and we ob ain ( 91 2 92 ) <1 max{li91ilp + 119211p, il - 91lw + il - 9211} We he e o e ha e li - 91 + - 9211,  =  l i - 91 i l e + l i - 9211, and hen - 91 = A( - 92) which also implies g1 = 92 . Rema k . In he case = 1di e en solu ions could be ob ained . We ealiza ha i g1 and g 2 a e wo di e en solu ions any o he unc ion g in he segmen de ined by g1 and 92 is also a solu ion . 1 .5 Lemma . i) I 0 <_ g < hen 0 <_ g* <_ and (P(g*) < <P(g) (g* is he non inc easing ea angemen o g) . ü) I he unc ional K,( ; ) has only one ex emal solu ion g hen - g is non inc easing . P oo . i) We apply he p oposi ion 1 o [6] (which is also aluable in he in e al (0, oo) ) and hen we ha e a This implies ha Ji - g* li q<_ l i - 9I l, and so <D(g*) < ~¿(g) . ii) I K,( ; ) = ~¿(g) hen g = g* . The unc ion g1 de ined by g1 = -( -g)* sa is ies -P(g1) _< $(g * ) by i) . Thus g 1 = g* and so = g* '= ( - g)* which implies ha he unc ion - g is non inc easing . Rema k . The lemma is also ue i we conside he co esponding K unc- ional be ween a couple o ea angemen in a ian unc ion spaces . P oo o he heo em L1 : I is ob ious om he p eceding lemmas . 9 THE K, . FUNCTIONALS OF INTERPOLATION  101 II . De e mina ion o he ex ema¡ solu ions when 1< <oo The main ool o his pa is he ollowing lemma . II .1 Lemma . Le g be an ex ema¡ solu ion o K,( ; . ) and assume supp = [0, a], 0 < a < oo .  The ollowing asse ions a e íe : i) Ei he g < a .e . on supp , o = 1 and g = , a .e . on supp . ii) I p > 1, ei he 0 < g(x) and g(x)p -1 jjgjj - = [ (x) - g(x)]q-1[l - g lI -q a . e .  on supp , o = 1 and g = O . iii) I p = 1, a > 0 and g :~ hen a < oo and (x) - g(x) = cons an = A > 0, a . e .  on [0, a], whe e a and A e i y ( a  -1   00 . ( I9)-1 - a~l  ñ9_  -1 I a~q + 1  q] Fu he mo e, i = 1, hen a < q' (q' is ¡he conjuga e exponen¡ 4+Q, = 1) and a = [ q' - a]-1IgIl X[a,oo)lIq i ) I g = 0, hen = 1 .  Mo eo e , i p = 1 hen ELq 1 L°° and q ' ~~gll ll . <- Il llq . ) I g = ( = 1) and p = 1 hen leng h supp = b < q' . P oo . i) Assume ha m{x ; (x) = g(x)} > 0 .  Then he e exis s n E [®[ such ha B n = {x E supp ; l/n < (x) = g(x)} and m(B n ) > 0 . Le cp be he unc ion de ined by cp(S) = 4>(gXB~ + ( - b)XBn) o 0 <_ b < 1/n . As c,(0) = 4>(g) and W'(0+) < 0 we necessa ily ha e ha > 1 and = g . ii) Le A n = {x E supp ; g(x) = 0, (x) > 1/n} and le W(ó) = (P(9 + 5XA  ) o 0 < b < l/n, nE N . I > 1 o 7~ g he same easons as be o e imply ha m(An) = 0, and so, 0 < g a .e . on supp . Conside now any measu able se A con ained in supp , wi h 0 <_ m(A) < oo . The unc ion cp(b) = <I>(9 + áXA) has i s minimum in S = 0, hence yp'(O) = 0 ( no e ha p > 1 ) . Thus A l l9I I P -I lgil -1 - l l - 9l l q -' I - 91q -1 = 0 and hen ii) ollows since A is a bi a y . iii) Le A be any compac in e al con ained in [0, a) . Le cp be he unc ion de ined by W(ó) = 4>(g +ÓXA) o Ial < in { (x) ; xE A} . I is clea ha W'(O) does exis s and ac ually W'(0) = 0 . The e o e we ob ain - ll - 9119 - ql - 9] q-1 = 0 . Since his exp ession is ue o any compac in e al con ained in [0, a) we deduce ha - g = cons an , a .e .x E [0 ; a] .  I A = (x) - g(x), hen 10 2  J . BASTERO, Y . RAINAUD, M .L . REZOLA g = ( - A)X(o,a] and consequen ly a and A ha e o e i y he equa ion (*) . No e ha a < oo as he unc ion - g E Lq . I = 1 we easily compu e ha 1  ía  0o (,) , j (119) - 1 - - ~ and hen ( q' - a) Aq = : ° q which implies iii) . i ) I g = 0 is an ex emal solu ion, hen E Lq . We epea he p eceding a gumen s by conside ing now he unc ion cp(ó) _ ~(SX ~ ~) whe e [c, d] C_ supp and 0 < S < (d) . Since cp has he minimum in b = 0, cp'(0+) >- 0 and hence d 0 <  lim á -1 (d - c ) 1P -  ,-1 l l i l q -q 6- .0+ c wha implies ha = 1 .  I p = 1 we he e o e ob ain 1 > i 1 l .iq-q (x) a .e . on supp . ) I g = ( = 1) simila a gumen s o hose appea ing in i ) show ) . This concludes he p oo o he lemma . Nex we will s udy he case p > 1 . II .2 In iew he p eceding lemma we ha e o conside he'class o unc ions g e i ying : i) g E LP, - g E Lq, ii)O<g< and ll - gllq -q [ ( x ) - g(x)lq-1 = llgli -Pg(x)P-1 a .e . on supp . Le A deno e he class o unc ions sa is ying i) and ii) . I is e y simple o check ha o hem q > (g) = ll - 9li q - q 1 ( - g) q-1 = llgl1P -P 1 g P-1 . q-1 Using he s ic 1y dec easing unc ion h de ined by h(y) - ( (x) - y ) q-1 y P-1 0 < y < (x), we see ha he ollowing ac s a e ue : i) I 9 E A, g is non inc easing . ii) A is o ally o de ed . Indeed, le 91, 92 be wo elemen s in A and w i e Mi = 1lgi l l -P l i - 9i l l9- , i = 1, 2 . I is clea ha M1=M2 ( espec i ely M1< M2) implies g1(x) = 92(x) a .e .x Esupp ( espec i ely g1(x) > 92(x) a .e . x E supp ) . Fu he mo e, o q > , he inequali y g 1 > g2 yields M 1 > M2 . Hence he se A has only one elemen which is necessa ily he unique ex emal solu ion o K ( ; ) . Le now assume p < < q .  I g deno es he in A we a e going o p o e ha g= min A and ha g is he unique ex emal solu ion o K,( ; ) . Indeed, we ealize ha o any wo elemen s o A, g1 < 92 we ha e 4)(g1) < <P(92) . THE K FUNCTIONALS OF INTERPOLATION  10 .3 Le g be de ined by g(x) = in {g(x) ; g E A} E LP  (LP is o de con inuous, see [5}) . De ine a = i IIgil, and le (gn)n be a sequence in A such ha g EA I I91 I Ip < ..... -< I I9n I Ip < .... . -> a .  The e o e 91 >_ 92 >- . . ........ > 9n > . . . . > 0 . Since ll - gnll' <4 )(gn) < <D(gl) we ha e ha g o = limg, n E LP and - g o E Lq . By passing o he limi in 11 .2 . i is easily checked ha g o E A , oo . E en ually we conclude ha g= g o E A . We summa ize all hese ac s in he ollowing heo em 11 .3 Theo em . The case p> 1 . i) I >_ q hen he caass A has only one elemen which is he unique ex emal solu ion o K ( ; )- i) I p < < q hen he leas elemen o A is he unique ex emal solu ion o K, ( ; ) . iii) I 1 < < p we know ha he unique ex emal solu ion o K ( ; ) is an elemen o A . i ) I 1 = < p ¡he solu ions e i y ¡he equa ion [ (x) - 9(x)] q-1 I Ig1Ip -1 = g(X) P-, I I - 9I I9 -1 Now g could be equal o 0 o . Fu hemo e he solu ion is unique excep i = AX[O a] . In his case = al/p -1 /q and K, ( ; ) = Aal/P . Mo eo e in ¡he ou cases, i g is an ex emal solu ion we ha e 1/  1/ I1 ( i ) = II9IIp-(pl ) U gp-11  = ll -9IIq-(ql ) (1 ( - 9) q-11 P oo . . We only ha e o p o e he las pa o i ) . I 91 7É 92 a e ex emal solu ion o K l ( ; ) hen all he poin s o he segmen [91,921 a e ex emal solu ions : So we may suppose gl :~ 0 7 É 92 and - gl 7 É 0 :~ - 92 . Since LP and Lq a e s ic ly con ex spaces he same easons appea ing in he lemma I .4 say ha g l = ag 2 , - 9 1 = b( - 92) o posi i e a, b . The e o e = cg2, o some c > 1, and hus g2(x)q-p = 1192119-1II92IIp-1 a .e . on supp . Hence = AX[0 a], > 0 and = a l/p-1/q, M 11 .4 Le us now conside he case p = 1 . We shall deno e by b =leng h o supp < oo . As i appea s in lemma II .1 ., he ex emal solu ions o K,( ; ), g, ha e he ollowing exp ession g = ( - A)X[o,a] wi h 0 < a < oo . I a > 0 hen A e i ies he equa ion IL1-iii)-(*) . We de ine he unc ion H(a, A) by a  -1  00 1( lq) - 1 H(a, A) _ ~~  - aA~  - Aq -1 [aAq -F J  qJ  ( > 1) o  a 00 (1/q)-1 H (a, A) = 1 - Aq-1 I aAq + ~  qJ  ( = 1) a 10 4  J . BASTERO, Y . RAINAUD, M .L . REZOLA o 0<a<_b,0< .~<_ 1 o , >1and0<A<oc, =1 . I aisa ixed posi i e numbe he unc ion H(a, A) is s ic ly dec easing in he a iable a .~ and H(a, 0) > 0 .  On he o he hand, H(a, 1  ) < 0 in he case > 1, alo and o = 1 lima_,  H(a,A) < 0 i and only i a < 9' . Hence, he equa ion H(a, a) = 0 has only one solu ion A = a o any 0 < a <_ b (case > 1) and o any 0 < a < min{ 9 ', b} (case = 1) . When a = band = 1 he co esponding equa ion H(a, A) = 0is hen 1 = b l l9 -1 and so, his equa ion has solu ion i and only i 9 ' = b . Fu he mo e, i g is an ex emal solu ion o he unc ional K I ( ; ) wi h a = b, hen b < oo, 9 ' = b, Kl ( ; ) = 1 l 11 1 (all he unc ions - a a e ex emal solu ions o he unc ional o 0 <_ A <_ (b - ) ) and H(b, (b)) = 0 . In o de o de e mina e he possible ex emal solu ions g we only ha e o check he alue o a equal o he leng h o supp g . The main ool o compu ing his alue a is he ollowing Lmma 11 .5 Lemma . Leí g be an ex emal solu ion o K,( ; ) and leí [0, a] _ supp g, a > 0 .  Then (a+) < Aa < (-_) . P oo . _Since g = ( - Aa)X[a,a~ > 0 we ind Aa < (a - ) . I Aa < (a+), we would ha e a < (x) o all x E (a, c) . By conside ing he auxilia y unc ion M) = '¿(9 + 6X[a,c]) de ined o 0 < b < a, we easily would check ha cp'(0+) = l i - gil' -9  ( , á -1 - . 9-1 ) .< 0 a and, consequen ly, g would no be ex emal solu ion o he unc ional K, . ( ; ) which con adic s ou assump ion . Hence he Lmma ollows . 11 .6 Rema k . As an immedia e consequence o Lemma 11 .5 . we ha e o s udy he se I = {x E (0, b] ; H(x, (x - )) _< 0 <_ H(x, (x+))}, because he leng h o he suppo o he possible non null solu ions belongs o I . Then we de ine he unc ion F(x) = H(x, (x)) de ined o 0 <x _< b . The ollowing p ope ies will allow us o compu e easily he ex emal solu ions o K,( ; ) . II .7 Lemma . The ollowing p ope ies a e ae : i) F( .) is non dec easing . Fu he mo e i F(xl) = F(x2) wi h x 1 <x2 hen is cons an in he closed in e al [x1, x 2 ] . ii) F(x - ) = H(x, (x - » and F(x 1 ) = H(x, (x+)) . iii) I xl < x2, H(xl, . (xl» :5 H(x2, . (x2))- i ) I is ei he emp y o an in e al and he unc ion is cons an on i s in e io . Fu he mo e I = {x ; F(x - ) < 0 < F(x+)} . P oo . i) Le xl < x2 be wo posi i e numbe s . Since is non dec easing we ha e ha z x2 (x2) T XI (x1) C ( x 2)( x 2 - xl) <  xl and he e o e I > q i is also clea ha xl (xl) q +  q > x l (x1) q +(x2 - xl) (x2) q +  q > x2 (x2) q +  q x  x 2 x 2 Hence (X1)q-1 IX, (X1)q + THE K, . FUNCTIONALS OF INTERPOLATION  105 x  -1 xl (xl) 0 q In he case 1 < < qwe ealize ha ( Iq) - 1 (xl) q 1 Ixl (xl)q +  q1  _ L  x 1 00 xl) -1 xl + oo ( L  ~q~( Iq)-1 Jx (xl) (x2) -1 1x, +J 00  ( l))q]( /q)-1 x2 ) -1 ~x2 + lo   ql ( lq) - 1 x2 (xl)  2) -1 1X2 + Jx2  ( ( 2))q]( lq)-1 and hus he i s pa o i) is p o ed . I F(x 1 ) = F(x2) and > 1 we ha e, in pa icula , ha x 2 ( x 2) - xl (x1) = o any x, x l < x < x 2 . Then (x) = (X1) = (x2) and is cons an in [XI, x2] . J .`1,2 q Suppos eno w ha =1 , he nw e in d J ( (x1) )= x2 -x 1 and he e o e we ha e ha is also cons an in [XI, x 2 ] . ii) This p ope y is easily compu ed om he special exp ession de ining he unc ion F( .) . iii) H(x l , (xi )) =  lim F(z) <  lim F(z) = H(x2, (x2) . <_ [ ¡x2 - x2 (x2) 0 ~ ~ (x2)q -1 IX2 (X2)' +  qJ x2 z ? (x-xl) (x)+(x2 -x) (x2) 112  J . BASTERO, Y . RAINAUD, M .L . REZOLA P oo . We shall p o e he heo em in h ee s eps . To s a wi h we shall es ablish he heo em o simple unc ions . n Siep 1 . Suppose ha = i : aiXI i whe e al > . . . > a n > 0, Ii's a e pai wise i=1 disjoin in e als wi h leng h(Ii) = mi > 0 . I is qui e clea ha 1C , 9 ( ; ) _ n min cp(x), being x = (xl, . . ., x n ), C = Jj[0, mil and W he con inuous unc ion 8=1 de ined by n  ) /P  ( n  9 /9 x  aPxi  +  ay (Mi - xi Le x = (11, . . ., - in) a minimum o his unc ion cp in C . I x EC hen a  (x) _ áx i 0, o all 1 < i <_ n . Easy compu a ions would imply ha a l = . . . = a n which is a con adic ion . Thus 1 E áC and, o ins an e, we may assume ha he e exis s j (1 <_ j < n) such ha Yj = Qj wi h Qj = 0 o = mj . Now we conside he unc ion W(xl1 . . . . xj_1, x7+1, ". ., x n ) = W(xly .. ., xj_1,Qj5 xj+1) .. ., xn) de ined o 0 < xi < mi, i :~ j . This unc ion 0 a ains i s minimum in (xl, . . ., 1 7 ) + l, . . ., 7n) . Repea ing he a gumen we would ob ain ha he e exis s ano he coo dina e j' such ha Tp = Qj, (£j, = 0 o mj,) and so on . E en ually we ob ain ha he e is a na u al numbe k, 1 <_ k < n, such ha j = (Q1, . . .,Qk_1,7k,Qk+1, . . ., n) whe e each 2i = 0 o mi and xk is equal o 0 a o mk o a solu ion o he equa ion in x : OX (~1, -- .A-1,x)4+1) . . ., £n) = 0 . In o de o p ecise he exac o m o x we ealize ha i x = (0, . . ., 0) o =_ (ml, . . ., m n ) he p oo o he S ep 1 would be inished . So, we may suppose -  (0, . . ., 0) and Y 7É (ml, .. ., n n ) . Le A be he well de ined posi i e numbe A =  4 [ i_1 ael ii /P-1 Le A be he se A = {i ; a ; - P >_ A} . The ollowing lemma is he main ool in he p oo o his s ep . A .5 . Lemma .  The se A :~ 0 . I k o =max A hen 1 = (ml, ". ., mko_1, xko, 0, ". ., 0)e whe e xko =mk o i aka P > A and  8- (x) = 0 o he wise . ax ko P oo .. We know ha x= (~ 1, . . .,Bk_1,xk,Qk+1, . . .,£n) o some k,1 _< k _< n . I j :~ k le h be he eal unc ion de ined by As Pj = 0 ( espec i ely 2j = mj) implies ha h'(0+) > 0 ( espec . h'(m i )_< 0) i is easy o check ha Pi = 0 ( espec . = mj) implies ha al - P _< A ( espec . al - P > A) . In he same way , i jk = 0 hen ak - P _< A, i !k =mk hen aq - P > A and o he wise ak _P = A . As x :~ (0, . . ., 0) he se A  ~ .  Hence i j  < k 0 ( espec .  j  > k 0 ) he co esponding j- h coo dina e has o be mj ( espec . 0) . Finally we only ha e o conside he di e en alues o ak o wi h espec o A and his concludes he p oo o he lemma . o By applying he lemma we ha e ha 1C,,9( ; ) = il II Q o K,,s( ; ) = li ji Hence o some xE supp . THE K FUNCTIONALS OF INTERPOLATION  113 _ h(y) = ~ P(~1, . . ., i) y, . . ., xk, . . ., ~n) . n 9 Iq + Cak o (mku _ xko)+ 5 a9m4/ ko+1 - I I X[0,x] I IP + i l X[x,-) I Iq whe e x = m l + . . . + mko_1 +xk o . In he he e cases he S ep 1 is p o ed . S ep 2 . Suppose now ha E L°° and he leng h o he suppo o is ini e . We can app oxima e by a non dec easing sequence o simple unc ions ( n)n con e ging o in he L°°-no m . We apply he S ep 1 o he e simple unc ions n and we ge ha K ,s( ; n) = II X[O,x ]IIp + ij X[x ,oo)IIq whe e x n E supp n . Since JC , s ( , n ) < K,,s( ; ) by passing o a subsequence i necessa y he e exis s x = lim x n and hus we ha e llm n ) = Ii X[O,x]IIp + iI X[x,oo)II`  ) . K,,s( ; ) = I I X[O,x] I IP + i l X[z,oo) I I q S ep 3 . I is a gene al non inc easing and non nega i e unc ion in LP + Lq we app oxima e by a sequence o unca ions o and we apply he ideas o he S ep 2 . 114  J . BASTERO, Y . RAINAUD, M .L . REZOLA In o de e de e mina e he poin x E supp o which 1C"-'( ; ) = 11 X[O, .J 1I P + ll X[x,oo) 11 q we conside he auxilia y unc ion cp de ined by Ux P) /P + C ¡oo q) sI q o 0 < x < b = leng h o he suppo o . This unc ion cp is con inuous and has de i a i e cp'(x) almos e e ywhe e . Mo e p ecisely, he e exis cp'(x+) and cp'(x - ) and hey a e equal excep a mos in he discon inui y poin s o . I is easy o check ha 0<x < b, whe e A .6 . P oposi ion . ~ I (x}) - q ll x[x,oo)11 q -q (x )Pl~¿(x) - (x q P 9 I1 x[0, .]H P -P ps 11 X[x,°°) 119 -q (cp'(x :1 ) deno es each one o he wo hand side de i a i es o cp, in he same way (x : ~) ep esen s he wo la e al limi s o ) . Hence we ha e ha i x 0 is a poin o minimum o he unc ion W necessa ily x0 = 0, o = b o (x~ )q - P >_ 4>(x0) > (xó )q - P . In pa icula we ha e he ollowing i) I ei he >_ p and s > q o >pand s >_ q he e exis s only one poin x 0 E supp o which ¡he minimum o cp is a ained . Fu he mo e xO = sup{x ; <D(x) < (x)q-P} and K ,s( ; ) = ll x[xo, .)11 q - q L p g ll x[O,x"] llp p (x)+ll x[xo~ .)ll gJ ii) I =p ands=q 00  ¡ min x x dx . 0 P oo .. i) Unde hese hypo heses ~¿ is a s ic ely inc easing unc ion . Le I = {x E (0, b) ; <P(x) < (x)q - P} . I = 01 means ha E Lq l L°°, = pand ll llóo P  <  E ll llq-9 .  Then 'p'(x :~) > 0 o all x and hen IC ,,( ; ) = 11 11 q . and THE I1', . FUNCTIONALS OF INTERPOLATION  115 I I = (0, b) hen b < oo and ~p'(x}) < 0 o all 0 < x < b . So IC,,,,( ; ) _ P . O he wise cp'(x ) < 0 o all xE I and ~o'(xl) > 0 i x 11 . Hence i) holds . ii) Now ~¿(x) _  and his pa o he p oposi ion is easily checked . Rema k . We no e ha ou 1C p , Q unc ional is exac ly he co esponding )C P , q unc ional used by Spa in [11, de ini ion (3 .1)] La e on we shall p o e ha o , s >_ 1 all he K,,,-mono onici ies a e equi alen in any in e media e space be ween LP and L9 and ha a weake esul is ue o he co espondig IC ,,,-mono onici ies . A .7 . P oposi ion . Le X be an in e media e space be ween LP and L9 and , s >_ 1 .  The ollowing asse ions a e ae : i) X is (1,K , .,)-mono one q X is (1, K) -mono one .  ü) I >_ p and s >_ q, X is (1, K, , ,)-mono one ~-¿ X is (1, /C,,,)-mono one . P oo . . i) Se E he unc ional de ined o E LP+L 9 and X > 0 by E(, ; ) _ in  11 - g11 a . I is clea ha I Ig11 P <_a K ,s( ; ) =in [A + E(i`l/ j)9J  (A .7 .1) K,, ( ; ) - A  1/s (see [11], lemma 3 .3) . Then K ,,( ; ) > K ,,( ;9) o all > 0 i and only i E(X ; ) > E(X ; g) o all A > 0 and hence i and only i Kj,l( ; ) >_ Kj,l( ; g) o all >0 . ii) The p oo o his pa is simila o he p e ious one by using a sui able modi ica ion o he unc ional E, namely £, de ined by £(X ; ) =  in  1I X[=, .)1w I1No, 311 P <_a Now he co esponding simila exp essions (A .7 .1) and (A .7 .2) o he unc ion- als K ,, and £ occu . Ac ually, he only hing we ha e o compu e is ha gi en A > 0 wi h £(A ; ) > 0 he e exis s > 0 such ha K,,,( ; ) = A + £(,X ; )s . I is clea ha he e exis s only one poin y, 0 < y < leng h supp , such ha li X[o,yj I P = A and li X[y,,, .) lI q = £( A ; ) . Since >p and s > q we ake =  qA ' -P  (y) P-9 and we apply P oposi ion A .6 . ps£(A ; ) 8- q Rema k .  Fo , s >_ 1 i is clea ha i X is (1, 1C , ,)-mono one => X is (1, K, )-mono one . Now we ecall ha a la ice homomo phism is a linea bounded ope a o be- ween Banach la ices which maps disjoin ly suppo ed unc ions in o disjoin ly suppo ed unc ions . 11 6  J . BASTERO, Y . RAINAUD, M .L . REZOLA A .8 . Theo em . Le X be an in e media e space be ween LPand Lq . X is (1,Kp,q)-mono one i and only i he ollowing in e pola ion esul is ue : I T E C(LP) n G(Lq) is a la ice homomo phism hen T E .C(X) and IITIIx-x _< max{IITIILy-L", IITIILI-- .La} . P oo . Fi s o all we ema k ha bo h asse ions imply ha X is a ea - angemen in a ian unc ion space . Suppose ha X is (1, IC p q )-mono one and ha T is a la ice homomo phism such ha max{IITIILD-DI, IITIIL9-L9} _< 1 . I EX, since T is a la ice homomo phism we ha e Kp,q( ;T )  a nT, IIgIIP+ IlhII9 < a + h , IIgIII+ ilhII9 qAh=0  q^h=0 = Kp,q( ; ) o all i > 0 . Thus T E X and IIT IIx < Il lix . On he con e se hand, le E X and g E LP + Lq such ha o all > 0 K p q ( ; ) >_ Kp,q( ; g) . Applying he lemma 4 .2 o [11] we ob ain ha o each e > 0 he e is a la ice homomo phism T E E.C(LP) n C(Lq) which e i ies TE( ) = g . Then by using he hypo heses we ha e g E X and IIgllx :5 IITIIx-xll IIx < (1 + E)II IIx Since his is ue o all e > 0 hus he p oo o he heo em is comple e . As a consequence o his esul we can es ablish he ollowing cha ac e iza ion o in e pola ion spaces wi h espec o he couple (LP, Lq) ; his co olla y is in essence an in e pola ion esul A .9 . Co olla y . Le X be an in e media e space be ween LP and Lq . The ollowing sia emenis a e equi alen : i) X is an in e pola ion space, ii) X is an in e pola ion space o la ice homomo phisms . P oo . We only ha e o show ii) => i) . We de ine a new equi alen no m II . II on X in he ollowing way M 1II =supJIT 1I whe e he in imum uns o e all possible la ice homomo phisms T E £(LP) n ,c(Lq) such ha max{IITIILD-LI, IITIIL9-L9} _< 1 . I is clea ha (X,111 .111) is an exac in e pola ion space o la ice homomo phisms . Thus (X,111 .111) ha e o be (1, IC p , q )-mono one . Hence he esul ollows by applying he heo em 5 .2 o [11] . Acknowledgemen s . The au ho s a e indeb ed o P o esso s J . Be gh, J . L8 5m and J . Pee e o he help ul co espondence which mo i a ed his wo k and o P o esso N . Kal on o some in e es ing ema ks on he inal pa o his pape . THE K, . FUNCTIONALS OF INTERPOLATION  117 Re e ences 1 .  J . BASTERO AND Y . RAYNAUD, On he K- unc ional o in e pola ion be ween LP and O licz spaces, Jou nal o App oxima ion Theo y 60, 1 (1990),11-23 . 2 .  C . BENNETAND R . SHARPLEY, "In e pola ion o Ope a o s," Academic P ess, 1988 . 3 .  J . BERGH AND J . LOFTROM, . "In e pola ion spaces . An in oduc ion," Sp inge Ve lag, 1976 . 4 .  J . HOLMSTEDTAND J . PEETRE, On ce ain unc ionals a ising in he heo y o in e pola ion spaces, Jou nal o Func . Analysis 4 (1969), 88-94 . 5 .  J . LINDENSTRAUSS AND L . TZAFRIRI, "Classical Banach Spaces II," Sp in- ge Ve lag, 1979 . 6 .  G . LORENTZ AND T . SHIMOGAKI, In e pola ion heo ems o Ope a o s in Func ion Spaces, Jou nal o Func . Analysis 2 (1968), 31-51 . 7 .  P . NILSSON AND J . PEETRE, On he K- unc ional be ween L l and LZ and some o he K- unc ionals, Jou nal o App oxima ion Theo y 48 (1986), 322-327 . 8 .  J . PEETRE, Nou elles p op ié és d'espaces d'in e pola ion, C .R .A .S . Pa is 256 (1963), 1424-1426 . 9 .  J . PEETRE, A heo y o in e pola ion o no med spaces, No as de Ma e- ma ica 39 (1968), 1-86, Rio de Janei o . 10 . J . PEETRE, A new app oach in in e pola ion spaces, S udia Ma h 34 (1970), 23-42 . 11 . G . SPARR, In e pola ion o weigh ed LP-spaces, S udia Ma h . 62 (1978), 229-271 . Jesús Bas e o : Depa amen o de Ma emá icas Facul ad de Ciencias Uni e sidad de Za agoza 50009 Za agoza SPAIN M . Luisa Rezola : Depa amen o de Ma emá icas Facul ad de Ciencias Uni e sidad de Za agoza 50009 Za agoza SPAIN Y es Rainaud : Equipe d'Analyse Tou 46, 4éme e age Uni e si é Pa is VI 75252 Pa is Cedex 05 FRANCE