Interpolation of linear operators in Calderôn-Lozanovskii spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

Mieczyslaw Mastylo

(Poznan)

Interpolation of linear operators in Calderôn-Lozanovskii spaces

Abstract. There are given necessary and sufficient conditions under some assumptions on the symmetric couples ^X and ^Y, that the symmetric spaces ^{( p ( X )} and i^{j / ( Y )} intermediate with respect to (<^{p 0 ( X}), q>x ( X ) ) and ( ф 0 { ? ) , ф х { ? ) ) , respectively, are positive interpolation with respect to (<p0 ( X ) , (px ( X ) ) and ^{( ф}0(F), ^фi (?)). In special case we get that q > ( X ) and <p(F) are positive interpolation with respect to (<p0(X), ^<px (X)) and (<p0(Ÿ), ^{( px} (Ÿ)), respectively, iff < p ( u , v )

~ в ( ( р 0 ( и , v) , v)) with some function 0e W.

1. Introduction. Let A

and A x be two Banach spaces. We say that A = { A 0, A X) is a Banach couple if both A0 and A x are continuously embedded in some Hausdorff topological vector space sé.

A Banach space A is called intermediate with respect to A if A0 n A x cz A c A0 + A x with continuous embeddings. Let A and В be two Banach couples and let T be a linear operator mapping A0 + A x into B0 + Bx. We write T: A -*■ В if the restriction of T to A{ defines a bounded linear operator from Ax into Bb i = 0, 1.

Let A and В be two intermediate spaces with respect to  and B, respectively. We say that A and В are interpolation spaces with respect to A and В if every linear operator T such that T: Я-+ В maps A into B. If  = В and A = В we simply say that A is an interpolation space with respect to A.

Proposition 1

(see

p.

Let A and В be interpolation spaces with respect to A and B. Then

II ^ c max П1 \\T\\

1-+

1}

for each T: A -* В with some constant c > 0 independent of the operator T.

2. The Calderôn-Lozanovskii space (p(X). A real function <p\ [0, oo) x x [

, oo) —» [

, oc) will be said to belong to the set if it satisfies the following conditions:

(i ) (p(Às, At) = X(p(s, t) for each and s, t >

(positive

homogeneous),

(ii) О < (p(s, t) ^ max [s/и, t/v) q>(u, v) for each s, t, u, veR+ — (0, oo).

% denotes the set of concave on R+, positive homogeneous functions ip: [0,

x[0,

-> [0,

We observe that ^ c fy.

R em ark 1. If ^ c= is a set of functions such that the function f( s , t) = inf (p(s, t) is positive for some s0, t0 >

, then / e °H.

Фе'в

Let (Q, X, p) be a complete cr-finite measure space and let us denote by L° = L°(Q, I , ц) the space of all equivalence classes of ^-measurable, real valued functions finite p-almost everywhere on Q, equipped with the topology of convergence in measure.

A Banach subspace X of L° is said to be a Banach lattice if x e L ° , y e X and |x| ^ |y| p-a.e.

"N imply

x g

X and \\x\\x ^ ||y||x.

Let A" be a couple of Banach lattices and let the function ipeflt. Then we denote by cp(X) = (p(X0, X ^ the space of all measurable functions xeL ° such that

\x\< X(p{\x0\, |хх|) p-а..e.

for some X > 0 and for some

.

X,-, with ||х,-||х. ^ 1, i — 0, 1. We put ||хЩ*)

= inf X.

We note that (p{X) is a Banach lattice intermediate with respect to X.

If we take (p(s, t) = s

- a f“, 0 < a < 1, we obtain the space Х1~ЛХ\ intro­

duced by Calderon [1]. The space (p{X) was investigated by Lozanovskii in [3].

Proposition 2.

Let X be a couple of Banach lattices and let (p0, (Pi, <pe #. Then

ф{Х) = ср(ср0{Х), cp^X))

with equivalent norms, where ij/(s, t) — q)((p0(s, t), (pi(s, f)).

P ro o f. We observe that ij/eW. If хеф(Х), then |x| ^ Xij/(\x0\, IxJ) a.e., for some A > 0 and for some x,e X h ||х,||х. ^ 1, i = Q, 1. Hence

|x| ^ X(p(y0, y j a.e., where yt = (pf |x0|, |xj|), ||у,|Ц(Х) < 1, i = 0, 1. This implies that xeq>((p0{X), (pAX)) and ||x||vtvo(Jp)iVl(jp)) ^ ||x | | ^ , whence {J/(X)cz (p((p0(X), (Pi(X)) with continuous embedding.

On the other hand, let xe(p(<p0(X)> Then |x| ^ Xip{|x0|, IxJ) a.e.

for some X > 0 and for some xfe<p,(X), ||х{||^ (^ < 1, / = 0, 1.

For an e > 0 there exist y0, y'0e X 0, y v, such that

|x0| ^ ( i + e ) ( p 0{\y0\, ly’il), ||y

llx

^

» I M l*! ^ b

l*il ^

+e) (Pi (|y'0|, l/il), ILV

II

< 1, \Ш\х1 <

so we have

M < A<p(jx0|, l*ll) ^ (l+fi)A<j!>(<j!>

(|y0|, lyj), (p! (|Уо|, l/il))

< 2{l+£)X(p((p0(x'0, x'J,

, x'l)), where

$ = imaxdy,!, |y,'|)e X{, ||x;||*. <

, i =

,

.

Hence хеф(Х) and ||x||*(Jf) ^ 2(1 + e )|ix ||^ o(Jfbvl(jr)). Since £ is an arbitrary positive number, we obtain \\х\\ф(Х) ^

||x||v(vo(jP)^ l(jP)) ; this implies (p((p0(X), (рх (X)) c= ф (X) with continuous embedding and the proof is complete.

Let E, F be Banach lattices. We say that a linear operator T: E -> F is positive if 0 ^ Tx a.e for each 0 < xeE.

Let X and Ÿ be two couples of Banach lattices and let X, Y be two Banach lattices intermediate with respect to X and Ÿ, respectively. We say that X and У are positive interpolation with respect to X and Ÿ, if every positive operator T: X -*■ Ÿ maps X into У. If X = f and X — Y we say that X is a positive interpolation space with respect to X.

Proposition 3.

Let X and Y be two couples Banach lattices. Then tp(X) and tp(Y) are positive interpolation with respect to X and Y.

P roof. Let Г: X -* Y be positive and let xe(p(X), so |x| ^ X(p(\x0\, l*il) a.e., for some A > 0 and .\', e X

||x(||x. ^ 1, i = 0, 1. Hence

!7x| s: T\x\ lT<p(|x0|, |x,|) « ATmaxi — , — \<p(u, v)

( U V J

^ 2

m ax\ ——° - i(p(u, v) a.e.

I и v j

for each u, v > 0. Consequently,

\Tx\ ^ 2X(p{T\x0\, TlxJ) = 2KXq>(y0, yx), where

К — max 'll Л1х

-у0,

^

-

J» Yi = T\Xi\E and

W ï ^ l . = whence

TxE(p{Ÿ) and ||73c||,(f)< 2 X ||x ||^ ).

R em ark

. Sestakov has shown in [9] that for x0e X 0, x 1e X li <pe#

and a positive operator T : X 0 + X x ~+ Y0+Yx, we have T(p(\x0\, (xj)

< (p(T\x0\, TlxJ) a.e., this inequality implies ||75c||v(y) ^ К ЦхЩ*, for each xeq> (X).

R em ark 3. Lozanovskii has given in [4] an example of couples of Banach lattices X and F such that the spaces Xo~a XI and У01_а F® are not interpolation with respect to X and F for each а е (

,

).

By Proposition 2 and 3, we get the following

Co r o l l a r y

1. Let X, F be two couples of Banach lattices and let tph фх, (pe$, i = 0, 1. Then the spaces (p((p0, (pi){X) and (р{ф0, *Ai)(F) are positive interpolation with respect to (q>0(X), (px{X)) and (ф0(У), фi(F)).

We say that a Banach lattice X of L°(Q, I , p) has the Fatou-property if supUxJI* < oo imply x = su p x „eX and ||x||* = sup||x„||x .

1

n

n>

Th e o r e m

1. Let X, Y be two couples of Banach lattices. I f Y0, Yx have the F atou-property and (ph фь (pe / = 0,1, then (p(<p0, (px){X) and

<р(ф0, «A

)(F) are interpolation with respect to (q>0(X), tpx (X)) and (ф0(? ), ф,(?)).

P ro o f. The spaces ф0(У) and ^ i(F ) have the Fatou-property (see [3]).

By a result of Ov&nnikov (see [7]), the spaces q>(<p0{X), (px (X)) and (р(ф0(Т),ф j(F)) are interpolation with respect to ((p0(X), (px (X)) and (ф0(?),ф i(F)).

Consequently, by Proposition 2, we get Theorem 1.

Pr o p o s it io n

4. Let tp0, <plt (peûU, ф0, фх, фе<% and let c be a positive constant. Then the following inequality

(1) <p(u, V) < \ (p0(u, v) (pi (и, v)

ф(э, t) ^ C maX(iAo(s, t) ’ фх (s, t) for each s, t , n, veR+

holds if and only if (p(u, v) ^ cx 6{(p0(u, v), (px{u, y)) and ф{и, v)

^ с20(фо(и, v), фх(и, v)) for some function d e W and some constants cx, c2 >

.

P ro o f. Let inequality (1) hold. Then

(p(u, v)

cf((p0(u, v), (px(u,

where

f(Ç,rj)= inf ф (s, t) max

s,t> 0

t ' r, \ Фо(э, t y ф

(s, t)J By Remark 1, we observe that f e <%.

Let

0(L V) = inf f{s, t)(£/s + ri/t).

s,t > 0

Then

0 6 * and f ( t ,

) 0({, ч )< 2 /« , ,), whence

<?(w,

) ^ сО(ф0(и, г), (м,

)) and

«А(м, У) ^ ?в(фо(и, v), ф ! (u, y)).

The converse inequality follows from the inequality 0(s, t) < max{s/n, t/y} 0(n, y) for each function OeW.

Theorem 2.

Let X and Ÿ be two couples of Banach lattices and let

фь (p, IД е # , i = 0, 1, fee such that

for each s, t, u, veR+ and some constant c > 0.

I f the positive operator T: X0 + X x -* F0 + Yl is such that T: ((p0(X), q>l (X))-> (ф0{7), ф1(Т)), then the operator T is bounded from (p(X) into ф(У).

P ro o f. Let inequality (2) hold. Then, by Proposition 4,

<p(u, у) ^ c! 6((p0(u, y), q>x{u, y)) and ф(и, v) ^ c20(<M«> v), ^ ( u , y)) with 0e ?/ and cb c2 > 0. Let хеср(Х). Then, by Proposition 2, xeO(q>0, q>i)(X)

= 0(q>o(X), cpx (X)). Now, if the operator T: + У

is positive and T: (<p0(X), (pi(X))-+ (ф0(?), фх(¥)), then, by Corollary 1, T is bounded from в((р0, (px){X) into в(ф0, фi)(F), so that

Consequently, the operator T is bounded from tp(X) into ф{ У).

By Proposition 4 and Theorem 1, we obtain

Corollary 2.

Let Y0 and Yx have the Fatou-property and let inequality {2) from Theorem 2 hold. I f the linear operator T : X0 + Xj -> T0+ is such that T : ((p0(X), (pl {X))-^ (ф0(?), ФА?)), then T is bounded from (p(X) into

Ф ( П

Corollary 3.

Let X and Y be two couples of Banach lattices and let

<Po, q>i, (petft be such that

№ I U ^

II 73с|1в(^

,^

)(У) ^

^ с

(|^о(Г),^

(К))

< С ъ 11 ^ 1 16>(«P0 (^),< p ! (Xs)) < C A |W l e f o , 0 ,V l ) (* )

^ c 5 M U * ) -

(p{u, V)

(p(s, t) ^ c max

for each s, t, u, veR+ and some constant c > 0. Then the spaces (p(X) and (p(Ÿ) are positive interpolation with respect to <Po{X) and (pl (Ÿ).

P roof. It follows from Proposition 4 and Corollary l.

R e ma r k 4. Let (p(s, t) = min [s, t\, ijj(s, t) = s + t. Then for each (pi% ij/h Oe %, i = 0 , 1 we have cp(s, t) ^ cx 9((p0(s, t), (px(s, t)) and i)/{s, t)

^ c20(i//o(s’ 0, ^i(s, t)). If the couples of Banach lattices X, Ÿ and the functions <px, / = 0,1 are such that Ф

n cp1 (ЛГ) and ф0_(?) + ф_1( У ) ^ У 0+У1, then <p(X) = X0 n X xf (p0(X) n q>x (X) c 0((po(X), <РЛЧ 0 (Ф

{У), Ф Л Г ) ) ^ ф 0(У) + ф1(У)Фф(У) = То+П. This implies that, in general, inequality (2) from Theorem 2 does not imply the spaces (p(X) and ф(Т) to be intermediate with respect to ((p0(X), (px(X)) and (Фo(F), ^i(F)), respectively.

3. The symmetric space cp{X). In the sequel let L° = L°(R+, X, m), where m denotes the Lebesgue measure defined on the <T-algebra I of all Lebesgue measurable subsets of R+.

A Banach lattice is said to be a symmetric space in the terminology of Semenov if xeL°, y e X and x* < y* imply x e X and ЦхЦ* ^ llyll*, where x*(t) — inf{A > 0: m\seR+: |.v(s)| > X] % t}. The fundamental function ФХ(Г) of a symmetric space X is defined for t > 0 as Фх (t) = ||X(o,olfx» where X(0,r> is the characteristic function of the interval (0, t).

The Lorentz and Marcinkiewicz spaces Л(Ф), М(Ф) are defined by the formulas:

where

Л(Ф) = {xeL°: |М|Л(Ф) = М(Ф) = [xeL°: ||х||л/(Ф) —

[ x*(s)dФx (s) < oo },

sup (Фх (s)x**(s)j oo|,

t

*

x* (.s) ds,

0

t > 0.

If X is a symmetric space, then

(3) 1МЦф) < M x for each x e X (see [2], p. 162).

Proposition 5.

Let X 0 and X x be symmetric spaces with fundamental functions ФХо, ФХх, respectively, and let (pefl. Then <p(X) is a symmetric

space and

(4) V(X)<= М(П (ФХо,ФХ1))

with continuous embedding, where <p*(u, v) = l/ç)(u~1, v~f .

P roof. By Lemma 4.3 in [2], p. 127, we obtain easily that <p{X) is a symmetric space. Let xecp(X); then |x| ^A<p(|x0|, (

I) a.e., for some A> 0 and for some х(е Х ^ Цх.-Цд. ^ 1, i = 0, 1. First, we observe that

c/>(x$*(r), xî*(t)) ^ i<jp(|A-0|, |*il)**(0, t > 0.

Hence, by inequality (3), we get

x**(s)(p*(<t>Xo(s), ^ j ( s ) ) < 2A, s > 0 . Consequently,

whence we obtain (4).

Corollary 4.

Let X 0 and X t be symmetric spaces with fundamental functions <PXq, ФХ1, respectively. Then

P ro o f. Let x;(s) = X(o,о(з)/фх{(*), 1 = 0> L Then ||х4||*. = 1 and *(0,0(s)

= <P*(&x0(t), <PXl{t))<p(x0(s), Xi(s)). Hence Ф<р(Х)(0 ^ <P*(^Yo(0>

The converse inequality follows from Proposition 5.

Proposition 6 (see [5], [8]).

Let X and Ÿ be couples of symmetric spaces, and let the symmetric spaces X and Y be intermediate with respect to X and Ÿ, respectively. I f X and Y are positive interpolation spaces with respect to X and

Ÿ, then there exists a constant c > 0 such that the following inequality

фг (0 f< 4 W

Ф*(з) ^ C maX| ^ * 0(s)’ Фдг, (S)J holds for every s , t e R +.

P ro o f. This proposition follows from Proposition 1 and from the fact that the operators Ts t: X Y defined by

Ts,t x (-) = - \x(u)duX(o,t){-) for s , t e R + *

have the norm im i * ^ = <Mf)/<Ms).

Indeed, by (3) we have

s x*(u)du<PY(t) < ~ \ ~ Ы Х.

&x(s)

For x — X(o,S), l|7^>f^(o.s)lir — Фу(0- Thus ||7^г|!*_у — Фу (0/Фх (s).

Theorem 3.

Let X, Y be couples of symmetric spaces. Suppose that {&x0/&xl){R+) = {Фу0/Фг1Н^+) = Я+. and suppose that (рь фь (p, ф е % i = 0, 1 are such that the spaces (p(X) ami ф{У) are intermediate with respect to ((p0{X), <Pi(X)) and (ф0(F), iAi(F)), respectively. Then (p(X) and ф(Т) are positive interpolation with respect to ((p0(X), (px (X)) and (ф0(У), ф± (F)) respectively, iff

(5) <p{u,v)^ )(po(a,v) q>!(u, !>)(

Ф( S, t ) ^ CmdXU o { s, t) ' ФЛ8, t) J for some constant c > 0 and for each s, t, u, veR+.

P ro o f. The necessity of inequality (5) follows from Proposition 6 and Corollary 4. On the other hand, if (p{X) and ф(У) are intermediate spaces with respect to ((p0(X), (pt (X)) and (*/^0(F), i/^1(F)), respectively, and inequality (5) holds, then by Theorem 2, we obtain our assertion.

Corollary

5. Let X and F be such as in Theorem 3 and let <p0, tpx, cps fyt. Then tp(X) and tp(Y) are positive interpolation with respect to (<p0(X), (p^X)) and (<p0(F), <MF)), iff (p(u, v) ~ в((р0{и, v), (px(u, v)) with some function в e J//.

R em ark 5. I f the symmetric spaces Y0, Yx have the Fatou-property and if the conditions from Theorem 3 hold, then tp(X) and ф(Т) are interpolation with respect to ((p0{X), (Pi{X)) and (ф0(Т), ^i(F)) iff inequality (5) holds.

Corollary

6. Let X, F be couples o f symmetric spaces such that (Фх0/Фх1)(К+) = Я+, (Фу0/Фг1)(Я+) = *+■ Then X t and Yh i = 0,1, are not positive interpolation with respect to (X o r^X i, X 0 + X x) and (YonYj, Fo+Fi), respectively.

P ro o f. Let (p0{s, t) = min [s, t], (Pi(s, t) — s + t, q>(s, t) = s. Then (p0{X)

= X 0 n X lt (pAX) = X 0 + X l9 <p(X) = X 0, (Po(Y) = Y0 n Yl9 <px{Y) = Y0+ Y1 and (p{Y) = Y0 with equality norms; moreover, inequality (5) does not hold.

Consequently, by Theorem 3, we get our assertion for X 0 and Y0. For X x and Yj the proof is similar, in this case we take (p{s, r) = t.

4. The Lorentz-Orlicz space Л (M, Ф). Let the positive concave function

Ф on R+ be such Ф(0+) = 0. Let M denote an Orlicz function (M is convex function M: [0,

-» [0,

such that M(0) = 0, M(t) > 0 for t > 0 and

lim M(t) =

We define the functional

, on L° by the formula:

t OO

x

q(x) = f M ( x * ( s ) ) d Ф ( s ) .

b

The functional

is a convex modular on L° in the sense [6]. Let ( L \ be the modular spaces

{ L \ = \xeL °: lim g

= 0}.

E- * 0 +

We easily verify that the modular spaces concide with the Lorentz-Orlicz space

A(M , Ф) = {x e L ° :

(

/

) < oo for some Я > 0}.

The space Л(М , Ф) is a symmetric space on R+ with the norm IWU

.*) = inf {A > 0:

{

/A) ^ 1}.

Let us remark that Л(М , Ф) has the Fatou-property. In the sequel, M ' 1 denotes the inverse function to the function M.

Proposition 7.

Let M be an Orlicz function on [0,

and let Ф: R+-*R+ be concave with Ф(0 + ) = 0. Then ф(Л(Ф), Z/30) = A (M , Ф) with equality of norms, where

q>(s, t) = 0

tM ~1 (s/t)

if t = 0, if t > 0.

P ro o f. Let хеф(А(Ф), L°°). Then |x| ^ Яф(|х0|, IxJ) a.e. for some Я > 0 and for some х 0еА(Ф), x ^ L 00, with ||х0||Л(ф) ^ 1, llx JI^ ^ 1. Hence

|xj <Яф(|х0|, 1) a.e. This inequality implies M |x|

Я (t) = M 'x^(t)

Я

and

thus

d0(t) ^ ||х0||л(ф) ^ 1»

.L00)’

Conversely, let х еА (М ,Ф ); then there exists а Я > 0 such that

J M (x*(t)/A)dФ{t) ^ 1, whence x0 = М(|х|/Я)е А{Ф) and ||х0||д о ^ 1 and о

|x| = Яф(|х0|, 1) a.e., so хеф(Л(Ф), L00); moreover, I M I^ ^ o o , ^ The proof is complete.

By Remark 5 and Proposition 7, we obtain the following theorem:

Theorem 4.

Let the functions Ф,: R+-* R+ be concave with Ф,-(1!+) = R+

and let M, N, Mh N ( be Orlicz functions on [0,

i = 0, 1. Suppose that Х = А (М ,Ф 0) and У = Л(1У, Ф^ are intermediate spaces with respect to Х = (Л(М0,Ф 0),Л (М 1,Ф 0)) and ? = (Л(1У0,Ф 1_), Л ^ Ф ^ ) , respectively.

Then X and Y are interpolation with respect to X and Y iff for some c > 0 М _1(0

N ~ l {s) c max Mj 1M M r‘w ]

N i l ( s ) ’

JV0- ‘ (s)J

for each s, te R +.

By Theorem 4, we obtain

Corollary 7.

Let Ф,: R+-+R+ he concave with Ф, (/?+) = R f and let Mh M be an Orlicz function on [0, oo), i = 0, 1. Then the Lorentz-Orliez spaces Л (М , Ф0) am/ Л(М, Ф^ are interpolations with respect to (Л(М0, Ф0), Л(М 1? Ф0)) am/ (Л(М0, Ф^, Л(М 15 Фх)), respectively, /jf

^br some concave function 0: R+ —> R+.

R em ark 6. In the special case Ф,(Г) = t, / = 0,1, Theorem 4 was obtained by Pustylnik in [8].

Problem.

Let X and Ÿ be couples of symmetric spaces and let (pef/.

Are always the spaces (p{X) and cp(Ÿ) interpolation with respect to X and Y, respectively?

Acknowledgements. This work is a part of the author’s dissertation prepared at the University of Poznan in 1984. The author wishes to thank Professor Julian Musielak for valuable comments.

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