t
LOREINTZ SPACES AND THE CALDERON-ZYGMVND THEOREM
By
Ky6ichi YosHiNAGA
(Received Nov. 30, 1968)
Special cases of the space L'q (denoted as L(p, g) by R. A. Hunt [12]) have first been studied by G. G. Lorentz [17]. Later the notion of the space LPq was introduced as intermediate spaces in the general interpolation theory of A.P.
Calder6n [3]. J. Peetre [20] identifies such spaces also as intermediate spaces for another interpolation theory of J. L. Lions and J. Peetre [16]. Yet in order to obtain deep informations upon the duality or some other problems relative to LPq, another straightforward investigation seems necessary. Such was done, as far as the present author knows, by Hunt [12] in the most extensive man- ner. One of the purposes of this paper is to give some supplements to those basic properties of LPq obtained by Hunt. As to the duality for the space LPa, an interpolation theoretic approach gives the isomorphism of (LP4)' onto LPtq' in the sense not of isometry but of equivalent norms. Now that in our case a com- plete analogy to the relationship between LP and LP' is expected, a closer ex- amination will be required. This is another purpose of the present paper. The notion of the space LPq of scalar-valued functions will be extended to that of the space LPq(E) of vector-valued functions. The corresponding problems in LPq(E) will also be investigated. Recently P. Kree [14], [15] has presented a systematic and extensive study of L. H6rmander's multiplier relative to 3rLP (the Fourier transform of LP). In his theory the inequalities of Calder6n- Zygmund [14, Th6orbme 3] play an essential r61e. This is an extension of H6rmander's main theorem [IQ, Theorem 2.1] according to which the original Calder6n-Zygmund theorem [2, Theorem 7] may easily be obtained. Problems of such types•have been studied by several authors: for example, J. Schwartz [22], B. F. Jones, Jr. [13], R. O'Neil [19], etc.. Let us call a theorem of such a type a Calder6n-Zygmund Theorem. The third object of this paper is to give a generalization of the Calder6n-Zygmund Theorem of Kree [14].
Section 1 is devoted to the preliminary remarks. The non-increasing rear- rangement of a vector-valued function is defined and its basic properties are given. With a view to employing in the study about the duality of LPq, I.
Halperin's level function [8], [5] are introduced. Fundamental properties of
such functions are described with some specializations for the facility of the
application. The vector-valued LPq(E) is defined in Section 2. Several basic pro- perties are given. Among others it is proved that the set of E-valued simple functions are dense in LPq(E). To obtain a satisfaetory result concerning the duality for LPq(E), Halperin's idea relative to the function spaee Mi.) [8] will furnish a powerful tool. Such is the space Lg'q'(E) introduced in Section 3, and some properties of fundamental importance will be given there. In Section 4 the dual space of the scalar-valued LPa(1ÅqpÅq oo, 1-ÅqgÅq cx)) is determined. Our main result is Theorem 3. We here note that this was already announced by Hunt [/ 12] omitting a full proof. We must also remark that Halperin's general theory of duality between L?.) and Mg.)[8] seems not to be applicable in our case, because the rearrangement w* of the weight function w is always non- decreasing. Section 5 is devoted to determine the continuous linear form on the vector-valued LPq(E) (1ÅqpÅqoo, 1ÅqgÅqoo), The method of the proof given to ,Theorem 4 is based on the technique used in the case LP(E) by R. E. Edwards
[4, pp. 602-607]. In the final Section 6 the multiplier about the L"q(E) are studied. The main purpose of this section is to obtain a generalization of the Calder6n-Zygmund Theorem of Kree [14]. This is done by supposing a fun- damental system {V,} of bounded neighbourhoods of the origin of R" satisfy- ing the conditions (i), (ii) and (iii) given just before Proposition 13. A propo-
sition such as Proposition 13 is named "covering lemma" by H6rmander [10]
and plays an essential r61e in the proof of the Calder6n-Zygmund Theorem. The rest is a mere version of Kr6e [14].
Except otherwise stated, the notations and the terminologies in this paper are essentially those of N. Bourbaki's "E16ments de Mathematique" and L.
Schwartz [23]. We also assume elements of A. Grothendieck [6]. Furthermore we shall often adopt the convention such as L'q(E)=LPq, Z(E)= Z etc., if E is the space of complex numbers.
gl. Preliminaries.
Throughout this paper E, F and G are Banach spaces and R" is the Euclidean n-space. Regardless of n the Lebesgue measure on R" is denoted by m and until otherwise stated all functions will be m-measurable. Let f(x) be an E-valued function defined on R". The distTibution function of f is defined by Zf(a)=:
m{.r; I]f(u)IIEÅrcr} where OinÅq'.'crÅqoo and li•IIE is the norm on E. If(cr) is non- negative, non-increasing and continuous from the right. The non-inereasing
rearTangement of f onto [O, oo ] is defined by f*(t) = inf {d ; Zf (d) -Åq t} , O -Åq t is{g oo . Since IIf(x)IiE is finite valued, we have Zf(a)-ÅrO as a--Åroo. Thus f*(t) is well
defined for O-Åqts{: oo exeept the case Zf(d)ÅrO for all OSaÅqoo, and when such
happens, f*(O) is defined as f"(O) =oo. f*(t) is clearly non-negative and non-
increasing on [O, oo].
It follows immediately from the definition that
(1.1) f*(Zf(6))-Åqd (O-ÅqOÅqoo)
and if there is no interval [d-e, a] (eÅrO) N&There Zf takes the constant value Zf(6), this inequality may be reduced to the equality. Owing to the continuity of Zf(d) from the right we also obtain
(1.2) Zf(f*(t))pÅq.Nt (Og"tnÅqoo)
and if there is no interval [t-E, t] (sÅrO) where f\' takes the constant value f*(t), this inequality may be reduced to the equality. By means of (1.1) and
(1.2) it is not diMcult to see that f*(t) is continuous from the right. We also remark that
(1.3) (f, +f,)* (t, + t2) pÅqfl (ti) +f f, (t2)•
For convenience' sake, properties concerning Rf and f* available in this paper are picked out below. Some familiar of them, given without proof, is cited from Hunt [12].
(Rl) m{t;f*(t)Ård}==m{x;lif(x)IIEÅrd} forever•yd20.
(R2) IfdÅrO isapoint of discontin2eity of Zf. Thenfor any t, af(d-O)År
t l}ii Rf(d), it follo2vs that f*(t) ==a.
An interval (u, v) is called a constant interval of f* if f* takes a constant value for all uÅqtÅqv. If a constant interval is not contained in a larger con- stant interval, it is called a maximal constant inteTval.
(R3) Jf tÅrO is the Tight ena point of a maximal constant interval of f*, then Rf(f*(t-O)-O)=t.
PRooF. One easily obtains lf(f*(t-O)-s)2t for all eÅrO and therefore R,(f*(t-O)-O)2t. In case Zf(f*(t-O)-O)Årt, take s, Zf(f*(trO)-O)ÅrsÅrt, and observef*(s)=inf{d; Zf(o)s:s};}})f*(t-O)-s for all sÅrO. This provesf*(s)
;}}rf*(t-O) and from this together with f*(s)paÅqf*(t-O) it holds that.f*(s)=
f*(t-O), which contradicts the hypothesis that t is the right end point of a maximal constant interval off*. This completes the proof.
(R4) ifOÅqrÅqoo and g(x)=Hf(x)llZ, then g"(t)=f"(t)'.
(R5) Given E-val2eeal fzanction f(x) and F-valuea fzLnction g(x) definea on Rn, it holds that
j,l1f(`v)1lEIl g(x)l1F (lx pmÅqj:(S)f *(t)g*(t) at
for any measurable set S(R".
(R6) Given toÅrO, theTe exists a measuTable set S(R" such that nz(S)==to and
S,llf(,v)lIE cix-j:ef *(t) at.
PRooF. If there is no interval [to-e, to] (sÅrO) where f* takes the con- stant value f"(to), we may take S== {x; Hf(x)liEÅrfX: (to)}.
In case that f*(t) =
f*(to) on I==[to-e, to], take the maximal constant interval (cr, B) off* contain- mg L Then one gets crÅqto-ÅqB and Zf(f"(to))=cr. Letting Si={x; "f(x)llEÅr
f*(to)}, S2 == {x; lIf(x)l]E2f*(to)} we see nz(Si)==aand m(S2-Si)=B-cr. There- fore by taking S3(S2-Si measurable with measure m(S3)==to-cr, it follows that S = SivS3 satisfies the desired conditions.
(R7) Letting l lf, (x)llE-År IIf(x)IjE (v --)F oo ) a.e., it holds that f" (t) thÅq- lim. f." (t)
v-co everywheTe. If in aaaition llf,(x)ilE-ÅqUf(x)IIE a.e. (v=1, 2, ••t), f,'k'(t)ndÅqf"(t) and f."(t)-Årf*(t) (v-År oe) eveTywhere.
PRooF. We only prove the first part. Letting E(6) ={x; IIf(x)IIEÅr6} and
E. (o) = {x ; ll f. (x)1 iE År o} , we see !, I.n. . E. (o) ) E(6) and therefore !, m." Zf. (a) År Zf (6)
p-" oe -
f"(t)SIinf{6Jli]Nm Zf.(6) År/.t}
p-}oo
-Åqinf lim {6; Zf.(a)-Åqt} =lim infV{if; Zf.(o)-Åqt}
v- oo lcc--. oo p2 pa
- lim inf ff(t) =- lim. f.\' (t).
pt-L'co P2@ v--,oa This completes the proof.
(Rs) pzetting f es (t) == l-i,-Sgf* (s)r dsl ;' (o ÅqrÅq oo ) an cl fbl (t) =f** (t), one obtains
i) f,",f(t) is a continuous non-increasingfunction anclf5.l(t).17År.f*(t).
ii) fW(t)=supl .ls) S,IIf(x)llsaxl; where the sup is taken over aa meas-
?eTable S, m(S)=t, as wetl as oveT all meas2LTable S, m(S)År-t.
, iii) (fi II- f2)M (t)' HÅqf l,\, (t)r +f 2*,#, (t)' ctncl in puaTtieulaTSg(f, + f,)* (s) as -Åq
S,ff (s) as + j,f: (s) ds.
(R9) I7roT anyfunetionf(t), clejinea foT O-Åq t-Åq co, non-negative, non-incTeas- zng and contznuous fTom the right, it hoeds that f(t)==f*(t), O-Åq-. t-Åq oo.
-1 1
(R IO) s,u.g tp f"(t) - gy,p o{zf (o)} i, o Åqp Åq c)o .
In this connection we shall prove an identity due to Peetre [21] and Oklander [18] which plays a fundamental r61e in the interpolation theory of Lorentz spaces [21]. Let LP(E) (lrÅqp-Åqoo) be the space of (classes of) E-valued functions f(x) on R" such that
-SIjllf(x)llZ dxl;Åq cÅro, invÅq,Åq o., 1lfllP-l.essenti.a,1.s.upllf(x)ilEÅq('O' P=OO'
L'(E)+LOe(E) is the Banach space of functions f=fo+fi, fo E L'(E), fi E LeO(E), provided with the norm
lIfll=inf(Hfolli+Hfil1eo), f=fo+fi
where the inf is taken over all expressionsf==fo+fi, focLi(E), fiELOO(E).
Setting for t );})O, f E L'(E)+Loo (E),
K(t, f)= inf( li folli+t11fi ll ..), f= fo +fi, fo E L' (E), fi E LOO (E), we shall prove
PRoposmoN l. K(t, f)=S:f *(s) as, O pmÅq tÅq oo .
PRooF. We begin by showing K(t,f)2}iiS:f*(s)as. Letting S(R" be meas- urable and m(S)==t, and denoting zs the characteristic function of S, we write
xsf==xsfo+xsfi. Then
j,llf(x)HE cl v E{Ijllxs (x)fo(x) 1lE (lx+ Slixs (x)fi(x)lIE clx
"ÅqlIfol1i+`1lfi11oo'
Such being the case for all S, m(S)==t, it follows by (R8), ii) thatj:f*(s)asS K(t,f) as desired. We next proveS:f*(s)as;}}iK(t,f). Omitting the trivial casef*(t) =oo, we may assumef*(t)Åqoo. Then it holds that
m{x;llf(x)llE -f*(t)}-Zf(f*(t)-O)-lf(f*(t)) l}) t - zf (f*(t))
by (R3). Therefore we may take a measurable subset Si( {x; llf(x)llE==f*(t)}
with m(Si)==t-lf(f*(t)). Then putting S ={x; llf(x)IIEÅrf"(t)}vSb we get
m(S)=t. Define
fo(x) ==i,1(X)Mf*(`) Hff((xX)llE' l[g;
and
fi (X) ==f(X) '-fO (X),
o by adopting the convention Holl. :O. It is now readily seen that llfill..g:fnj'(t) and Zf, (a) = Zf (o +f* (t )). Therefore
fx(,)..If*(s)-f*(t), oÅq,Åqt,
tO, t-Åqs,
and so
llfolli-j,uf, (x)Il. ax -ÅqSgfi(s) ds -jgf*(,)a,-,f*(,).
Consequently
K(t, f) hÅq i1fo I1i + tl1fi H .. f{:S:f* (s) as•
This completes the proof.
Owing to this identity the Lorentz space may be defined by means of the interpolation theory [20], [21]. But the investigation in such a direction is not the present purpose and no further discussions along this Iine will here be .glven.
In accordance with Halperin [8], some explanatory remarks about the Ievel function will now be offered. Let v(t) (OÅqtÅqcx)) be a non-negative function and let OÅqpÅq oo, OÅqgÅq oo and O-Åq aÅqb iÅq oo. Define
R(a,b)..[,k'`k'g.`,i,$`S-'at• zÅq..:
th,oo
An interval (a, b) is called a level interval (of v with respect to p, g) if R(a, t):{;
R(a, b) for all aÅqtÅqb..If the level interval is not contained in a larger level
interval, it is called a maximal level interval. It is readily seen that every level
interval is contained in one and only one maximal level interval and therefore the set of all maximal level intervals of v with respect to p, g is a denumerable family {I.} of non-overlapping intervals. vO(t), the level fzenction (of v with respect to p, g) is defined by:
tR(a., b.)t'pq-i, for a.ÅqtÅqb., I. =(a,, b,),
VO(t)= i.(t), for all other tÅrO•
If vO =v then v is called a levelf2Lnetion (with respect top, g). It is known that vOO=vO , i.e. vO is a level function and
(Ll) vO ean be eharacteTized among the levetfztnctions wszLch that jgv(s)as wtÅqSgw(s)ds (tÅrO) as the onefor which j:w(s)as attains the minim2Lm value for eveTy OÅqtÅq oo.
Halperin [8] proved a fundamental theorem which may be given in our
case as follows.
THEoREM A. Let 1ÅqpÅqoo, 1-ÅqgÅqoo. If u(t) vaTies oveT all non-incTea- 11 sing non-negative fzLnctions on OÅqtÅq oo with 1ItMPre u(t)l1,s:1, then
SUPj:u (t)v(t) at=[v],.,., i
where
[.],,, .. 1,Iii`iS",O(lt)lq dtle) i,Åq, ...g.' .Åq, oo•
11 11 ancl + ,-- == 1, + , == 1.
PP 99
In consequence of this theorem it is now an easy matter to see (L2) [v + w]ptq, -Åq [v ]piqi + [w]prqi.
(L3) j:v (s) as isg jgw (s) ds foT aa t Åro implie$
j:a (t) v (t) at "ÅqS:u (t)w(t) at
foT any non-increasing non-negative u(t) and therefore [v]p,,t:E{g [w]pt,t.
PRooF. It is sufficient to show the first statement. To prove this we may
N restrict u of the form: u(t)== Z u,(t),
p =1
u,(t) .. IC"' O-Åq tÅqa, (O, a,-Åqt,
where OÅqaiÅq---ÅqaNÅqoo, c.ÅrO, v=:1, •••, AT. The statement is then almost evident. Thiscompletestheproof.
(L4) Letting v.(t):sggv(t), v.(t).v(t) (v.oo), OÅqtÅqoo, it holas that [v.]pt,,
"År[V]piat (V---), oo).
g2. ThespaceLPg(E).
Iff(x) is an E-valued function with domain R", we write SIj,eetZ-if*(t)qatli, oÅqpÅqoe,oÅqgÅqoo, ]Ifllpq=l,.p ,;f*(t), oÅqp-Åq oo,g= oo.
'tÅro
PRoposiTioN 2. Letting llf.(t)llE s{:gllf(x)llE, llf.(x)llE.IIf(x)llE (v-co) a.e.,
it ho lds that ll f. Ilpq .S-S l1fi lpq anel llfv ll pq -' l1fl lpq (v -År oo )•
PRooF. Owing to (R7) we see fY(t)-Åqf*(t) and ff(t)--Årf*(t) (v--boo), OÅq
tÅqoo, and so by Fatou's lemma the proof is complete for OÅqpÅqoo, OÅqgÅqoo.
In case OÅqp-Åq. oo, g= oo, lfif.Hpq-Åqllfllp,"4 is obvious• If tli,II!e2.llfvlip4 Åq,aÅqIIfllp4,
then there exists a toÅrO such that toPfes(to):E{gfiu.g tbfl(t)ÅqaÅqto)f*(to) for infinitely many v. This is a contradiction and the proof is complete.
Let
,.,"(IS,OOtZ-'fr,\(t)qatl$, OÅqpÅqoo,oÅqgÅqoo, llfil'a - IEyg ,s'fi.l (t), oÅq.p [s{g; oo, g=: oo'
Then it follows that
PROposmoN 3• 1lflipq-Åq RflISr,' -Åq(p"-P . );HflIpq foT eitheT OÅqrÅqpÅqcxD, OÅq r"ÅqgÅq oo oT OÅqrÅqp "Åq oo, g = oo.
PRooF. The first inequality is obvious from (R8), i). In proving the second
'
for OÅqrÅqpÅqcÅro, OÅqr-ÅqgÅqoo we shall make use of a well-known theorem of
Hardy [9, pp. 245-246] to get .
Llf"sr,) = lj,eOt'pqm-i (-}-j:f*(s)r as)1 atl l'
.. I.g e,e(j:f*(,)r d,)7- t-q(ii--})-i dtli
:{i:( ,(r/-g/-',/,) );-lj:f*(t)q,f-i d,le
= (-[7J{lr );lIfilpq•
If OÅqrÅqpÅq cÅro, g= oo the second inequality is shown as l1fl1S'..' -= ?.u..,p t;"ii(S:f*(s)r ds);'
:E{ 9}l..5' `; haipoe(j:s ; as)l = (Ii5J{Iir )iiiifil,oo.
When OÅqrÅqp== oo, g== oo, the latter becomes 1IfIida'da="f"... This completes the proof.
PRoposmoN 4. (Calder6n) iIfl1p, -Åq( Z )iq-;1lfiI,, foT OÅqpÅqoo,OÅqgÅqrAÅq oe.
PRooF. We first observe that
lifllS,= j:sS-if" (s)q as );2i j:sZ-if*(s)q ds
2f*(t)qpt-,q- ,
9
from which it follows that f*(t)-Åq(-p9-)t t-S'llfilp,. Therefore if rÅqoo, we get 1lf l1s, =- j:tJr" f*(t)r-qf * (t)a at
-Åqj:,;-i(( g )t'4g t-"gllfH$Eq)f*(t)q dt
10 • K. YosHINAGA
=( IZi )S-'l1fllsEq1lf"s,.
This proves llfllp,-Åq( Z )}-;llfllp,. In case r== cÅro,
llfllpoo == ggg tSf*(t)-Åq s,yg t$( g )} t-$IIfllp,
=: ( g )}NflIpq This completes the proof.
The Lorentz space LPq(E) is the collection of all E-valued functions defined on R" with llfllp,Åqoo. We remark that if OÅqpÅqoo,OÅqg:E{I: oo, it is not difficult to see Zf(o)Åqoo for each oÅrO, fEL"q(E). We also note that L'"(E)=LP(E) which follows from (Rl) and (R4). Since IIfllp, is positive homogeneous and
1lfi -f2Hpq -Åq2fi Cq(Iifillpq+ l1f21Ipq), Cq =max (1, 2T), L"q(E) becomes a topological vector space by defining f,.f (v-Åroo) in LPa(E) if and only if llf, -fllp,-O (v-År oo). It is known by Hunt [12] that we obtain the following
PRoposmoN 5. If OÅqpÅqoe, OÅqgsl; oo, LPq(E) is compeete with Tespect to
the metric a(fi, f2)=:11fi -f2HSr,'', OÅqr-Åqg•
PRooF. It holds by (R8), iii) that
(fi +f2)f.f (t)' s;f f,\, (t)r +f ;,\, (t)r,
and so if OÅqrKg, it is not diMcult to see
Ilfi + f2 I[ Sz' r "Åq l[f, ll Sz' r+ lIf2 ll Sz' r.
This together with Proposition 3 proves that ifOÅqrÅqpÅqco,OÅqrSIgÅqcx) or O ÅqrÅqp-Åq oo, g== oo, a(fi, f2)= llfi-f2IIS','' is a distance function giving the
topology of L"4(E). To see that LPq(E) is complete with respect to a(fi,f2) it is enough to show that for any sequence {f,} (L"q(E), Iif, -f,,Ilpq--ÅrO (pt, v.oo), we may find anfc L'q(E) such that llf.-fllp,-ÅrO (pt.oe). By Proposition 4 ac- companied by (RIO), it follows that
i
SUP d[Zf.-f. (d)]P= li f. -f.llp...O (pt, v-)b oo), crÅro
and consequently Zf,-f,,(o)-ÅrO (", v-Åroo) for each oÅrO. Therefore by means of
an ordinary procedure [7, p. 93, Proof of Theorem D] one obtains a subsequence
{f,.} and a functionf such that f,.(x)-Årf(x) (rc--)oo) almost uniformly. Given
EÅrO, take pt=pt(e) large enough to get 1lf.-f,,1lpqÅqe (y2}iipt)• Putting g.=f,,-- f. g=f-f. we see g.(x).g(x) (rc-År(x)) a.e.. Then by (R7) it holds that g*(t)
[slgli.m. gf(t), tÅrO, which, by Fatou's lemma, leads to
rc-}oo
l1 gl ip4 sg Iim. il grc Hpq == 1ir . I1f,.'f,,1lpq s: s,
rc-oe rc-ee
and so llf-f,,llp,Åqe for ,a)}ii/L(s). This completes the proof.
In case of 1ÅqpÅqoo, ls{l;gHÅq oo, letting r=1 we may infer that llfllSi,' is a norm leading to the topology of L"q(E) defined by llfllp,. Therefore in this case LPq(E) is a Banach space.
Observation of the proof of Proposition 5 tells us
PRoposmoN 6. if either OÅqpÅq oo, OÅqg Sg oo oT OÅqp -Åq oo, g = oo, f, .f(v.
oo) in LPq(E) impliesf,(x)-Årf(x) (v.cx)) in measzeTe ana therefore a subsequence {f,.} may be fo2Lnd so that f,.(x)-Årf(x) (rc-År cx)) almost zenifoTmly and henee a.e..
PRooF. By assumption together with Proposition 4 it follows that f,.f (v-Åroo) in LP'o(E) and hence t}(f,-f)*(t)---ÅrO (v.oo) uniformly in tÅrO. Then for any eÅrO, (f. -f)'(t).O (v-År oo) uniformly in t ;;}iis, which leads to
m{x ; lIf, (x)-f(x)l1EÅre} = Tn {t ; (f, -f)* (t)Åre} -ÅrO (y --)• oo ).
The rest is well-known. This completes the proof.
On the other hand it holds that
PRoposiTioN 7. Let OÅqpÅqoo, OÅqg-Åq oo and ass2Lme {f,} is a seque7zce of funetions 2vith s2eppoTts containea in a measurable set S of Jinite measure. lf foT all v, Hf.(x)llE.ÅqM a.e. ana f,(x)-Årf(x) (v.oo) in measure, then fc LPq(E)
and f,-Årf (v.oo) in LPq(E).
PRooF. We need only provef.-Årf(v-Åroo) in LPq(E). To begin with, let us examine(f.-f)"(t). Clearly
f sg m(s), oÅqdÅq oo,
Zfv-f (d)i .. o, 2M-Åq d, t
and for any given eÅrO, there exists an integer v(e) such that Zf,.f(a)Åqe for vÅr-v(E)andepÅqcr. Therefore
=O, m(S) .Sl:l t, Åq
(f' -f)" (t)iS{: 2M, OÅqtÅq o.,
N-Åq e, sg t, v(e) S v•
12 K. YosHiNAGA
Consequently if gÅq oo it follows that
IIf. -fllp, -Åq ISgtSi (2M)q at + j7(S'tS-i eq dtl Li
=( : )4' {(2M)qef+sq(m(si-p-eqpm)} iq' for v(s)-Åqv, and the proof is completed for this case.
In case of g= oo, we get for v(e)thÅqv
i
]lfv -f Ilpeo =sup tP(f, -f)* (t) tÅro
-Åq max(m(s); s, EpLi 2M).
This eompletes the proof.
By an E-valued simple funetion we mean a function which can be written in the form
N
f(x)= ZXs, (x) e.
v=1
where e. F 1!l, {S,} are pairwise disjoint sets of finite measure and xs,(x) is the charaeteristic function of S.. The set of all E-valued simple functions with compact supports is denoted by Z(E).
The main purpose of this section is to prove
. THEoREM 1. IfOÅqpÅqoo,OÅqgÅqoo, E(E) is dense in L"q(E). Preeisely,for any fc LPq(E) we may fZnd a sequence {f.}(Z(E) so that Bf,(x)IIE-ÅqIIf(x)IIE a.e. anel f..f (v--Åroo) in LPq(E). Consequently theTe exists a subseq2eence {f,.}, f,,(x)-Årf(x) (rc--Åroo)a.e..
PRooF. We first show that we may restrict f to the case where it has the support of finite measure. To this end letting s,= (x; IIf(x)IIEÅr-,1--l and put- ting g,(x)=f(x)xs,(x) we shall show g.-Årf (v-,,oo) in LPq(E), that nz(S,)Åqoo beingobvious. Since
1
{o, ,-Åqo,
Zf-g"(O)=izf(o)mzf(,vi ), oÅqoÅq vi ,
it follows that
(f-g.)*(t)==f*(t+zf( ,1 ))-Åq ,1 , oÅqtÅqoo.
Therefore taking any aÅrO fixed, we may write
llf- g. "p, = Ij ,Oe tSH 'f* (t + zf ( ,i -))qatl tt
Sl Cq {A at + Ba}
1-q
where C, =max(1,27T4-),
A. = Ij:tS' if* (t + zf ( ,i ))qdtl }
KIj:tS-'f*(t)4atl;.o (a-oo)
uniformly in v, and
B.-(jgtS-if*(t+a,( } ))4atle
.: -}nv(jg,s-i dt)}- -J-- (-p-, -)!- ev}.
Thus it turns out to be filf- g.Hp,-ÅrO (v-År oo).
Assuming now f to have the support S of finite measure, we next show thatfmay be approximated in LPq(E) by a sequence from Z(E). Given 6ÅrO, 6 take a closed sphere Kas m(S-K)Åq 2 . Then we may find a sequence {h.}(
Z(E) so that supph.(K, Ilh.(x)llE:i{gIIf(x)llE and h.(x).f(x) (pt-oo) a.e. in K [1, p. 190, Corollaire 1 de Th6oreme 3]. It now follows that there exists a com-
6 pact Ki(K, m(K-Ki)Åq 2 such that h.(x)-Årf(x) (pt.oo) uniformly on Ki and each h,,(x) is continuous on Ki [1, p. 187, Theorbme 2 (Egoroff)]. Setting T =S
-Ki, one finds m(T)Åq6. For anyeÅrO there exists an integer pt =pt(e) such that llh.(x)-f(x)llEÅqe on Ki and so {x; Hh,,(x)-f(x)llEÅrs}( T. Therefore Zf-h.(e)
1 -Åqm(T)Åq6 which shows us (f-h.)*(t)-Åqs for t2}li6. Taking 6=s= v and writing f,==h. in question, we observe that llf,(x)IIE-ÅqIlf(x)llE, uER", (f-
11
v for t2}r v and therefore (f-f.)*(t)--ÅrO (v.oo) for tÅrO. Conse- f.)* (t) S:
quently, since
(f'fv)* (`) -Åqf*( S )'f'( S )HÅq 2f*( S)
14 K, YosHiNAGA
it follows that
Hfmf,llZ, =j,cotS-'(f-f,)*(t)qat--Åro (v-)Foo).
This completes the proof.
CoRoLLARy. The set CoÅqE) 'of E-valuecl continuozLs function$ on R" with eompact suppuorts is dense in LPq(E).
PRooF. It is sufficient to prove that each f6 Z(E) may be approximated in LPq(E) by a sequence of functions in Co(E). Owing to the inequality ilf-gH
i -Åq2J Cq(11fllp,+IIgllpq) we may restrict f to have the form f ==xse where S is
bounded measurable and eE E. Then take an open bounded set U containing S and select a sequence {h,} of Iower semi-continuous numerical valued functions
in such a way that supp h. ( U, zs(x) :f{g h, . i(x) -Åq h,(x) -Åq 1, y = 1, 2, • • •, and h.(x) -Årxs(x) (v--År oo) a.e. [1, p. 151, Corollaire de Theorbme 3]. Then it holds that by Proposition 7, h.-Årxs (v-Åroo) in LPa. Thus the problem is reduced to show that every bounded lower semi-continuous non-negative function h(x) of compact support K is approximated in LPq by continuous functions of support K. And this is proved as follows. Since it is known that
h(x)== S.UP ga (X),
where g. runs through the set of all continuous non-negative funetions -Åqh(x), we find {g,}, O-Åq g.(.x)-Åq g..i(x)s-h(x) such that
S g, (x) ax -År jh (x) dx (y ., oo ).
Then it is not difficult to see g.(x).h(x) (v--ÅrcÅro) a.e., and the proof is finished again by Proposition 7.
g3. ThespaceL.P'gf(E).
In • this section we suppose l hÅq g Åqp Åq oo, -1- + 1, == 1, rl -+ -1. == 1. Let
PP 99
f(x) be an E-valued function with domain R" and Iet us write 1lfUS'q' == [f*]ptqt
ÅqIS,"OtSi"if*o(t)q'atl}, 1ÅqgtÅqoo,
=i i
ksup t5'f*O(t), g' == oo.
tÅro
According to (R8), iii) it follows from (L2) that iI fi + f2 l I' 2'q' ptÅq [ff + f lj ]ptq!
-Åq [f IC]pt,i + [fi]p!q• == 1lfi H2'4' + H f2 Ilg'q"
Similarly to Proposition 2 we now obtain the following
PRoposiTioN 8. Letting IIf.(x)llE-Åqllf(x)llE, Hf,(x)ilE.H,f(x)llE (y--,Foo) a.e.
it hOla$ that Ilfvli2tq'mÅq lifIl2'q' and 1Ifpll2'q"lifliB'q' (V-' OO)'
PRooF. Simple applications•of (R7) and (L4).
PRoposiTioN 9. Let f(x) be E-valuecl f2Lnetion with domain R". If g(x)
vaTies ovecr all fiLnctions of LPq with UgUp, -Åq1, then supjllf(x)1IE l g(x) l clx= l]fHSi,i•
PRooF. Owing to Proposition 8 and making use of a well-known theorem [1, p. 190, Corollaire 1 de Theorbme 3], we may first restrict f to have the com- pact support and next to be a simple function contained in Z(E). then it holds that
supjllf( u)llE I g(x) l (lx = supj,Oef *(t) g* (t) dt
= s.pj:. (t)f * (t) at,
where u varies over all non-negative, non-increasing functions on OÅqtÅq oo with
11 Iit-p-a u(t)ll,-Åq1. By means of Theorem A this proves the statement.
PRoposiTioN 10. (Halperin) Jf the E-vatued fzLnctions fi and f2 aTe diffeTent from zeo on clisj'oint sets, then it holds that Iifi+f211St,t2max(llfili2iq/, lif2112iq,)•
If in aadition 1Åqgf Åq oo, then i1fi +f2Il29,!! ll}ii ilfi ll29,'•+ il f21l29,'t•
PRooF. We need only prove the second statement. In this case we may
plainly suppose OÅql[fill2t,,Åqoo, OÅqIlf2lI2i,,Åqoo. Then for any eÅrO, Proposition 9 implies that there are numerical valued functions gi, g2 with
Hgillpq = IIfi ll29,'7i, j 1 gi (x) 1 Il fi (x)I IE dx År ll fi]I296• -- s, i= 1, 2•
It may clearly be supposed further that g,(x)=O whenever fi(x) =O (i=1, 2).
Then it follows that
16 K. YosHINAGA
(3•1) S l gi (x)+ g2 (x) l 1l fi (x) +f2 (x)i1E ax År ll fill29,'r+ 1l f21 fi S•q,'t-2e.
On the other side it holds that
[l gi llSq + lI g2 IIS, -= j,ee tS-i (g f (t)q + g lj (t)q) dt
=j,cotrpq'-i(I gi l q* (t)+ l g2 1 q* (t)) at
by (R4), and therefore on account of (R8), iii) and (L3) it results that 2S ,Oetvpq"-i(1 g, l q+ 1 g2 l q)* (t) at
.. j,OetS-'( 1 g, + g, l q)* (t) at
=11 gi + g21IZa•
Consequently one obtains i
l1 gi + g2 l1p4 -Åq {11fi lI2i4,'t+ I1 f2 U2tq,',} q•
Making use of this in (3.1) and letting E-ÅrO it follows from Proposition 9 that
i
l1 fi +f2 Il2'q' År- (l1fi l129q" + il f2 ll29q't) q7
as desired. This completes the proof.
PRoposiTioN 11. Let f(x) be an E-valued function with elomain R" and let
(f(x) for llf(x)llE-Åq N
f"(X)=IA[ilffrm(/-X)llff forl1f(x)l[EÅriV•
Then it holels that HfNl12t,t.ÅqllfH2t,t anel i1fNlI2t,i----Årllf]12•,i (v--Åroo)• Jf in paT- tiezelar i1fIl2!,•Åq oo, then IIf-fNll2t,t-ÅrO (v-År oo).
PRooF. We need only prove the last statement, because the first two are obvious from Proposition 8. To begin with we remark that Zf(d)-ÅrO (d-oo).
This is seen as follows. Supposing the contrary, there exists a positive number
a such that Zf(6)Åra for all dÅrO. Then f*(t)=oo for OÅqtÅqa and so R(O, a) =
oo. Therefore (O, a) is a level interval of f* and f*O(t)=oo for OÅqtÅqcr which
contradicts the assumption IIfli2t,•Åqoo. To obtain an estimate of llf-fuII2i,•,
we note that Zf-f.(a) == Zf(IV+o) -Åq Zf(IV) and so
• •-• -,,m,.,*,,,=I3,*(t)mN• 9,Åq,]Vf.Zf,(iV)•
Hence it follows from Theorem A that
1If- fNIi2!ai = [(fww fN)*]p•qt == supj,X'(")u (t)(f*(t)-7v) at
-Åq sup j,N'(N)u (t)f *(t) dt,
and so by virtue of (Ll) and (L3) it turns out true that -Åq sup j ,N'(")u (t)f*o(t) at.
Applying H61der's inequality it results that •
' llf- fNII2•,/ if{-. lj,X'`"'f*O(t)q' tSf-i atl ,7' -År o (N --År oo ).
'
This completes the proof. •
The space Lg'q'(E) is the collection of all E-valued functions f defined on R" with llfU2.,iÅqoo. Since llfll2•,• is positive homogeneous and ilfi-f2112•,iAÅq llfil12t,t+Ilf2112t,!, Lg'q'(E) becomes a Banach spaee. The completeness is shown by a usual technique, namely: letting {f,} be a Cauchy sequence in Lg'q'(E), a subsequence {g,} is picked out so thatÅí lig,.i-g,llBt,!Åqoo. Putting ' v=1
go (x) = 11 gi (x)I1E+ X 1I g, +i(x)- g, (x)1IE,
v=:i
and observing that 11gol12t,tÅqoo, the set S== {x; go(x) =oo} is seen to be negligi- ble, because the assumption that m(S)ÅrO leads to Z.,(o) Årnt crÅrO for all aÅrO and, as in the proof of Proposition 11, a contradiction ligoll9i,i =: oo will be obtained.
Thus
,, Il
oe 1- 1'
f(x)== gi (x)+ Z (g,+i(x)- g, (x)) v=1
is defined a.e. with llfl12,,tÅqoo. It now follows from llf-g,Ii2•,t.O (y--,oo) that llf-f,H2•,/-ÅrO (v-Åroo). Thus the completeness of Lg'a'(E) is proved.
THEoREM 2. Z(E) is dense in Lg'q'(E).
PRooF. We must approximatefc Lg'q'(E) by functions in Z(E). To begin
with we show that for any given eÅrO, there exists a bounded measurable set
18 K. YosHiNAGA
(sphere) S such that llf-xsfll2t,,Åqe. On account of Proposition 9 one may obtain a numerical valued function g, IlgllpqK1 so that
j l g(x) l 1lf1l29,' t- 'llf(x)ilE clx År lIfi129,; -eq'.
Then, letting S be a large sphere with the centre at the origin of R", it holds, by Proposition 8, that
j I g(x) H1 )c sfllS9,'7i11xs (x)f(x)llE (l ,c År llfil2iq,1 -Eq',
and hence ilxsfll29,'/Årilfll29,'t-eq'. This together with Proposition 10 leads to l 1 f- xsf il2•q,'• -Åq l[f1 I29,'t - Iizsf Il296t Åqeq'
as desired.
Assuming f to have the support S and to be bounded, as we may by Pro- position 11, it now follows that f may be approximated uniformly on R" by a sequence in Z](E) [1, pp. 193-194, Proposition 11]. So that, given eÅrO, one may find hE Z(E), llf(x)-h(x)IIEÅqe, xE Rn. Thus
(f- h). (,)1 == o, m(s) ,,{: ,, (Ss,
oÅqtÅqm(s) and therefore Theorem A shows us that
fi lf- h1l 9t,• == [(f- h)*],t,t = : supS,"" u (t) (f- h)* (t) at
-Åq s sup Ij,aotg-i . (t)q dtlt(s:(S)tg;-i atl}
==s( :I )T4irm' m(s)}'.
This completes the proof.
g4. ThedualspaceofLPg.
Throughout this section we suppose 1ÅqpÅqoe, IHÅqgÅq oo orp=g== 1 and let
1 + 1, =1, 1 + 1, ==1. The duality relative to LPq may be studied on the
basis of the interpolation theory of Banach spaces [3], [16], [20], [21]. But to
obtain the isometry between (LPq)' and LP'q' a straightforward examination
seems to be necessary. Such a treatment was partly.announced by Hunt [12].
We here give this in full.
THEoREM 3. in the sense of eqzeivalent noTms, it holds that (LP4)t==LP'q'. In ease that 1ÅqpÅq oo, 1ÅqgÅq oo or p== g=1, (LPq)'=Lg'q' is tTue even in the sense of isometirie isomorphism.
PRooF. We only prove the theorem for 1ÅqpÅqoo, 1-ÅqgÅq oo, because the casep=g=1 is well-known. Let lE(LP4)' and let IIIII be the norm of l. For any
measurable subset S(R", m(S)Åq cÅro, one knows zs E LPq and
xx(t)=I2i :.(Etitls).
Therefore putting pt(S)=l(xs), it holds that
11
1 ,et(S) l :f{II lllMlxs1lpq-lilH( : )q m(S)i.
Hence pt is absolutely continuous with respect to m and the Radon-Nikodym Theorem [7, p. 128, Theorem B] then gives a locally integrable function g(x) such that
pt (S) = j, g(x) dx
for every measurable set S. We now prove gE LP'q' in several steps.
(i) For any bounded function f having the support of finite measure, it holds that Sf(x)g(x)dx SI illll11flIpq•
To prove this take, as we may by Theorem 1, asequence {f.} in Z such
that if. (x) l s: lf(x) i , f.(x)-f(x) (v. oo) a.e. and f, •f (v -År oo) in L'q. Then it is easily seen thati Sf,(x)g(x)dx =ll(f.)ls:Hlllllf.lip, and letting v-Åroo we obtain the desired result.
(ii) Z.(o)Åqoo for alldÅrO.
Take f(x)==xs(x)l gg((X.))-/, m(S)Åqoo, in (i) and we get
j.I g(x)l ax .Åq lli"( : )b' .(s)'p-.
Thus setting t== rn(S) it holds by (R8), i), ii) that
2Q • • K, YosmNAGA
g* (t) -Åq g** (t) =sup.(is )S.i g(x)l dx ' •
.' -Åq-"lllll( iCil )},-illTi.
Therefore by means of (Rl) we get Z.(6)=m{t; g*(t)Åra}
-Åqm{ts illli( : ),,'t-S-Åra} Åq oo
as desired.
(iii) Given O==toÅqtiÅq•••ÅqtNÅqcx) and aiÅra2År•••ÅraNÅrO, setting u(t)=
N Z] a.xi,(t), I, ==[t,-i, t.[ it holds that p=1
Su (t) g"(t) dt MÅq lEi11 ll tl-ai' u(t)Eld:
' " To prove''the 'statement it is enough to find a numerical valued function f
de fined on Rn so that f* == u and l (f) == j ,ee u (t) g*(t) dt. f is de fined as follews :
a) if I. is contained inamaximal constant interval of g*, then m{x;lg(x)1
=g"(t,-i)} ll}it,-t..i and one may find. .a ,measurable S, ( {x; 1 g(x)l == g*(t,-i)},
M(Sv) =t.-t.-1; L
b) if I, is..n.o..t,,,o, ntained in a single maximal constant interval•of g*, ,wherg4s t.-i ls e.ontained in a maximal constant interval, then cr is defined to be its right' end point, otherwise cr =t,-i, and Similarly, in case t. is contained in a maximal constant interval, then B is defined tb be its Ieft end point, otherwise
B==tv' ''.'.•, ,•, . ,
'
.Clearly, it holds that •t.-i-Åqcr-ÅqB.ÅqtJ•.and g*(.a-O)Årrg*ÅqB). Then taking measurable sets :
tt tt
t L. ---'4 -.' . 'tlt' 1
[.
A,( {x; l g(x)1 == g*(a-O)}, m(A.)==q-t,-b B. ={x; g*(B)Åqlg(x)1Åqg"(a`O)}, C. ( {x; l g(x)l == g*(B)}S nz (C,)== t,-B,
and setting S, =A,vB.vC,, one may infer that m(S,)=t.-t.-i by (R3). Put f(x) = Åí a, xs (x)-g(X) .
v-i ' lg(x)l
Then it is not diflicult to see f*(t) =u(t). It remains to show l(f)== j,OOu(t)g*(t)at.
To this end we first remark that l(f) == .g, a.j.,l g(x)l ax.
Let us observe J,==j,,lg(x)1ax; ' Case a): J, == g"(t,-i)m(S.)
== g*(t,.i) (t, - t,-i)= S,. g"(t) at,
Case b): J, = S.,1 g(x)l cix+j.,l g(x)1 (ix+S.,1 g(x)i ax == g*(a-o)m(A,)+S,B-"g*(t+a) at +g*(B)m(c.)
- S,",-,g*(t) at + S:g*(t) at + j2" g*(t) dt
== j,,g*(t) at.
Thus Z(f)==.llila.j,,g*(t) at= j:u(t)g*(t) dt as desired.
(iv) For any non-increasing non-negative function u(t) it holds that j:u(t) g*(t) at is{: Hll-t}-e u(t)ll,.
This is seen by taking a sequence {u,} of functions like u given in (iii) in such a manner that u, (t) S; u, .i (t) sg u(t) and u. (t) -År u (t) (v --)F oo) a.e.. Then
applying the Lebesgue's dominated convergence Theorem to
S:uv(t) g*(t) (lt -Åq 1lllH1t}-; u.(t)l1q,
the statement is proved.
The final step is divided into 3 cases.
(v) Case 1Åqp AÅq gÅq oo.
Since g* is non-increasing while tS-' is non-decreasing, it is not diMcult to
see that no Ievel interval of g* with respect to p, g can really exist and con-
sequently g*O=g*. Hence by virtue of Theorem A it follows from (iv) that
22 ,, ,,.• K.YosHINAGA
ilgllp•q•=[g*]ptqt-Åqillll• On the other hand for any fE L'q and gE LP'q' it holds
j lf(x) i l g(x) l dx -Åqj,eOf *(t) g*(t) at
pmÅq [lgllp'4tllfHpq•
Therefore l(f)==gg(x)f(x)clx is defined for anyfEL"q and we get alinear form lE (L"q)X with Illll"Åq[igllpt,t. We have thus proved the theorem for 1ÅqpÅq
gÅq oo.
(vi) Case 1ÅqgÅqpÅqoo•
Owing to Theorem A it follows from (iv) that Hgll2t,•=[g*]pi,tinÅqlllH and hence gE Lg'q'. Conversely now given any gE Lg'q', let
l(f) =jf(x) g(x) clx, fE LPq.
Then according to Proposition 9 it holds that ll(f) l -ÅqS lf(x)I l g(x) l ax -Åq Ilfilpql1gl12iq/,
and therefore we get a continuous linear form l E (L"a)' with UIUtuÅqllg"2t,t. We have thus proved (LPq)'==Lg'4' in the sense of isometric isomorphism for 1ÅqgÅq pÅqoo. We now show Lg'q'=LP'q'. Taking any gcLP'q' and putting forfcLP4,
fi(,f.t-E[hf.`S,)g.(/{),gZ'.i,tS.1,t,90e.tg'XU,Z".Z,il",e,a!s`,OI.M,,O,2,Li.W.i8h,ALilUi,Y,g!"(.q's
==h(x) a.e., that is LP'q'(Lg'q' with llgl12t,,-Åqllgllp,,!, gcL"'qi. As the injection i(g) == g of LP'q' into Lg'qi carries Z onto Z] dense in LPtq' and in Lg'q' respectively, it follows that `i, the transposed of i, is also a continuous injection of (Lg'q')' into (LP"q')X carrying Z onto Z. Since clearly (LPq)t'=(Lg'q')' and since (LP'4')t = LPq is already known by (v) (1Åqp!ÅqgtÅq oo) it turns out that `i is an isometric isomorphism of (LPq)!' onto LP4. It is now an easy matter to see that i defines an isomorphism of LP'4' onto Lg'q'. This proves the theorem for 1ÅqgÅqpÅqoo.
(vii) Case 1 = g ÅqpÅq oo.
i
As is already known in (ii), it holds that g*(t)-Åqllll[•p•t-if', tÅrO. Therefore gELP'co and llgllpt..[sglp•lllll. On the other hand any gELP'oe defines a linear
form .
l(f)==jf(x)g(u)apc, fELpi,
and it follows that
i l (f) l .:f::j ,'Of*(t) g*(t) at Sg il gl lptee Hfllpi•
[E'his proves the theorem for !=gÅq.pÅqc)o with HlIi-ÅqligilptooiÅqp•Iilii. This com-- pletes the proof.
g5. The dual space ofLP9(E).
To simplify the matters, throughout this section we suppose 1ÅqpÅq oo, 1Åq gÅq oo and E contains a countabee dense subset.
We say an Ef-valued not necessarily measurable function g(x) defined on R" is scalaTwise measztrabte if the numerical-valued function Åq g(x), eÅris meas- urable for each eEE. Then it is known by [4, p. 575, Proposition 8. 15. 3] that, if Econtains a countable dense subset, g(x) is weakly measurable, i.e. measur- able with respect to the topology a(E', E) of E' and besides, lig(x)IIE! is a nu- merical measurable function. Therefore g*(t) and consequently
liglE,,,,==(j,OetZl'"ig*(t)q'dtl}', lIgll9y,t=[g*]pt,t
may be defined for such funetions.
THEoREM 4. Szeppose that l is a eontineeoLes linear form on LP4(E). Then there exists an E-valzted zveakly measierable f2Lnction g(x) defineal on R" seeeh that
l (f) == j Åq g(x), f(x)År dx
foT eaeh fELPq(E) and ilgllpt,t foT 1Åqp:{:gÅqoo (resp. ilgl12i,• for 1ÅqgÅqpÅqoo) is the norm of l. ConveTsely, by means of this integTal, any E-val2Leel weakly measu,rabte funetion g(ac) defZned on R" with llgUp,,iÅq oo foT 1ÅqphÅqgÅq cÅro (resp.
IIgli2,,iÅqoe for 1ÅqgÅqpÅqcx)) deLfZnes a eontineeous lineaT foTm on LPq(E) with - morm llgHpt,i for 1Åqp ff{:gÅq cÅro (Tesp. 1lg]l2•,• for 1ÅqgÅqpÅq oo).
PRooF. Letting lllli be the norm of l we begin by the remark that ll(f)1-Åq IIIMIfIlp,,fE LPq(E). Since f•eE LPq(E) for any e( E, fc Co, it follows that e.
I(fe) is a continuous linear form on E, and therefore we may write l(fe)=
Åqpt(f), eÅr, /t(f)E E'. Observation of the inequality I Åq,et (f), eÅr I -Åq IIIII llfellpq=III]HlfilpgllellE
shows us that II!t(f)IIEt-ÅqllllMfllp,,fc Co and as Co is dense in L'a (Corollary of
24 K. YosHINAGA
Theorem 1) we may extend pt, by continuity, from Co to LPq so that ll ,a (f)11E• "Åq llill llfl lp,, f E L" q•
Then given e E E, Theorem 3 tells us that there exists a function h, E LP'a' so as to satisfy
(5.1) Åq,et(f), eÅr=Sf(`v)h,(x)dx, fEL"q, llhellptqi-ÅqillllllellE•
N Let f(x)=Zg,zs.(x)e,, where S=Siv•••vSN is a finite partition of S into v==1
measurable sets S, of finite measure and e, E E, IIe,llE=:1, 6, any complex num- ber,v=1, •••, IV. Then it holds that
IV N
l(f) == E e. I(zs. e.) =Z 6, Åqg(S.), e, År
v=1 v=1
where we write pt (S.) = pt (xs,) as usual. By assuming l 6M l;}ii 1 SN-i l ;;}i: • • • År- l ei l ÅrO, it is not diMcult to see
O, M(SiV ''' V SN) -Åq t, (
f*(t) =1l8, l, m(S,+iV•••vSN) .s{;tÅqm(S,v•••vSN), v= !, •••, IV- 1, xieNI, OÅqtÅqm(sN),
and therefore
1ifl1pq == Ij,M(S'tS-'f*(t)4 atl}
= (})${ I 6. I qm(S.)g' + 1 S.-, l 4(m (S.-,V S.)S -m(S.)gP) +
+ ••• + l 6, l q(. (s)qp -.(s,v...v s.)s)} }.
N Since l.4,S.Åq,a(S,), e.År1-ÅqllllHlfllp4 holds for any e,EE, lle,llE=1, it follows that
N
E I {i'y I IIvet (Sp)llE' -Åq lllll IIfl lpq
v=1
for every {6,}. Then setting l6il =d•••==l8Nl, this becomes
,g,lI,et(S,)li.t-Åq 1llll( : )e .(s),i
for any finite partition S=Siv•••vSN by disjoint, measurable sets S. of finite measure. We introduce now the positive set function o defined for all measur- able set S of finite measure by the relation
N
o'(S)=sup : jI,a (S.)"Ei, v= 1
the sup being taken over all finite partitions of S into rneasurable sets S,. Then it appears that
o(S) s{ "lti( : )t m(S);,
and by a standard procedure we may verify that d is finitely additive. There- fore according to a well-known Theorem [4, p. 295, Ex. 4.39] one may conclude that by a positive locally integrable function L(x) it holds that
c(S)= j .L (x) ax
for every measurable set S of finite measure. Since ll/t(S)!IE!pmÅqa(S), ne(S)Åqoo, it now follows that
NN
IiZf(pc,) ,et (S.)HE' -Åq Z If(x,) i H,ee (S,)IiEt
v=1 v=1
N
-Åq Z ]f(x.)Io(S.) v=1
for everyfE Co and for every finite partition {S,} of the support of f, so that we see
(5.2) [l ,a (f)IIE! lhÅqj lf(x)IL( M) (l sc
for every f E Co and therefore for every f c Loe with compact support. Since L is measurable, given any compact set K(R" and any eÅrO, there exists a compact set K'(Kwith nz(K-K')Åqe such fhat the restriction of L on K' is continuous and thus L(x)iE{gc, xE K' by a constant c. Then for each fE Co with support
(K' and SIf(x)laxveÅq1, it holds that Il,a (f)1lEi s{gj lf(x) l L(x) cix s{: c.
We may now say that: given any sÅrO and any compact subset K(R", there
exists a compact subset K' (K such that m(K-K')ÅqE and for which
26 K. YosHINAGA
, {pt (f);fE c,, ,.ppf( Kr, g lf(.)1 d. -Åq 1}
v
is a weakly (d(E', E)) relatively compact subset of Ei. Then according to [4, p.
599, Proposition 8. 19. 11] it results that there exists a scalarwise locally in- tegrable function g defined on R" with values in Ei such that
(5.3) pt (f)- jf(x) g(x) ax
for each fE L"O having a compact support. Furthermore comparison with (5.1) shows us that Åqg(x), eÅr =h,(x) a.e. so that for each eEE
llÅq g(•), eÅrllptqt -Åq illIl l1eI1E•
Given any bounded measurable set S(R", it then follows that j,IÅqg(x), eÅr1ax=SÅqg(x), eÅrf(x)ax
= Åq ,et (f), eÅr-ÅqHeHEl1,et (f)lIE•
1 where we set f(x)= l Åqg(.), .År 1 Åqg(x), eÅrxs(x) by adopting the convention Trol=O• BY virtue of (5.2) it results that o
S, l Åq g(x), eÅr l (lx islg lIelIEj.L(x) (l Jc
for any bounded measurable set S. This proves l Åq g(x), eÅr 1 -Åq llellE L(x) a.e..
For any compact set K(R", we denote ÅëK the set of all functionsf of the type
N
f(x)=Zf.(x)xs,(x)e,, Rfllpq-Åq1, p=1
where K=Siv•••vSN is a partition of K into measurable sets {S.} and f, E Leo, e, 6 E, lle,llE==1 (v=1, •••, IV). For eachfc ÅëK it holds by (5.3) that
N
l(f) == Z Åq pt (f. xs.), e, År p-=1
= .g, j fv(x) Xs,(x)Åq g(x), e. År ax,
and therefore
(5.4) l(f) == SÅq g(x), f(x)År ax
for each fE OK.
We next show that (5.4) remains true even for each fc LPq(E). Since g(x) is scalarwise measurable, appeal to the assumption that E contains a countable dense subset shows us that g(x) is' weakly measurable and Hg(x)llEt is measur- able. Hence for any measurable set S(R", m(S)Åqcx?, and for any eÅrO, there exists a compact K( S with m(S-K)Åqs and such that the restriction of l[ g(x)llEt on K is continuous and the restriction of g(x) on K is weakly (a(E', E)) con- tinuous. The proof of (5.4) for fE LP4(E) may be carried out if we can show the next
(5•5) Sll g(x)llEtllf(x)IIE (lts{lll 1111i
for each fc ÅëK. For, if (5.5) is true it holds in particular that jil g(x)llE' i f(x) I clxS HlI1
ior each fEZ, ilflilp, f{gl. Given fELPq, iifllp,Sl, we may find {f,}(Z,
l f, (x) I thÅq I f(x) 1 , f, (x).f(x) (v -År oo) a.e. (Theorem 1). Therefore jl1 g(x)lIE' xK (x)i f, (x) i ax -Åq 1llil,
and so letting v.oo we get
jli g(x)xK(ov)llEt i f(x) 1 clx -Åq lill1.
This proves Il g(•) xK (•) HEt (f L"'q', II gxK Iip•,, .:s{ lllll for 1 Åqp Sg g Åq oo (resp•
l1g(•)xK(•)1lEtELg'q', llgxKli2•,• E{IIIllil for 1ÅqgÅqpÅqoo)(Theorem 3). Since sÅrO is arbitrary, it follows that llgxsllpt,tS:lll" for 1Åqpis{l[lgÅqoo (resp• llgxsHBi,tis{;
lllll for 1ÅqgÅqpÅqoo) for every measurable S with m(S)Åqoe. Thus we may conclude that lIgllp,,tgglll11 for 1Åqp-ÅqgÅqcÅro (resp• l1gl12t,tSHlIl for 1ÅqgÅqpÅq
oo). Such being the case it is now almost evident that (5.4) is true for each fc Lpq(E).
Let us finally prove (5.5). We begin by the remark that L (5.6) sup1jÅq g(x), f(x) År axi -supll(f)1-Åq lilll
where the sup is taken over all fEÅëK. Take any fcÅëK and consider the
28 K. YosHiNAGA
integral j.]lg(x)1IEtllf(x)1IEax. In doing so we adopt the following conventions;
a : a partition K== Siv•••vSN of K into measurable sets {S,}, (x.): a system of points x. E S. (v==1, -••, N),
(e.) : a system of elements e, E E, lie.liE==1 (y == 1, •••, IV),
(e,): asystem of complex numbers, i5.i=1 (v==1, •••, N).
Since IIg(x)IIE, is continuous on K and g(x) is weakly continuous on K it helds
(5•7) jll g( v) li E']lf(x)H .E (i x Sl sup .li.il, Il g(xv)liE'j ,,Ijf(pc)IIE d,c,
where the sup is taken over all {(x,)} and over all sufiiciently fine {d}. As for the sum under the sup on the right side of this inequality we see '
(5•8) ,ll.il,llg(xv)llE'j.,"f( u)HE (ix -Åq sup,IIIi, l Åq g(x.), epÅr l j,,11f(x)HE d,c,
the sup being taken over all {(e.)}, and finally we get
(5•9) .g, 1 Åq g( -x .), e, År 1 j,,lif(x)HE apt .:i{:Il sup l.IIIi, s, Åq g( pc .), e. Årj, li f(x)]l (lx l ,
p
taking the sup over all {(e.)}.
Putting
(5.10) A(d, (x,), (e,), (6,)) =,$i,e.Åq g(x.), evÅrS'..llf(x)UE dx and
N
h(x) = Z 4, zs,(pc)Ilf( u`)liE e,,
v=1
one observes that llh(x)llE=IIf(x)llE, therefore hE OK, and that jÅqg(oc), h(x)År cix == ,Åí.,j,,Åq g(x), e,Årg.Ilf(`u)llE cix•
By fixing (e,), (6,) and passing to the refinement A' of d:
S,=S.1V•••VS,N, we may write
jÅqg(x), h(u)Årav == X,.j.,f( g(x), e.Års,,,l1f(x)lIE cix,
where we set e.. =e., S,,,==S,. Given eÅrO, take d' sufficiently fine, then it holds that
lA(d/, (`v mp), (ev.), (6v.)) l Åq i lli,;,j,,5 g(x), e.År6vpllf(x)IIE cl`v 1 +E
= 1SÅq g(x), h(x)Årdx 1 +E, and owing to (5.6) it follows that
lA(d, (v.,,), (e,i,), ({F,,,)) 1 Åq1llH+e•
Such being the case for each sÅrO, we may conclude (5.5) from (5.7)-(5.10).
We have thus proved that any l6(L"q(E))' may be put in the form l(f)=jÅqg(x),f(x)Årax, fELP4(E),
with weakly measurable g, lIglip•4t:slllfil1l forllÅqp-ÅqgÅqoo (resp• 11glI2t,tS:lilll for 1ÅqgÅqpÅqoo). As for the converse of this statement, there will be no need to say. This completes the proof.
CoRoLLARy. If, in adaition, E is Tejlexive, by Te-moTming LP'q'(Ei) if neces- saTy it holds that (L"q(E))'=LP'q'(Ei), in the sense of isometric isomoTphism.
LPq(E) is reflexive.
PRooF. In this case E' contains also a countable dense subset, and so any E'-valued weakly measurable function is measurable [4, p. 574, Theorem 8. 15.
2]. The rest is obvious from Theorem 4.
g6. Multiplier.
An operator T acting on an F-valued function space and having values that are G-valued functions is called quasi-lineaT if T(f+g) is uniquely defined whenever Tf and Tg are defined, and if
ll T(f+ g) (x)"G -Åq K(iI Tf(x)1lG+ ll Tg(x)ll G) a•e•,
where K is a constant independent of f and g. If, in particular, K==1 and for any constant c, T(cf) is defined together with Tf and
IIT(cf)(x)ilG=lclHTf(x)HG a•e•, i
then T is called sublinear.
Hunt proved the following generalization of the Marcinkiewicz-Lorentz
30 K. YosHINAGA
•Theorem on interpolation of operators [11], [12].
THEoREM B. Let (p,, g,), (p;, gl) (v= O, 1) be given paiTs of numbeTs s2Lch that
OÅqp.Åq cÅro, OÅqg, [f{ oo oT p,=g.= oo 2Vith po Åqpi
and
OÅqpgÅq cx), 0Åqgg -Åq oc oT pg=:g; == oo with p6 =}Åq p{•
Jf T is quasi-eineaT anel
ll Tfilp6q6 -Åq Bolif1ipe,qo, II Tf li pfq{ -Åq Bil1fIlpiqi,
tlzen
II Tf ilpgs -Åq Bellfllp,q,
where gS:s anel for OÅqeÅq1
1 1-e 0 1 1-e e
=---"M""V +- ) """ 't' == -L 1 + -r7 .
-Pe Po Pi Pe -Po Pi
i ij r=min(g, go, gi), then Be= O((e(1-e))-7) as e-)0+O aoza e-1--o.
The purpose of the present section is to obtain a generalizatien of the Calder6n-Zygmund Theorem [2].
Let (a, b)-Årab be a given continuous bilinear application of ExF into G.
We assume that ab =O for each bcF implies a=O. Owing to the eontinuity, one finds rÅrO such that llabllG.ÅqrliallElibllF for all ac E, bE F. We next assume that su and iJf are Banach spaces such that ycg)F( M( y'rop"xF, yopG( Jf ( yi'xfi-NG with continuous injections: ,SPXF-ÅrM-År,SefttF, ,90XG--g,.)iT--ÅrYi!x)G. Suppose further that YCDF and YXG are dense in x' and X respectively. If u is a con- tinuous linear application of M into X and if it holds by a TE ,9e!C 'xE that u(q)
= T*q for all q c Y(E{)F, then sueh a T is uniquely determined. The collection of all such Tis denoted by {g'. It is a mormed linear space with norm HTIi2g=
supllTx:qli.pg-, where the sup is taken over all op EY(2i)F, I!/qilm-Åq1. As ,SPXF is dense in M, it is not diMcult to see IIT"(es=supilu(S)"f, the sup being taken over all Sc ,if, ]IS[1ew-Åq1. The Fourier transformation is denoted by y'"lp':
T(fu) ---) ,2c,z T( S) =: Se-2T'Xfi T(x) ax
and its inverse transformation by :6kny :
T( S).[}ti"'i' T(hi) = S e2n'x-Y T(y) a y.
-0'(M), the Fourier transform of M, becomes a Banach space provided with the
norm lITIIgT(m)=l1,s`'b(T)llm for TE ,:atR(.?2f')and similarly for ,2`k"(.if). ..a' = .2`,z((if) is a
normed linear space equipped with the norm IITII.f= II3'J(T)ilv for Tc .-di'. An element of a is called a multiplieT of -dir(.if) into 3o'(X").
In order that we may study the multiplier about the Lorentz space it is neeessary to prove the following
PRoposmoN 12. If 1ÅqpÅqoo, 1ÅqgÅqoo oT 1ÅqpmÅq oo,g== oe, then by a con-
tinuous in3'eetion it holds that LPq(E)( .ge"c'2Xgth E.
PRooF. Since Y is a complete nuclear space with the property of ap-
proximation, it holds that ,9n'(iSE==,s;P'eE==.s;Pe(,9e; E) [23, p. 47, Corollaire 1 de Proposition 11], [6, Chap. II, p. 34, Theoreme 6]. Therefore to prove the state- ment it is enough to show that by means of the application q-ÅrSf(x)q(x)ax any bounded subset C=:{q}. of Y is mapped into a bounded subset of E by any given bounded subset B= {f} of LPq(E);{jf(x)q(x) dui ; fE B, q c C} is a bound- ed subset of E. To prove this let us first remark that by boundedness of C
there exists, for any integer k a number cr ÅrO such that l q (x) 1 -Åq cr (1 + l x i )-k for all xER" and for all q7 E C. Then it follows that
Zg(d)-Åq' m{x;a(1+jx1)-kÅr6} ' =tun(( Zl )i-"1)"
where tu. is the volume of the unit sphere {x;lxlmÅq1} in R". Therefore one obtains
q*(t) E{Il a(( i. );+ 1)ff le•
Consequently it holds that
lljf( v) q7 (x) (lxilEKS ilf(vc)llE l g)( Jv)1 apc
-Åqj :f*(t) q* (t) dt HÅq ajf*(t)(( .`. )S + i)-kat.
In any case, H61der's inequality shows us that
32 K. YosHiNAGA
l1jf(x) q7 ( u) ax1lE -Åq a1lf1lp,lItili7'-ilT'(( -alS-)n-i + 1)-le ll ,,
where i +-l-=i, 1 +-lÅr- =i. Taking k sufficiently large we may come to
PP 99 the desired conclusion. This completes the proof.
The proof of the Calder6n-Zygmund Theorem usually given ([2], [10], [13], [14], [22]) is based essentially upon a fundamental "covering lemma" dUe to Calder6n-Zygmund [2] and H6rmander [10]. In order to give it inaform available for our purpose, let us suppose that we are given a fundamental sys- tem {V.}, v=O, Å}1, Å}2, •••, of bounded, not necessarily open, neighbourhoods of the origin of R". Assume furthermore that:
(i) V, -i( V. for every v and V V. =R", p
(ii) each V, admits a partition: V.==V,-i,iV•••VV,-i,N,, where V,-i,x==
T.,x( V.-i) by means of a measure preserving transformation T,,x (Z == 1, ..., IV.) and IV, -Åq Ar for every v by a fixed natural number IV,
(iii) for each v there exists a natural number 3==B, such that setting T,,x,
••• T,-.+i,x.(O)=:xo, it holds that T.,x,••• T,..+i,x,(V,-.)( V,-,,+B+xo for all pt, pt = O, Å}1, Å}2, ••., and for all possible {Zi, •••Z.}.
Under these circumstances the "covering lemma" says
PRoposiTioN 13. Let " be an E-vaeued continuous function with elomain R"
and letsÅrO be given. Then it holds that
i) u=:v+,ze=Olw, with v, w,ELi(E), jw,(.)ax.=o,
ii) ilvlli -Åq Ilulii, Åí Ilwxlli inÅq 2I fi ulli,
X=1
iii) IIv(x)llEisgNs everywhere in R",
iv) theTe exists a di$j'oint countable family {SN} of measuTable sets Sx hav- ing an expTession Sx == T.,x,••• T,-.+i,N.( J7,-.) such that szepup wx( SN and
lilli]m(SN)-Åq ,1 j,1]u(x)l]Eax, S=YSx,
v) seeppu, suppv anel all SN aTe containea in a ceTtain compact szebset C(
Rn.
PRooF. To begin with take l7, such that suppu( V, together with nz(JÅr",)År
-,1- j"u(pc)[iEax and set C== jJ7.. Inspecting the partition V, = V.-i,iv•••v V.-i,N,, pick out all l= li satisfying
m( Vp -i,xi) -Åq 1, j.,-, ,,Ii" (x)llE dx'
Then it follows that
sin ( V, -i) -Åq Sl .,-,,,,1la (`t)iiE (l x
Åq sm ( V,) pmÅq. s IVm ( V. .i).
Define v(x) on VV,-i,N, as Nl
v(x)= m(jl},-1)jv,rv,,,,u(y)dor, xc J7,-1,x,, and set
wxi(x)==Io",(X)-"(X)' .Xtfi,III{Ji,'",i.
Then it holds that 1[v(x)11Es{sNfor xEVV.-i,x,. For each of the remaining Z, Xl
denoted by Zl, it follows that
m( V'v-i)Årml,mj.,-, ,.ll"(x)llE dx' ,l
Observe now the partition V,-i,xf=V,"2,x{,iV•••VV.-2,xf,N,-i where V,-2,x(,x=:
T,,xl T.-i,x(V,-2) and so m(V,r2,xf,x)=m(V.-2). Pick out from 1, -••, N..i all Z =Z2 satisfying
M(Vv-2•x{•x2):E{{I l Sv,-,,,f,,,ll"(X)llEaX,
define v(x) on VV,-2,xf,N, as
N2
V(PC)=-nt-(-Jll;pm2) jv,-,,,;,,,U( Y)dY' JU E J/"-2'X('X2' "
and set
wxf,x,(x)-Io",('X)-"(X)' .Xtg,IllWi2,'/i'i"2
Then it holds that IIv(x)IIEE{{lsN for xcVV.-2,xl,x,. As fQr the rest of 1, •••, X2
N,-i, denoting any one of them by ZS one obtains "L(J7`"-2)År l S.,-,,,f,,YI"(X)llEdX'
i
