The X-functional for a Banach couple (AEq;K, ÀEvK)Abstract. We give estimations for the X-functional for a Banach couple

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXV (1985)

Miec z y sl a w Ma s t y l o (Poznan)

The X -functional for a Banach couple (AEq;K, ÀEvK)

Abstract. We give estimations for the X -functional for a Banach couple {ÂEq.k , ÂEi.k) under some assumptions on the spaces E0 and E x. We apply them in the proof of the reiteration theorem (Theorem 2). The proof of this theorem was given by Gustavsson in [1], but our proof is different.

1. Preliminaries. Let À = (A0, A t) be a Banach couple, i.e., A 0 and A t are two Banach spaces continuously embedded in some Hausdorff topologi­

cal vector space с/. We define the Banach space

Aq~i~Ai = \q,e s$ \ a — üq-\~£Ii, ü q eAq, üi£ y 4 j |

with the norm

I M U o + A i = i n f { | | a 0 | L 0 + | | u i l U 1 : a0EA0, a^EAu a0 + at = a).

For each ü eAq + A^ and for any 0 < t < oo we define the К-functional of Peetre

K(t , a; A) = K ( t , a; A 0, A t)

= inf J||a0L 0 + f HfljLj: a0EA0, qxeA x, a0 + a i = a ) . Let L° be the space of real-valued Lebesgue-measurable functions on R+ = (0, o o ) (with equality almost everywhere) and let a subspace E of L° be a Banach function space such that m in(l, t)e £ . Then the space

^E-.K ~ (Aq> AiIe-k = {ae Aq + A^ : К ( •, a, A )e E]

is a Banach space with the norm

I M U — II^IIeja: — a -> A ) \ \e- E : K

For any measurable function /: R+ -> R+ we define a submultiplicative function M: R + -►[(), o o ] by the formula

M ( s , f ) sup 0 <t < 00

fist) f i t ) '

94 M. M a s ty lo

If M is finite and measurable or non-decreasing, then (by Lemma of Hille- Philips in [2], p. 241, the additive version) it is possible to define the indices of function /:

P o ( f ) = sup

0 < s < 1

In M( s , f )

In s lim

s->0 +

In M ( s , f ) In s

P 00( f ) = inf

S > 1

In M ( s , f )

In s lim

S~-> ^GO

In M ( s , f ) In s

We say that a function /: R + -+R+ belongs to a function class âSK if it satisfies conditions given below

(i) / is continuous and non-decreasing, (ii) M ( s , f ) < oo for every s > 0, (iii) f m in(l, s l) M ( s , f ) — < o o .

о s

We note that a function f belongs to âSK if and only if it satisfies (i) and

о < Р о ( Л < P o o ( / ) < ! •

Let F be a convex Orlicz function on [0, oo), i.e., a finite, continuous, convex and increasing function satisfying conditions F(0) = 0 and F(f) -> oo as t -* oo and let f e & K. The weighted Orlicz space L*f consists of mea­

surable functions x on R + for which lx(f)l Xf{t)J t о

À = Я(х) > 0.

The Luxemburg norm in L*f is defined by

is finite for some

||*4 = i n f { A> 0 : j f ( H

If E = L*f , then we write (A 0, A j ) ^ = Âf<F.K. When F(u) = up, 1 < p < oo, we write Af<p.K, and when f ( t ) = te, 0 < в < 1, we write shortly А в р.к .

Let / be a quasi-concave function on R +. The weighted space Щ consists of measurable functions x on R + for which

M u p f = sup

J t > 0

1^(01 f i t )

< 00.

If E = Щ-, then we write (A 0, A = Af a0.K. When f (t) — te, 0 < в ^ 1, we write AetaoiK.

N o t a t i o n . f ( t ) ~ g { t ) if and only if cx f ( t ) ^ g(t) ^ c2f {t ) for every t > 0 and some c t , c2 > 0.

The letter c will denote a constant which need not be the same in different cases.

XA denotes the characteristic function of a set A c Æ + .

2. Main result. We give estimations for the К -functional for a Banach couple (AEq.k , Â Ei;K) under some assumptions on the space E 0 and E x.

Th e o r e m 1. Let  be a Banach couple, let (Eh || -\\E.), i = 0 , 1, be Banach function spaces on R+ and let functions f \ Я+ -> R+, i = 0, 1, be such that

(1) I I W O 'A I Ie,. - llz[s,oo)ll£i - 1 / m , i = 0, l,

(2) ll/i Z(o,<5(f))tl£0 ^ ct’

(3) H/oXt^fbaojIUj ^ CA

for some increasing function Ç: R+ -»Я+ such that Ç(R + ) — R+. Then (4) K(t, a; Â Eq.k , A Ei;K) ~ 1!^(*, a ‘, a ’ Â) Xmt),co)\\Ei for every a e ÂEq.k + Â Ev<K.

P ro o f. Let

F(t, a) = ||K ( , a; 4 )x (0>(S(f))||£o + t||K ( *, a;

for every a e  Eo;K +  Ei;K and t > 0.

First we will show that F(t, a) ^ cK(t, a; Â Eq;K, ÂEi;K) for some c > 0.

If a = a0 + a1 is any partition of a e ÂEq.k + ÂEl.K with и, е Л £.;К, i = 0, 1, then F(t, a ) ^ \ \ K { - , a0; Л)х(0.$(0)||£о + ||К ( S ; Â) x(o,m)\\E0 +

4-t||K (-, a0; +t\\K(-, a^ ; Л );^ (0>00)||£1

= 7i + / 2 + / 3 + / 4.

Let us denote J { = ||аг||£..£ = ||X (-, at\ A)||£., i = 0, 1.

From the inequality

(5) K( t, a ; Â) ^ m in(l, t/s)K(s, a; Â) and (1), we get

(6) K (s, af ; Â) cJi f (s), i = 0, 1, for every 0 < s < oo. From (2), (3) and (6) we obtain

I l — ll-K( cii '•> A)X(o,ç(t))\\e0 ^ c^ i ll/i Z(o)(*(t))ll£o ^ C^ 1 h — f a0> Â) oo)l If j ^ tcJo\\fo ^ cJ0.

For I x and / 4 we have the obvious estimates : ^ J 0 and / 4 ^ fJi- Thus F(t, a) ^ c(|ja0||£o;A: + f 11^11^.*). If we take infimum over all decompositions

96 M. M a s ty lo

а = üq + ü!, where а ,е Л £..£, i = 0, 1, we have (7) F(t, a ) ^ c K ( t , a; A Eq;K, AEi;K).

To show the converse inequality, let us note that from the definition of X-functional for the Banach couple A we get: for all t > 0 there exist a0( t ) e A 0, such that a = ao(0 + a i( 0 anc* llao(f)IL0 +

+*lk*i(0IL1 < 2 K (t>a ; Â).

We now define a'0 and a\ by a|(t) = a,-(£(£)), i — 0,1. Then a'0 + a[ = a and

(8) K( s , a'0(t); A) < | K ( 0 I L o = ||«o(«(0)|Uo 2К (4(f), a; A), (9) K (s, a\ (t); Â) s |K (OIL, = s | h (4(0)|L, Ы (0~ ‘ Щ (0, a; Â).

By the triangle inequality we get

(10) K(s, a'0(t); Â) ^ K(s, a; Â) + K(s, a[(t); Â), (11) K (s, a\ (t); À) ^ K (s, a ; Â) + K (s, a'0(t); Â),

(12) K( t, a; ÂEq:K, A Ei:K) < ||K (s, a'0(t); T)||£o + r||K (s, ai(t); Â)||£l

^ ||K (s, a'0(t); Â )/(0^(t))(s)||£o + ||K(s, a0(t)‘, ^4)X[^(t)>00)(s)||£o4- + £||K(s, a\{t)-, Л)х(0.«о)WlUi + r 11^ (s > ai ( f); ^ )fe ),ao )(s)|k

= Г1 + Г2 + Г, + Г4.

Let

T<o = 11^(s, ai >f)X(o>(*(t))(s)ll£0» L i = ll-K(s> • From (1) and (5) we obtain estimates

(13) K(s, a; Â) ^ cL0f 0(s) if 0 < s < £(£), (14) K ( s , a ; A ) ^ c L l f 1(s) if s ^ £ ( r ) .

From (9), (10) and (1) we get (15) = ||Х ( 5,а Ь ( 0 ;Я ) х <s>lk

=S ||K (s, a; ^)X(0,i(,))L)il£0 + ||K (s. a\ (0 ; <4)x(0,{(„,M ||t0

< z . o+ ||2 s 4 ( t r , K ( ê ( 0 .e ;^ ) z ,o A(t)) M l k

= L 0 + 2<i;(t) 1 K (£(t), a; ^4)||sx(0^(t))(s)||£o

^ L 0 + 2cL0 £ (t) 1/o(<^(t))l|5/(o,^(f))(s)||£0 ^ c L 0.

From (8), (11), (14) and (1) we get (16) /4 = ai W; ^4)X[ç(t),00) (*^)|!

^ f||K (s, a ; 4 )x K(f), x)(s)||£ l + t||K (s , flo(0 ;

^ tLi + t\\2K(Ç{t), a; 4 )x K<l)>oo)(s)||£l

< t Li +2 t\ \K( s , a\ ^ ) t o ) >00)(s)||El ^ ctLi, (17) Г2 = | j. K ( s , a 'o (t)‘,-4)x^(f),<»)(s)||£o

^ | | 2 - K ( £ ( t ) » я ; • ' 4 ) X [ ^ ( t ) , o o ) ( ^ ) | |e q ^ c L o M W 1 I I X [ 4 ( t ) ,о о)11е0

^ cLq,

( 1 8 ) /3 = f j | X ( s , a \ (t) ; Л) Х( о,<^j(^o) ( ,^s)||.^e1

^ t\\2sÇ(tyl K(Ç{t), я; ^ )x (o.€(o)(s)|Ui

= 2t Ç i t y 1 K(Ç(t), я; Â )||s/(0^(f))(5)||El

^ 2t£ (t)~1 c L 1 f i (<* (0) l|sz(o,«»)) {s)\\El ^ ct Lx.

Finally, from (15), (16), (17) and (18), we have (19) K( t, я; ÂEq.k , Â Ei;K) ^ cF(t, я).

From (7) and (19) we obtain (4).

3. Examples. Now we give examples of weighted Orlicz spaces which satisfy conditions (l)-(3) from Theorem 1.

Example 1. Let F, be convex Orlicz functions on [0, 00) satisfying condition A2 for all u > 0 (in short F ; e d 2) [6] and let f g , / = 0, 1.

We assume that the function т = f i / f 0 is an increasing and

(20) р о( т) >0.

Then, the spaces i = 0, 1 fulfil conditions (l)-(3) from Theorem 1.

P ro o f. Let E( = L^j., i = 0, 1. From Proposition 3.2 in [2] and Theorem 3.2 in [6] we get

(21) 0 < p0{ Fo f ) ^ Poo(Fo/) < 00, (22) - o o < Po(Fo 1//) < p 00(F o 1//) < 0 for F e A 2 and / e !%K.

Since t / f { t ) e 0SK, 1 = 0, 1, we get (by (21), (22) and Corollary 3, in [4], P. 80)

(23)

(24)

S

5

~ Roczniki PTM — Prace Matematyczne XXV

98 M. M astyto

for every s > 0 and for each A > 0. This implies that pC(0,s) (0 ^ II X[s,oo)ll£t-

M s ) ’ i = 0, 1

From (20) we obtain that x{R + ) = R + and that the function т has the inverse t^ 1. Let £(r) = T - 1 ( f ) , then Ç(R+) = R + .

It is easily seen, that 0 < p0(x) < Poo(T) < 1, hence т е if* and (by (23) and (24)) we get

l l / l X (0 ,t - 1(,))IIe o ~ ll ^ ° ^ [t ~ 1(i),oo)ll£ l ~~ ^

respectively. Thus, we obtain conditions (l)-(3) from Theorem 1.

Ex a m p l e 2. Let /„ / = 0, 1, be quasi-concave functions on R +, i.e., 0 < f ( t ) < m ax(l, t/s) f (s ) , and let the function x —f j f 0 increases, x{R + )

= R+. Then, Щ . , i = 0, 1, satisfy conditions (1)—(3) from Theorem 1.

From Theorem 1 and Example 2, we have

Co r o l l a r y 1. Let f be quasi-concave functions on R + , i — 0 ,1 . I f the function ^t =fi!fo increases and x{R + ) = R+, then

(25) K( t, a; A f Qt, :K,

о

K{s, a; A)

sup — ——— + t sup

s J o \ S ) s ^ r ~ 1(t)

K ( s , a; A) fi(s) for every a e Âfo^ :K + Afvoc:K.

In particular case we have Corollary 2.

(26) К (f, a ; Л 0х;А:, л К ) ^ ^v(t, a', Â) for every a e A 0^ :K + A X^ , K.

From Theorem 1 and Example 1, we get

Corollary 3. Let Fi e A 2 be convex Orlicz functions on [0, oo) and f e 0 S K, i — 0, 1. I f an increasing function x = f j f 0 is such that p0{x) > 0, then

(27) K(t, a; ÂfoFo.K, À fltF l.K)

l|K(*, a; À)x( O . t - i f r ) ) 1| ^ д + 1 ||Щ ^ ; / Г ) х [т_ 1(() J u 1 Vi for every a e A fo<Fo;K + AfltFl.K.

Corollary 4. Let /q, / ^ 1 < Po, Pi < oo, t/ie function x —f j f 0 be an increasing and p0{x) > 0. Then

(28) K(t, a; Afo<P0.K, A f l Pi;K) r ~ht)

K(s, a ; Â)IP 0 M s ) for every a e A f0tPo;K + AfliPi;K.

ds \ П^ро

+ t K(s, a; A)

fl(s)

~*P1 ds s

I/Pl

R e m a rk 1. In a special case if f 0(t) = te°, fi{t) = t°l, where 0 < Q0

<Ol < 1 , we have Theorem of Holmstedt (Holmstedt [3], Theorem 2.1).

4. The К -functional for Lorentz spaces. Let a Banach function space X be a rearrangement invariant (r.i.) space on R + , as defined in [4]. A fundamental function Фх of the r.i. space X is given by Ф*(0 = H^o.oIIa'- It was shown in [4], [7] that X can be equivalently renormed in such a way that Фх is concave. Therefore we can assume that Фх is concave. Let us put Pi(X) = р,(Фх), i = 0, oo.

The r.i. Lorentz spaces associated with r.i. space X are defined as follows Л{Х) = {a<=L°: ||а||л(Х) = | а*{8)с1Фх {8) < oo},

M ( X ) = |a e L ° : ||a||MW = ^sup y ^ J a*{s)dsJ< °° j>.

о Let 0 < P o ( * K P . W < 1, 1 ^ q < oo ; then

00

Ц Ф Х, q) = j aeL° : \\а\\ЦФхл) = Q [a*(s)Фх (s)]« y j < oo|, 0

oo

£ *(ФХ, q) = j ae L° : \\а\\^Фх,9) = ( J [ a * * ( s ^ x (s)]« y j < oo|, where

a**(s) = s ~ 1 j a*(t)dt.

Here, as usual, a* denotes the non-negative, non-increasing rearrangement of a which is equimeasurable with a, i.e.,

m {t e R + : \a(t)\ > y] = m [teR+ : a*(t) > y}

for all у > 0 (where m is the Lebesgue measure).

The space Ь*(ФХ, q) is r.i. space for each X and 1 ^ q < oo.

Lemma 1. I f f and p0(X) > 0, then ае(Л(Х), Т°°)/><г;К if and only if

Ç ds

C ( X , f ) : = [a*(s)<M s)((/o<PJf)(s))-1] , y < oo.

0

P ro o f. We assume that a e Л(Х) + L® and C(X, f ) < oo. Since фх 1(0

K{t, а; Л( Х) , L“ ) = J a* (s) <№,(*) 0

(29)

100 M . M a s t y l o

(see [7]), we get

фх 1(0

I f V dt

a*(s)d<Px (s)J — (0

о oo фХ 1(,)

<

(0

* 1 ( \

ds\q

a*{s)<Px (s) — \ — s J t

00 1

= | ( j ^ j j a* (0x 1( t)u)0x ( 0 x 1( t ) u ) ^ j J .

0 0

From the Minkowski’s inequality, we obtain 1 00

1 f i t )

(30) ^

о о

Putting s = ФхЧОи» we get from (30) 1 00

a*(s)

а*(Фх 1(^и)Фх {Фх 1^)и) q d t \ du t ) и

(31)

IMI/,9;K ^

s J и

о о

From the inequality ( ( / оФ х)(х/м)) 1 ^ M ( u , f o 0 x) (( f o 0 x)(s)) 1 for s, и > 0 and (31), we get

I\a\\%;K^ C ( X , f ) M ( u , f оФх)4 du

It is easy to show that

M ( u , f оФху — < oo,du

hence, \\a\\ftq.K < oo or ae(A{X),

Conversely, assume that ае(Л{Х), L®)/>e;JC, then from (29) we obtain K ( t , a ; A { X ) , L») ^ 1а*(Фх 1 (0)-

If we put s = Фх 1 (0, then from Theorem 2.4 in [7] we get

\\a \ \ } , q ; K

ta* (Фх 1 (г)) f i t )

q dt

^ m C ( X J ) ,

о

where m > 0. Hence, we have C ( X , f ) < со.

In the case, when X = L1, we have

Corollary 5 (Gustavsson [1], Lemma 3.1). I f f e then a e(L*, L®)yt<J;K if and only if

(32)

00/*

J0

ds I y < « -

Co r o l l a r y 6. L e r 0 < Po(Xi) ^ / ? * № ) < 1 , 1 ^ qt < oo, i = 0 , 1. I f an increasing function т = Фх0/Фх1 ls suc/z that р0(т) > 0, then

(33) K (t, а; Т(Ф*0, q0), (Ф ^ , q j ) ~ K (f, a; Т*(Ф*0, q0), Ь*(ФХ1, qt))

T _ 1(t)

0

[a * (s)0 ï o (S)], c -

° s

1/90

I + t ' f d s \ l/qi

[a* (s) **,(»)]“ y ]

t “ *(1)

for each а е Ц Ф Хо, <?0) + L(4>Xl, q j . P ro o f. First we can notice that

K(s, a; L1, L°°) = J a*(u)du о

for a e Û + L™ (see [7]).

By Corollaries 4 and 5 for а еЬ ( Фх , q o) + L ( 0 x , q j , we have (34) K( t , а; Ц Ф Хо, q0), № Xi, ?1)) ~ K(t, а; q0), L * ( 4 y , „,))

~ K(t, a; (L1, L CD)s/0Xo(s),qo;K, UA

r _ 1 0 ) s

Ф*о(*) s

\«i

T_ !(t)

— I0 + tI 1.

The function a* is non-increasing, J a* (u) du ^ sa* (s), so о

[a*(s) Фх0(«)]«о ds s J

\ 1/90

From the generalized Hardy’s inequality (see [5]), we get

л><

T - 1 (t)

\ / r d s \ l/9°

M ( s - ' ^ Xo) d s ) M [a*(s)#*0(s)], 0 y j ■

0 0

102 M . M a s t y l o

By inequalities 0 < р0( Х :) < p ^ i X J < 1, we get Фх. е ^ к , i = 0, 1, hence

1 CO

j M ( s ~ \ <PXQ)ds =

^

< P Xq)

о 1

The remaining term of (34) is treated in the same way.

In the particular case, when L(<PXj, q{) = L(rilPl, </,), г = 0, 1, 1 < p0

< pt < oo and 1/oc = l/po—l/Pi, we have the theorem of Holmstedt [3], Theorem 4.2 (by Corollary 6).

Corollary 7. Let be r.i. spaces with the fundamental functions Фх., i = 0, 1. I f the function x = ФХо/ФХ1 increases and x(R + ) = R+, then (35) K ( t , a ; M ( X 0), M (X,))

~ sup (a**(s)$Xo(s))+t sup (a**(s)4>XlU))

0 <s<t— s? t— h * )

for every a e M ( X 0) + M ( Xf ) .

P ro o f. Let  = (L1, L°°), /( s ) = s /0 x.(s), i = 0, 1. Since

5

K(s, a; Л) = J a*(u)du о

for all s > 0 and a e L 1+ L (X), we have

INI/,.,®* = sup (а**(5)Фх,(«)) = И1м(х{). * =0,1-

0 < s < oo

Taking r(s) = / 1(s)//0(s) = 0 Xo(s)/0Xl(s), we get (35) by Corollary 1.

Counterexample. We note that for E0 = L® n and E x = L^jt, con­

ditions (2) and (3) from Theorem 1 do not hold.

5. The reiteration theorem. To show the reiteration theorem we will use Corollary 4.

Let a function class consist of all continuously differentiable func­

tions / : R+ -+R+ such that inf ,

t>о f ( t ) = af > 0,

7 w " = b / " 1

Theorem 2 (the reiteration theorem). Let  be a Banach couple, f e M * , 1 ^ P, < oo, Bi = (A0, A x)fbP .'K, i = 0, 1. I f there is a constant m such that a function x = f j f 0 fulfils the condition

tr'jt)

X (t) ^ m > 0, (36)

then

(37) (B0, = (A0, A 1 )g<P'K

with equivalent norms for (ре.Аф, where g(t) = f 0(t)(p(z{t)), 1 ^ p ^ c o . P ro o f. Let 1 ^ p < oo, 1 ^ Pi < oo, i = 0, 1. The inequality

tg'(t)

implies, that деШф- Since Bt is of class CK( f , A), i = 0, 1 (see [1]), we get K(f, a ; Â) ^ K { t , a0; A) + K(t, A) ^ c(/o(OI|aoll/o>pjc+/i(O I|aill/1,p;K) for a e ( B 0, B ^ p . K, a — ao + ay with а,еБ,-, i — 0, 1. Taking infimum over all decompositions a0 + ax = а, а{е В ь i = 0, 1, we get

(38) K(t, a; A) < cf0(t) K(x(t), a; B).

The function т has the inverse t' 1 and x(R+) = R+. If we put r = r _1(s), then from (38), we get

(39) IM|gi, * ^ c K(x(t), a; B)~f J0

(p(x(t)) - 00

K (s, a; В p sx

tpis) x'

^

INI(B0.Bi)V(

The last inequality is a consequence of the following one

< J_

t ~ 1 ( s ) m From (39) we obtain that

(40) (Bo, ^l)<p,p:K <= (Aq, A^g p-x

with continuous embedding.

Conversely, let a e ( A 0, A f ^ p . K. From Theorem 3.2 in [6] we have [PoW, Роо(т)] <= [aT, b j .

From this inclusion and (36) we get p0(x) ^ ar ^ m > 0. Moreover, (ре,Аф

104 M. M a s ty lo

and t( R +) = R+, so by Corollary 4 we have

00 T-1 (f)

(41) IMUo.,,! W ~ ( j

* K{s, a; Â)

J fois)

P0 d s \ l'P0 у d t V 1"

- )

T J +

+ (p(t)

0 t -1(0

K{s, a; A)

Â(s) J

"тПтГ—'-

We will change the variables in 70 and in the following way: first we put s = x ~ 1(t)v and next t = t(m). Then

/о <р(т(и))

<p(t (m))

K(uv, a; A / o M

K(uv, а; Л) _ /о)"»)

P0 d » Y " ° y UT>) А Л 1"

t; / / t(u) mJ p° d v \ >IP0J d u y

have

1° If 1 < Po < p < oo, then according to Minkowski’s inequality we 1 00

0 о

1 00

0 0 1 00

K(uv, a; A) f 0(uv)q>(x{u))_

K(s, a; Â) J 0 (s)<p(t{s/v))

" d u \ Polp d v \ llP0

U J V J

” d j \ Polp d v J P0

K(s, a; A) 0 о

1

/o(s)(<P°t)(s)

d v \ llP0

M(v, (pот) ' d s Y 0"’ d v J P0

( f d v \ llPo

= у [M (p , <г>от)]Р0 — J M (Ao,AlttfP:K- 0

It is easy to see that (po i e ^ , hence,

C = ( [ l M { v , ( p o r ) Y ° — ) ° < o o ,

v /

о

and

<42>

If 1 < Pi ^ p < oo, then

h

1 о

ao ao

t(u) К iuv, a; Л )!p difv'’1"’ dv jp(xiu)) f i (uv)

- U У^{I V}

ris/v) К is, a; Л у \ P' IP dv

_(p(zis/v)) A(s) s ) V

1 0 00 00

<

1 b)0

A/fl - 1 T \ K{s, a; A) M l v \ 1

q> от J f 0{s) (p(x{s))

p d f j llp d v J Pl

— С ||я||(ло,Л1)0.р;К’

Let us note that for a function F(s) = ^t(s) / (ç>о т) (s) , we have

(43) ^ =

(>0,

F (t) t(t) \ <p(s)/

where s = r(f). From (36) and (43) we obtain tF'{t)

(44) F{t) > m ( l - b v) > 0.

From (44), we get p0(F) ^ aF ^ m(l -b ,,) > 0, hence, l

С = [M (u, т/<рот)]^pi dv v

i/p i

< 00

Consequently, we get

<45> / . « C | M | (, 0. , l W . From (41), (42) and (45) we obtain

( 4 6 ) i ^ o , A l ) g , p ; K ^ ( ^ 0 > B l) < p ,p ;K

with continuous embedding.

2° If 1 < p ^ ’p0 < oo, then 0 < p0(/o) < Poo(/o) < 1, since / о ^ ^ .

106 M . M a s t y l o

Hence, for 0 < s ^ 1, we get

К {uv, a; A) "d v r K{uv, a; A) uv

fo( w) » uv f 0 {uv)

p dv v

(К (us, a ; A ) \ p f / uv \ p dv ^ 1 fK{us, a; A)

^ 1 / J Vo(^)y U /о(

)

0 ns

that is (47)

which implies

К {us, a; A) ^ C J j/p/ 0(ns),

K{uv, a, A) / o W

PO dvj p 0

1 ^

= ( J fo(»v) P0(K (uv, а ; Л))"0 p(K(uv, a ; A j f ~ )

« ( |/ o M " ' ,0(K («i>,a;i)y’(CJj"’/ 0(«t>)f0' ,' * j ° = C /0.

From the inequality 00 _, 1

, «

K{uv, a; A)

u

P0 dv \ plp° duY'*

v j 7 '

0 0 we obtain

00 1

(4 8 ) I o ^ C \

= C

= c

Г / Г K{uv, a; A)

и

0 о 1 00

К {uv, a; A) _/o (uv) <p (t (n))_

0 0 1 QO

ivr

JU

p d v \ d u \ l/p v J и J

p d u \ d v \ l/p

и J v J

p ds \ dv\ 1/p s J v J

о о

1 00

^ c

p ds^j d v ^ p 0 о

1

d v \ l^p

= C( I [M(v, (pox)Y — \

о о

= C M \ {Ao,Al)(l'P:K.

If 1 ^ p ^ Pi < oo, then

K (s, a; A) g(s)

d s ^ j ,p

Ji = K(uv, a ; A) ]p dv f i (м у)

l

>( K( us, a ; Л)У

K(uv, a; A) / i (му)

p dt;

и

1 Y dv 1 / K (us, a; Л)^р

1 (м у) 7 У

с V

for s ^ 1, hence,

К (us, a M ) ^ C J } /p/i(Ms) for s ^ 1, w > 0, and

00

1

JK(ut;, a; A) Pl dv f i (м у)

J

pIp i

I < C J t .

By the inequality 00

0

00

1

K ( u v , a ; A ) T 1 Л Л Р/Р1 d u V lp f i (uv) v ) и )

we get the next one (in the same way)

(49) ' . < c

From (48), (49), (46) and (40) we obtain (37).

If Pl = p = oo, then the proof is similar.

I wish to thank Dr. L. Maligranda for his suggestions and very helpful comments that made possible the preparation of this paper.

108 M. M a s ty lo

References

[1] J. G u s t a v s s o n , A function parameter in connection with interpolation o f Banach spaces, Math. Scand. 42 (1978), 289-305.

[2] F. H i lie, R. S. P h il li p s , Functional Analysis and Semi-Groups, Amer. Math. Soc.

Providence (1957).

[3] T. H o lm s t e d t , Interpolation o f quasi-normed spaces, Math. Scand. 26 (1970), 177-199.

[4] S. G. K r e in , Ju. I. P e tu n in , E. M. S e m e n o v , Interpolation o f linear operators, Nauka, Moskwa 1978 (in Russian).

[5] L. M a lig r a n d a , Generalized Hardy inequalities in rearrangement invariant spaces, J. Math.

Pures Appl. 59 (1980), 405-415.

[6] —, Indices and interpolation, preprint.

[7] R. S h a r p le y , Spaces Ax(X) and interpolation, J. Functional Analysis 11 (1972), 475-513.

INSTITUTE OF MATHEMATICS

ADAM MICKIEWICZ UNIVERSITY, POZNAN