S. V. Astashkin- Extrapolation properties of the scale of Lp-spaces

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Extrapolation properties of the scale of -spaces

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Sbornik: Mathematics 194:6 813832 c2003 RAS(DoM) and LMS

Matematicheski Sbornik 194:6 2342 DOI 10.1070/SM2003v194n06ABEH000740

Extrapolation properties of the scale ofLpLpLp-spaces

S. V. Astashkin

Abstract. A new class of extrapolation functors on the scale ofLp-spaces (1 0. Then T can be defined in the Zygmund space L(log L) (see the definitions

below) as a bounded operator from that space into L1. The dual result for the spaceExp L1/ dual to L(log L) also holds. If T is a linear operator bounded in Lp forall sufficiently large p and TLpLp = O(p) (as p ) for some > 0, thenT: L Exp L1/.

General approaches of extrapolation theory have been developed since the 1990s.This is primarily due to Jawerth and Milman [3][5], who initiated a systematicinvestigation of the natural limiting spaces associated with scales of interpolationspaces, which can serve as a basis for estimates of the norms of the operators inthese spaces. For a bibliography of numerous applications of extrapolation theoryto analysis one can also consult [3][5].

Note that the tools (the extrapolation functors) used by Jawerth and Milman andbringing about new limiting spaces are the sum and the intersection of families of

Banach spaces (see, for instance, [4], 2 or [5], Chapter 2). This allows one to obtainas extrapolation spaces only spaces of the form (X0, X1)Jl1(1/) and (X0, X1)Kl(1/),where (X0, X1) is an arbitrary Banach couple and is a quasiconcave function. Herewe overcome certain limitations of these methods and introduce a much broaderclass of extrapolation functors (at any rate on the scale of Lp-spaces) closely con-nected with the real interpolation method. As a result we obtain as limiting spaces

AMS 2000 Mathematics Subject Classification. Primary 46M35, 46E30.


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814 S. V. Astashkin

almost all symmetric (or rearrangement invariant) spaces close to L and L1.Crucial here are the extrapolation relations for the K- and J-functionals for the

couples (L, Exp L

) and (L1, L(log L)1/

), respectively, established in this paper.En route, for the norms of spaces in the above-mentioned class we find fairlysimple equivalent relations involving only the Lp-norms of the function. As a con-sequence one obtains a series of extrapolation results similar to Yanos theorem.

In what follows we consider spaces of functions on [0, 1]. For generalizationsand refinements of Yanos theorem in the case of functions on a space with -finitemeasure we refer the reader to [6], [7].

We shall now discuss in this introduction facts and observations prompting theidea of the existence of an equivalent expression for the K-functional for the Banachcouple (L, Exp L2) depending only on the Lp-norms of the corresponding func-tions.

In 1923, Khinchine [8] proved the following inequalities, which later becameclassical:

Ap

k=1

|ak|21/2

k=1

akrk

Lp[0,1]

Bp

k=1

|ak|21/2

, (1)

where the rk (k = 1, 2, . . .) are the Rademacher functions on [0, 1], that is,

rk(t) = sign sin(2k1t) (t [0, 1]),

ak R, and Ap and Bp are constants depending only on p [1, ). These inequal-ities were the starting point for many researchers, who repeatedly generalized andimproved them. It is important for us that the constant Bp on the right-hand sideof (1) approaches infinity as p

. More precisely, Bp

p as p

, that is,for all p 1 we have

C1

p Bp C

p

with some constant C > 0.In 1993, Hitczenko [9] proved inequalities similar to (1) but having one consid-

erable advantage: the constants in them are independent ofp. In these inequalitiesthe l2-norm of the sequence of coefficients a = (ak)k=1 must be replaced by itsK-functional K1,2(t, a) = K(t, a; l1, l2) for the Banach couple (l1, l2). Let us presentthe precise statement: there exists a positive constant c such that for all p 1 anda = (ak)

k=1 l2,

cK1,2(

p , a)

k=1

akrkp

K1,2(

p , a). (2)

Here and throughout fp = fLp[0,1].Later, the present author showed [10], [11] that

C11 K1,2(t, a) K

t,

k=1

akrk; L, Exp L2

C1K1,2(t, a), (3)


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Extrapolation properties of the scale ofLp-spaces 815

with constant C1 > 0 independent of a = (ak)k=1 l2 and t > 0. It follows byrelations (2) and (3) that for some C > 0 and all a = (ak)

k=1 l2 and p 1 we

have

C1k=1

akrk

p

K

p ,

k=1

akrk; L, Exp L2

C

k=1

akrk

p

.

If in place of the Rademacher sum one substitutes in the last relation an arbitrarymeasurable function f, then the right-hand inequality fails (see Remark 1). How-ever, if one replaces fp by a quantity with a more regular behaviour in p, thenone obtains correct inequalities:

C1

p supqp

fqq K(

p , f ; L, Exp L

2) C

p supqp

fqq

.

The proof of slightly more general inequalities (for K(p,f; L, Exp L)) is themain subject of our 2 (in 1 we present definitions and notation required inwhat follows). On this basis, in 3 using the real K-interpolation method wefind equivalent relations for the norms of symmetric spaces close to L involvingonly the Lp-norms of the functions (more precisely, their asymptotic behaviour asp ). Finally, in 4 we obtain dual results on the J-functional for the cou-ple (L1, L(log L)1/) and on spaces close to another end-point of the symmetricspace scale, the space L1.

1. Definitions, notation, and auxiliary information

An extended exposition of the theory of symmetric function spaces and operator

interpolation theory can be found, for instance, in the monographs [12][15].A major role in various analytic questions is played by the Zygmund spacesL(log L) and Exp L (see, for instance, [16]). They consist of all measurablefunctions f: [0, 1] R with finite quasinorms

fL(logL) =10

log2

2

t

f(t) dt and fExpL = sup

0 0 and f(t) is a non-increasing rearrangement of thefunction |f(t)| ([14], Chapter 2, 2). In each of these spaces one can introducequasinorms making it symmetric (or rearrangement invariant).

Recall that a Banach space X of measurable functions on [0, 1] is said to besymmetric if the following conditions hold:

(a) ify = y(t) X and |x(t)| |y(t)|, then x = x(t) X and x y;(b) ify = y(t) X and x(t) = y(t), then x X and x = y.An important, and the simplest, example of symmetric spaces are the Lp-spaces

(1 p ) with the usual norm:

xp =1

0|x(t)|p dt

1/pfor p <


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816 S. V. Astashkin

andx = esssup

0t1|x(t)| for p = .

Orlicz spaces are their natural generalization. Let N(t) be an increasing convexfunction on [0, ), N(0) = 0. Then the Orlicz space LN consists of all measurablefunctions x = x(t) on [0, 1] such that the norm

xLN = inf

u > 0 :

10

N

|x(t)|u

dt 1

is finite. In particular, if N(t) = tp (1 p < ), then we obtain the Lp-spaces.Other examples of symmetric spaces are Lorentz and Minkowski spaces. If (t)

is an increasing concave function on [0, 1], then the Marcinkiewicz space M()consists of all measurable functions x = x(s) on [0, 1] such that

xM() = sup0


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An important method for finding interpolation spaces is the real interpolationmethod based on the use of the Peetre K- and J-functionals:

K(t, x; X0, X1) = inf{x0X0 + tx1X1 : x = x0 + x1, x0 X0, x1 X1}

andJ(t, x; X0, X1) = max{xX0 , txX1}.

For fixed positive t the first functional is a norm in the sum X0 + tX1 and thesecond is a norm in the intersection X0tX1 (ifX is a Banach space and a > 0, thenthe space aX contains the same elements as X, but with the norm xaX = axX).If we fix x X0+X1 (respectively, x X0X1), then K(t, x; X0, X1) (respectively,J(t, x; X0, X1)) is an increasing concave function of t.

We recall a well-known result on the duality between K- and J-functionals ([12], 3.7). Let (X0, X1) be a Banach couple such that X0 X1 is dense in X0 and X1.If the sum X0 + X1 is endowed with the norm K(t, x; X0, X1) (t > 0), thenits dual space is isometric to the intersection X0 X1 endowed with the normJ(t1, x; X0 , X

1 ). Using this fact, the already mentioned duality between Lorentz

and Marcinkiewicz spaces, and the formula for the J-functional for the correspond-ing couple of Lorentz spaces ([14], Theorem 5.10) one easily obtains the followingresult on the K-functional for a couple of Marcinkiewicz spaces:

K(t, x; M(), M()) sup0 0 (see also [19]). Hereand throughout, an expression of the type F1 F2 means that cF1 F2 CF1for some positive c and C; usually, the constants c and C are independent of all orof some arguments of the functions F1 and F2.

Let E be a Banach lattice of two-sided number sequences a = (aj)j=. If

(X0, X1) is an arbitrary Banach couple, then the space of the K-method (X0, X1)KEconsists of all x X0 + X1 such that

x = (K(2j, x; X0, X1))jE < .

The space of the J-method (X0, X1)J

E contains all x X0 + X1 admitting therepresentation

x =

j=

uj (the convergence in X0 + X1), (5)

where uj X0 X1. The norm in (X0, X1)JE is set to be

inf{uj}

(J(2j, uj; X0, X1))jE,

where we take the infimum over all sequences {uj}j= such that (5) holds.


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818 S. V. Astashkin

If E is a Banach lattice of two-sided sequences and = (k)k= is a non-negative number sequence, then E(k) is the space of all sequences a = (ak)k=

such that (akk)k=E and aE(k)=(akk)E. Let E(l) = l l(2k)(respectively, {0}= E(l1) = l1 + l1(2k)). Then the map (X0, X1) (X0, X1)KE(respectively, (X0, X1) (X0, X1)JE) defines an interpolation functor: namely, forarbitrary Banach couples (X0, X1) and (Y0, Y1) the triple (X0, X1, (X0, X1)KE ) isan interpolation triple with respect to (Y0, Y1, (Y0, Y1)KE ) (a similar definition holdsin the case of the J-method). The set of all such functors is called the real K-(respectively, J-) method of interpolation. It is important to note that in the case ofthe K- (or J-) method its parameter Ecan always be assumed to be an interpolation

space with respect to the couple l = (l, l(2k)) (respectively, with respect tol1 = (l1, l1(2

k)); [13], Corollaries 3.3.10 and 3.4.6).In particular, for 0 < < 1, 1 p we obtain the classical interpolation

spaces

(X0, X1),p = (X0, X1)Klp(2k) = (X0, X1)Jlp(2k),

the properties of which are studied in detail in the monograph [12].

2. The KKK-functional for the couple (L, Exp L)(L, Exp L)(L, Exp L)

Lemma 1. For arbitrary > 0,

K(t, f; L, Exp L) t sup

0 0.Proof. By relation (4) and in view of the equalities L = M(0), 0(u) = u and

Exp L = M(), (u) = t log1/2 (e

1+

/u) (see 1), we obtainK(t, f; L, Exp L

) fM(t),

where t(u) = u max

1, t1 log

1/2

e(1+)/

u

,

(6)

with constant depending only on > 0.Let M(s) be the dilation function of the function , that is,

M(s) = sup0


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We consider three cases:(a) (u) t; then

Ft(u) =(u/2)

(u)M

1

2

;

(b) (u) < t (u/2); in that case we also have

Ft(u) =(u/2)

t

(u/2)

(u)M

1

2

;

(c) (u/2) < t; then Ft(u) = 1 M (1/2).

Thus, in view of (7) and (8), supt>0Mt(1/2) 21M (1/2) < 1. Hence

proceeding as in the proof of Lemma 1.4 in [14] we obtain that for all 0 < s 1,

s0

t(u)u du Ct(s),

with positive constant C depending only on . Hence, by the definition of the normin a Marcinkiewicz space,

fM(t) sup0


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820 S. V. Astashkin

Proof. We set k = max(1, 2/). Then for each q 1,

fq =1

0

(f(s))q ds1/q

2kq121kq0

(f(s))q ds1/q

2k21kq

0

logq/2

2

u

du

1/qsup

0 0 such that for all q 1,

fq C()q

1/ sup0


10/21


11/21

822 S. V. Astashkin

3. Description of symmetric spaces close to LLL

In what follows we denote by ek (k = 0,

1,

2, . . .) the standard unit basis

vectors in the space of two-sided number sequences, that is, ek = (ekj ), ekk = 1,ekj = 0 (j = k).Definition 1. Let F be an intermediate Banach lattice with respect to the Banach

couple l = (l, l(2k)), which means that there exist continuous embeddings

(l) F (l). (15)

Let LK,F be the set of measurable functions f on [0, 1] such that the sequence

af =k=0

f2kek

belongs to F.

It is clear from the definition that LK,F is a symmetric space with norm

fLK,F = afF.

In particular, LK,l = L and LK,l(2k)

= Exp L (the last equality follows

from the expansion of N(u) = eu 1 in a Taylor series). Hence

L LK,F Exp L .

Moreover, it will follow from Theorem 2, our main result in this section, that LK,Fis an interpolation space relative to the couple (L, Exp L

) ifF is an interpolation

lattice with respect to the couple l.We start with an auxiliary result. Consider two projections in spaces of two-sided

number sequences:

P+a =j=0

ajej and Pa =

1j=

ajej , where a = (aj)

j=.

Lemma 2. Let F be a Banach lattice of two-sided number sequences such that(15) holds. Then there exists a positive constant C such that for each sequence

a = (aj)j= satisfying the condition |aj| 2j|a0| (j = 1, 2, . . . ),aF CP+aF.

Proof. Since a = Pa + P+a, it follows that

aF PaF + P+aF. (16)


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By the assumptions of the lemma,

PaF |a0|

1j=

2jejF

C1|a0|

1j=

2jej(l)

= C1|a0|;

moreover,P+aF |a0| e0F |a0|C12 e0(l) = C

12 |a0|,

where C1 and C2 are the constants of the embeddings in (15). Hence it followsby (16) that

aF (1 + C1C2)P+aF,which proves the lemma.

Theorem 2. Assume that > 0 and let F be an interpolation Banach latticewith respect to the couple l = (l, l(2k)). ThenL

K

,F= (L, Exp L)

K

F(with

equivalence of norms).

Proof. It is sufficient to demonstrate that, with some constants depending onlyon F, for an arbitrary function f on [0, 1] we have the equivalence

afF bfF, (17)

where

af =k=0

f2kek and bf =

k=

K(2k, f; L, Exp L)ek.

First of all, L Exp L with constant 1, therefore for k = 1, 2, . . . we have(b

f)k

= 2k(bf

)0

= 2k

fExpL

. Hence by Lemma 2,

bfF P+bfF. (18)

Next, it is well known ([13], Remark 3.3.8) that if F is an interpolation space

with respect to the couple l, then

aF aF, where a = (ak)k=, ak = supj=0,1,...

{min[1, 2kj] |aj|}.

HenceafF afF

and by the obvious inequalities af P+af af,

afF P+afF. (19)

Since the sequence of af increases, it follows that

P+af =k=0

2k supjk

f2j2j

ek


13/21

824 S. V. Astashkin

and therefore by Theorem 1,

P+afF P+bfFwith universal constants. Hence relation (17) is a consequence of (18) and (19).

Remark 2. Keeping the notation of Theorem 2 we shall now show another way ofproving the embedding X LK,F by interpolation arguments. In fact, let T bean arbitrary linear operator bounded in both L and Exp L

. Then the sublinearoperator

Qf =k=0

T f2kek

is bounded from L into l and from Exp L into l(2k). Since the functorsof the real method interpolate sublinear operators and F = (l, l(2

k))KF ([13],

Lemma 3.3.7), Q is bounded from X = (L, Exp L

)K

F into F, that is, in ournotation,T fLK,F = afF CfX .

Thus, the triple of spaces (L, Exp L, X) is an interpolation triple with respectto (L, Exp L,L

K,F). In particular, if we take for T the identity operator, then

we obtain the embedding X LK,F.Interpolation in the couple (L, Exp L) is described by the linearK-method [19]

(see also [10], Proposition 1). This means that each interpolation space X withrespect to this couple can be represented in the form X = (L, Exp L)KF, where Fis a Banach lattice of two-sided number sequences. Here, as mentioned in 1, wecan assume that F is an interpolation space with respect to l = (l, l(2

k)).Hence by Theorem 2 we obtain the following result.

Corollary 1. Assume that > 0. Then for each interpolation space X with respectto the couple (L, Exp L

) there exists a Banach lattice F such thatX = LK,F (withequivalence of norms).

Remark 3. We point out that many recent authors investigating function spacesclose to L used the asymptotic behaviour of the Lp-norms of a function asp . For instance, in [20] the author studies the convergence of Walsh seriesin the spaces Gp, (p > 1, 1), one of the equivalent norms in which has thefollowing form:

fp, =

1

fxx

pdx

1/p.

In particular, [20] investigates the position of these spaces on the scale of Orliczexponential spaces. Namely, it is shown that for p > 1, q > 0, and 1 such thatp1 + q1 = and each r > q,

LNr Gp, LNq ,

where as before, Nr(t) = etr 1 (r 1) and Nr(t) = exp(t + t1/rr )r exp tr ,

tr = 1/r 1 (0 < r < 1).


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Using Theorem 2 and the results of [10] it is easy to show that the Gp, areLorentz spaces. In fact, fix p > 1 and 1 and consider arbitrary (0, p/(p1)).An immediate verification shows that Gp, = L

K

,lp(2k), where = (p 1)/p, (0, 1). Since lp(2k) is an interpolation space with respect to the couple l([12], Theorem 5.6.1), it follows by Theorem 2 that

Gp, = (L, Exp L)Klp(2k).

However, the last space (see [10], Example 2) coincides (with equivalence of norms)

with the Lorentz space p(), where (s) = log1p2 (2/s).

One can deduce from Theorem 2 a whole series of results similar to Yanostheorem. Using the last observation we present one of them as an example.

Theorem 3. Let T be a linear operator bounded in Lp for each p 1 and let

1

T

LxLx

x

q

dx < for some 1 and q > 1. Then T is a bounded operator from L into the Lorentzspace q(), where (s) = log

1q2 (2/s).

Proof. By Remark 3 and the assumptions of the theorem,

T fq()

1

T fxx

qdx

1/q

1

TLxLxx

qdx

1/qsupx1

fx = Cf.

In the last section of this paper we establish dual results on symmetric spaces

close to L1. The key role here is played by the J-method of real interpolation.

4. Description of spaces close to L1L1L1

We recall the definition of the intersection and the sum of a countable familyof Banach spaces. Let Xk (k = 1, 2, . . .) be Banach spaces continuously linearlyembedded in a topological linear space T. Then their intersection is the Banachspace k=1Xk of all x

k=1 Xk such that x = supk=1,2,... xXk < . In the

construction of the sum

k=1 Xk we assume additionally that there exists a Banachspace X0 embedded in T such that Xk X0 with constant 1 for all k = 1, 2, . . . .Then we denote by

k=1 Xk the set of x X0 representable in the following form:

x =

k=1

xk (xk

Xk), where

k=1

xkXk 0such that

a (l1), a =

k=

ak (convergence in (l1)), ak = (aki ) (l1)

yields

i=

max(1, 2ki)

|aki

|kG c

a

G. (21)

Proof. It is well known ([13], Corollary 3.4.7) that G is an interpolation space with

respect to l1 if and only if(l1, l1(2

k))JG = G. (22)

If (22) holds, then for the proof of (21) it is sufficient to use the definition of the

norm in the space (l1, l1(2k))JG and the asymptotic equality

J(t, b; l1, l1(2k))

i=

max(1, t2i)|bi| (23)

with constants independent of b = (bi)i= (

l1) and t > 0.We now claim that (21) yields (22). First of all, we observe that

(l1, l1(2k))JG G

for each lattice G intermediate relative to the couple l1. For if a = (aj)j= G,

then we have the representation a =

k= akek, where the ek are the standard

basis vectors. We have akek (l1) and J(2k, akek; l1, l1(2k)) = |ak|. Hencea(l1,l1(2k))JG aG. We point out in conclusion that in view of relation (23)the reverse embedding in (22) is an immediate consequence of condition (21).

Theorem 4. The following asymptotic equality holds for arbitrary positive > 0with constants independent of g

L(log L)1/ and k = 1, 2, . . .:

J(2k, g; L1, L(log L)1/) gUk , (24)

where Uk =

jk 2jkLrj , rj = 1 + 2

j .

Proof. By Theorem 1,

K(2k, f; L, Exp L) fVk ,


16/21


where Vk = jk2kjL2j . The constants in this equivalence are independent of

k = 1, 2, . . . and f Exp L . Hence passing to the dual spaces, in view of theduality between K- and J-functionals (see 1) we obtain

J(2k, g; L, (Exp L)) gVk

with constants independent of k = 1, 2, . . . and g (Exp L). Since L(log L)1/and L1 are subspaces of (Exp L) and L, respectively (see 1), it now followsby Lemma 3 that

J(2k, g; L1, L(log L)1/) gUk

for all g L(log L)1/. Here

Uk =jk

2jkLsj ,

where k = 1, 2, . . . and sj = 1 + 1/(2j

1).For all > 0 and j = 1, 2, . . . we have

usj+1 < rj < sj , where u =22 1

2.

It now follows by the definition ofUk that Uk = Uk and gUk gUk u1 gUk .Definition 2. Let G be an intermediate Banach lattice with respect to the Banach

couple l1 = (l1, l1(2k)), which means that there exist continuous embeddings

(l1) G (l1). (25)Let LJ,G be the set of measurable functions g on [0, 1] having a representation

g =k=1

gk (convergence in L1), gk Lrk , rk = 1 + 2k , (26)

wherek=1

gkrkek G. (27)

It is easy to verify that LJ,G is a symmetric space with norm

gLJ

,G= inf

k=1

gkrkekG

,

where the infimum is taken over all representations (26). Moreover, LJ,l1 = L1 and

LJ

,l1(2k)= L(log L)1/. Hence for each Banach lattice G with property (25) we

obtainL(log L)1/ LJ,G L1.

We claim that, in addition, LJ,G is an interpolation space with respect to the couple

(L1, L(log L)1/) if so is G with respect to l1 = (l1, l1(2

k)).


17/21

828 S. V. Astashkin

Theorem 5. Assume that > 0 and let G be an interpolation Banach lattice

with respect to the couple l1 = (l1, l1(2k)). ThenLJ,G = (L1, L(log L)

1/)JG (with

equivalence of norms).Proof. Now let Y = (L1, L(log L)1/)

JG and let

J(t, h) = J(t, h; L1, L(log L)1/)

for arbitrary h L(log L)1/ and t > 0.First of all, by the definition of space of the J-method

fY = inf(J(2k, fk))kG,

where the infimum is taken over all representations

f =

k=

fk (convergence in L1), fk L(log L)1/

. (28)

If g LJ,G, then we have the representation (26) with property (27). In thiscase we immediately obtain by Theorem 4 that

g Y and gY C1k=1

gkrkekG

.

Hence LJ,G Y and gY C1gLJ,G .For the proof of the reverse embedding we shall show first that in the calculation

of the norm in Y one can content oneself (up to an equivalence) with representa-

tions (28) in which fk = 0 for k = 0, 1, 2, . . . .Since L(log L)1/ L1 with constant 1, it follows that

J(2k, g) = 2kgL(logL)1/

for k = 0, 1, 2, . . . . Hence if C2 is the constant of the embedding G (l1), thenk=0

J(2k, fk)ek

G

=

k=0

2kfkL(logL)1/ekG

C12

k=0

2kfkL(logL)1/ekl1(2k)

= C12

k=0

fkL(logL)1/ . (29)

In place of the fixed representation (28) we now consider another one:

f =

k=

fk, fk = fk for k = 2, 3, . . . ,

f1 =

i=1

fi, fk = 0 for k = 0, 1, . . . .


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Then in view of (29) and since G is a Banach lattice, it follows that

(J(2k, fk))kG (J(2k, fk))kG +k=0

fkL(logL)1/e1G

(1 + 2C2C3)(J(2k, fk))kG,

where C3 is the constant of the embedding (l1) G.Now let f Y. Then, as shown before, there exists a representation (28) such

that fk = 0 for k = 0, 1, . . . and

fY C14k=1

J(2k, fk)ek

G

. (30)

By Theorem 4, for each k = 1, 2, . . . we can find a representation

fk =j=k

gk,j, where gk,j Lrj , (31)

and in addition,

J(2k, fk) C15

j=k

2jkgk,jrj . (32)

Again, since G (l1), it follows by (30) and (32) that

k=1

j=k

gk,jrj

k=1

j=k

2jkgk,jrj C5

k=1

J(2k, fk)

C5C2

k=1

J(2k, fk)ek

G

C5C2C4fY < . (33)

This shows, in particular, that the double series

k=1

j=k

gk,j

is absolutely convergent in L1 and has the same sum as the corresponding repeatedseries, that is, f. Hence

f =j=1

gj (convergence in L1), where gj =

jk=1

gk,j.

Here, in view of (31), gj Lrj and

gjrj j

k=1

gk,jrj . (34)


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830 S. V. Astashkin

Consider now the sequence

bg =

j=1

jk=1

gk,jrj ej . (35)

In view of (33), we conclude that

bg =k=1

bk (convergence in (l1)), where bk =

j=k

gk,jrj ej.

It follows by (32) that bk (l1) for each k = 1, 2, . . . . Hence by Lemma 4,since G is an interpolation space with respect to the couple l1,

bgG C6

k=1

j=k

max(1, 2jk)gk,jrj ekG

= C6

k=1

j=k

2jkgk,jrjekG

.

Hence it follows by (34), (35), (30), and (32) thatj=1

gjrjejG

bgG C4C5C6fY,

so that f LJ,G and fLJ,G C4C5C6fY.Interpolation in the couple (L1, L(log L)1/) is described by the real K-method

([21], Corollary 5.10; see also [22]). In view of results on the connection between K-

and J-functors [13], it is also described by the J-method. In other words, ifY is aninterpolation space with respect to (L1, L(log L)

1/), then Y = (L1, L(log L)1/)JG

for some interpolation Banach lattice G with respect to the couple l1 = (l1, l1(2k))

(see 1). Hence Theorem 5 has the following consequence.Corollary 2. Assume that > 0. Then for each interpolation space Y with respectto the couple (L, Exp L

) there exists a Banach lattice G such thatY = LJ,G (with

equivalence of norms).

We now present another concrete extrapolation result.

Theorem 6. Let and A be positive quantities such that for 1 < p < 1 + alinear operator T is bounded in the space Lp and TLpLp A log2(1/(p 1)).Then T can be defined in the Lorentz space 1() corresponding to the function

(t) = t log2 log2(16/t) such that it becomes a bounded operator from 1() into L1.Proof. Assume that > 0. Then 1() is an interpolation space with respect tothe couple (L1, L(log L)1/). Moreover, one can prove the following equality, to betreated as an isomorphism, with constant depending on > 0:

1() = (L1, L(log L)1/)Jl1(wk), (36)

where the weight wk is k if k < 0 and wk = 1 ifk 0.


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In fact, let us show that the corresponding dual spaces coincide (up to anisomorphism). By the duality theorem for the real interpolation method [13],

((L1, L(log L)1/)Jl1(wk))

= (L, Exp L)Kl(uk),

where uk = 1/(k + 1) if k 0 and uk = 1 ifk < 0. Since (1()) = M() ([14],Theorem 5.2), (36) is a consequence of the equality

M() = (L, Exp L)Kl(uk),

which is proved in [10], Example 1 for = 2. The case of arbitrary positive > 0is perfectly similar.

We now select such that 2 < . If g 1(), then in view of (36) andTheorem 5, there exists a representation

g =k=1

gk (convergence in L1), gk Lrk , rk = 1 + 2k ,

such thatk=1

gkrkk < .

By the assumptions of the theorem

k=1

T gkrk k=1

TLrkLrkgkrk Ak=1

gkrkk.

Hence

T g =k=1

T gk (convergence in L1)

and in view of the equality L1 = LJ

,l1and Definition 2, T g1 Cg1() with

some positive constant C depending only on .

Remark 4. As is known ([5], Chapter 2), from the values F(X0, X1) at a couple(X0, X1) of the functors in a sufficiently broad class {F}0


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832 S. V. Astashkin

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(2000).

Samara State University

E-mail address: [email protected]

Received 8/OCT/02Translated by IPS(DoM)

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S. V. Astashkin- Extrapolation properties of the scale of Lp-spaces

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