Peetre, Jaak

English

Peetre Jaak

University of Szeged

.255 On an interpolation theorem of Foias and Lions By J. PEETRE in Lund (Sweden) Introduction Let X be a locally compact space provided with a positive measure p. We denote by L'¿(E), where and £ is a positive ^-measurable function and E a Banach space (or, more generally, a field of Banach spaces over X; we do not consider this generalization here in order not to complicate the notation), the space of ¿¿-measurable functions a with values in E such that II CellE is of /i-integrable p th power (if or p-bounded (if p = °°). We provide L"^E) with the norm. (1) II U(E) ( / I M I E ^ ) 1 ' " ( i f /> -< - ) X /¿-sup HCalU (if P — co)-A function H(z0, zt) defined, measurable and positive for z0 ^ 0 , zx is said to be an interpolation function of power p if and only if whenever n is a linear mapping from some space, containing Lga(E) and -££,(£) as linear subspaces, into itself such that the restriction of n to L i t(E) maps L^(E) continuously into itself ( / = 0 , 1) then the restriction of n to Ln^0 i i i )(E) maps -£//(CO,?>)(£) continuously into itself. E. g. zo~°Zi with O < 0 < 1 is an interpolation function of power p for any p (see example 2). In [1] FOIA§ and LIONS found a sufficient condition for a function to be an inter-polation function of power p (in the above terminology). In the present note we give two constructions of interpolation functions of power p. in a sense dual to each other The first of these constructions leads to a condition essentially the one of FOIA§ and LIONS (see remark 2) while the second leads to a condition in a sense dual to the first one. It is also shown that under some auxiliary restrictions both constructions are equivalent. In particular this leads to a simple condition which is independent of p (see theorem 4). The general ideas underlying these results were briefly discussed in [2] (cf. also [3]). 256 J. Peetre § 1 Let us set (2) J(t, a) = (||fl||^o(E) + / ' MlZf,<«,)''a € lJ0(E) fl Lf, (£), 0 < / < », and (3) *(/,<») = inf ( I K I I i ^ + ' l a J / V » ) 1 ' ' ' ^ L l ( E ) + L l { E ) , 0 « = 00 + « l Let a = a ( / ) and /J =/?(/) (0 < ? < » ) be two positive functions measurable with respect to dt/t. We denote by S„ the space of elements a £ Lj-0(E) + L^(E) such that there exists a function u = u(t) (0 < /<<») measurable with respect to dt\t with values in L^0(E) fl H ££,(£) such that (4) a = / « ( / A i n Lf 0 (£ )+ Z.c',(£)), a(/)7(/ , « / ( 0 ) 6 ^ , o and by Tp the, space of elements o f_ L?0(E) + L^(E) such that (5) ¡](t)K(t,a)£Lr„. (¿-I denotes Ln with respect to the measure dt/t.) We provide S„ with the norm (6) ||fl||s. = inf| |a(/)y(/. i/(/))||Lf, a = f u « ) - -o ' and Tp with the norm (7) ' M r , = ll/3(0*(' .fl)ll tï . T h e o r e m 1. Each of the spaces and Tp is an interpolation space with respect to L£0(E) and L^(E) ; /'. e. whenever n is a linear mapping from some space, containing /-£,(£) and L^(E) as linear subspaces, into itself such that the restriction of it to maps L%.(E) continuously into itself (/ '=0, I) then the restriction of n to Sr or Tp maps S,, or Tp continuously into itself. Moreover, if (8) II n a W a ^ M M , . ^ a ^ L l ( E ) (/ = 0,1), where M0 and M, are positive constants, then (9) \\na\\^M \\a\\, a£Sa or Tp with || || =|| ||Sti or || || r , where M is a constant that depends only upon A/0 and M{. Proof , i) We have Since 7Ta = J nu(t)~ On an interpolation theorem 257 we therefore get M s , 3 > m a x ( A f 0 , M,)\\a(t)J{t, w(/))|U and, by rriaking vary u, (9) follows in this case, with Af = max (M 0 , Mj) . ii) We have K{t, na)^{Mp0\\a0\\UoiE) + t''MnVn llif,'®)1". Making vary ci0 and a, we get K(t, no) ' max(M0 , Af,) K(t, a). Therefore (9) follows in this case, again with M = max(M0 , Mt). R e m a r k 1. If a and /? satisfy inequalities of the form (10) we may replace M = max(M0, Mt) by . M = = (cf. [3]). In particular if g(s) = &(s) =s~ we get T h e o r e m 2. We have Sx = /.?l?0ir,)(£') and Tp — Li;u-o ilJ(£) with equality of yq P) norms, where \q p, ( °° , \ — o / <i) (11) F(z0, zi) = ( / (zS + / p z ? ) - ( , / p ) ( a ( 0 ) " " f j and -(12) C ( z 0 , z 1 ) = ( / ( z 0 - ' + r i z r ' ) - ( W , ) ( / » ( 0 ) , ' y ) '• E x a m p l e I. If a(t)=P(t) = t-° ( 0 < 0 < J ) we get F(z0 , z,) = czj)-<,z'j, G(z0 , z,) where c and d are constants. P r o o f , i) We have to minimalize the expression 5» = / (II" (0II lcp0 (£) + /" || i/ (i) || (E)) (a (i))" -y- = 0 ' oo o x 1 where a= j u(t)dtjt. We claim that it is sufficient to consider u(t) of the form (f (t) a wi = 1, cp(t)^0. Indeed given any nil) let us set . o cp(t) = ll«(0ll£ / I W O M ' / ' then 258 J. Peetre CO Then f <p(t)dt/t=\, (p(t)^O and moreover | |(p(t)a\ \Em\ii( t ) \ \ , : so that o which proves the assertion. Thus restricting ourselves to the case u(t) = (p(t)a we obtain after a change of the order of integration I" = I I (?(t)Y(Co + f ^ M O Y ^ - WaWldp. x o The problem is now reduced to minimizing (for each x£X) the expression oo /(<p(0)"(Cpo + f p C ? ) ( « ( 0 ) P T 0 ' where J cp(t) dt/t = 1, (p(t)^O. Choose o <p(t) = Ci))'(CS + t>tiri1,p\a(t))-<; g v = / ( / • (Co , -C i ) r ( c s + ^ c f ) 1 " ' ( « ( 0 ) ' w + P Y = № , C i ) r " ' = 0 so that min S ^ F i U , ^ ) ) " -On the other hand, using HOLDER'S inequality № , Ci))p = (F(Co, C0)p ( / 9» (0 (CS + ipC?)1 /p«(0(CS + / p C ? ) ~ ( 1 / P V ( O r 1 y ) s ^ ( f g o , C0)p / (<p(0)p(Cpo 4- i 'CiXaCOr-^-^CCo, i,))~p- = g*, 0 which finishes the proof. ii) We have to minimize the expression X„,„, = / ( l |y0(0l lP f o (E )+iP | l^ (0l |PLf 1 )E ) ) ( i3(0)P^ = 0 OO = JJ(eo\\vo(t)\\pE+tptnvi(t№dp(my^ ox where a = voity+v^t). We claim that it is sufficient to consider v0(t) and v,(t) of the form i/^CO« a n d « M 0 a w i t h 'AoW + ' M O = h ^oiO ^ ( i ) SO. Indeed given u0(i) and v^ t ) let us set . J ; 0 ' ' ! 1 1 - , , • # . « ) - — ^ ^ l » o ( ' ) I . + I » . ( ' ) l l i ' b o O l i + l ! » ! « ! ! , ' On an interpolation theorem 259 Then iA0(0 + '/ ' i(0 = 1= "AoWsO, i M O ^ O and moreover IIil/0(t)a\\E^\\v0(t)\]E , II"AICO^HE—H^ICOLLE so that which proves the assertion. Thus restricting ourselves to the case v0(t)=\l/0(t)a and vi(t) = xj/1(t)a we obtain after a change of the order of integration = ¡J ({Mt)y{po+(t>(t)y>pti)(my^-MpEci». X 0 The problem is now reduced to minimizing (for each x£X) the expression 0 ' where i/'oW + 'AiCO = 1> •/'oW —0> "MO —0 fr°m which the result in this case (see (12)) easily follows as in the preceding case. Combining theorem 1 and theorem 2 we get The o r e m 3. Each of the functions F(z0, z¡) and G(z0, z as defined by (11) and (12) is an interpolation function of power p. E x a m p l e 2. By example 1, with O < 0 < 1 is thus an interpolation function of power p for any p. This leads to the interpolation theorem of STEIN and WEISS [4]. R e m a r k 2. It is easily seen that the condition provided by (11) is essentially the one found by FOIA§ and LIONS [1]. The only significant difference is that these authors allow (a( t )~ q dt / t to be replaced by an arbitrary positive measure d£, (not necessarily absolutely continuous with respect to dt/t). It should be possible to extend our approach to cover this generalization too. § 2 . We conclude by pointing out some relations between the functions F(z0,z and G(z0 , Zj). Since they are both homogeneous of degree 1 it suffices to consider the functions f (z) = F(z, 1) and g(z) = G(z, 1). We have then after a change of variable / ~ , \-(l/9) (13) / ( z ) = z (1 + f ^ « ^ ( « ( r z ) ) -and (14) g(z) = *(/(»+' -«r^di{<?))»-Let us consider the special case a(t)=fl(t). By HOLDER'S inequality we obtain or (15) 5 t /00 f ( z ) = Cg(z) 86 J. Peetre where C is a constant, 0 < C < « . Assume next that a satisfies (10) where (16) Then we get so that (17) Aza(z) ^ f ( z ) where A is a constant, 0-</l-<<*>. Assume again that fi satisfies (10) where dt (I + t-")-^""(a{t))p o (18). /  t~") lpl">((r([))" -Then we get f (\ +t-«)-W»\(i(tz))>> — =§/ (1 -j- t'")~<"/"»(a(t)Y — (fi(z))'' ô f o 1 so that (19) g(z)^Bzfi(z) where B is a constant, 0 T h e r e f o r e in the special case a(t) = [i(t). ¿>(f) = = <r(t) assuming also (16) and (18) we get by (15) (20) A za (z) =S/(z) ë Cg (z) â CBzfl (z). (Note that A^CBV) In other words the functions f(z),g(z) and za(z) are here equivalent. Finally we make a few observations concerning the conditions (16) and (18). We note that, since all functions of the form (1 + /'')'•'" are equivalent, they may be replaced by the conditions dt < oo /M'-TMtFT and, after a change of variable, We note also that in view of HOLDER'S inequality if one of these conditions holds for two different values of the parameter, p or q, than it holds for all intermediate values. On an interpolation theorem 261 We may sum up these results as follows. T h e o r e m 4. Assume that c/.(t) satisfies (10) where (23) / min {l, 1 } e ( j ) sup min {l, -}} Q ( j Then z0a(z0/z,) /.? equivalent to an interpolation function of power p for any p. R e m a r k 3. Conditions of the type (23) arose in a similar context in [3]. References [1] C. FOIA§ and J. L. LIONS, Sur certains théorèmes d'interpolation, Acta Sci. Math., 22 (1961), 269 -282 . [2] J. PEETRE, Nouvelles propriétés d'espaces d'interpolation, C. R. Acacl. Sci. Paris. 256 (1963), 1424-1426. [3] J. PEETRE, A theory of interpolation of normed spaces. Notes, Universidade de Brasilia, 1963. [4] F.. STEIN and G. WEISS, Interpolation of operators with change of measures, Trans. Amer. Math. Soc., 87 (1958), 159-172. (Received August 20, 1963) 

On an interpolation theorem of Foiaş and Lions

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On an interpolation theorem of Foiaş and Lions

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