Orlicz spaces for which the Hardy-Littlewood maximal operator is bounded

Publicacions Matemátiques, Vol 32 (1988), 261-266 . Abstract by following condition be satisfied ORLICZSPACES FOR WHICH THE HARDY-LITTLEWOOD MAXIMAL OPERATOR IS BOUNDED DIEGO GALLARDO Let M be the Hardy-Littlewood maximal operator defined by Mf (x)  = up  1  19 lfldx  ,  ( .f E Lloc(R 1 )), where the supremum is taken over all cubes Q containing x and ¡Q1 is the Lebesgue mensure of Q . In this paper we characterize the Orlicz spaces L*, associated to N--functions 0, such that M is bounded in L* . We prove that this boundedness is equivalent to the complementary N-function ip of <k satisfying the A 2 -condition in [0, oo), that is, sup s ~ o 0(2s)/zy(s) < oo . 1 . Introduction It is known that for the Hardy-Littlewood maximal operator, on R, defined (1 .1)  9(t ; f) =  sup It-rho T - t If(s)Ids, to act boundedly in a symmetric space E it is necessary and sufficient that the IIUTIIE = o(T) as T -> oo, where a, are the dilation operators defined by Q r f (t) = f (, r -1 t) . (See Ch . II, Theorem 6 .10 in [2]) . In this paper we consider the more general Hardy-Littlewood maximal operator M, on FBn, defined by (1 .3)  Mf(x) = sup 1  IfIdx  ,  (f E Lio,(R - )), xEQ IQI Q This research has been supported by a CAYCIT Grant (PB85-0434) . 262  D . GALLARDO where the supremum is taken over all cubes Q containing x and ¡Q1 is the Lebesgue measure of Q . (Cube will mean a compact cubic interval with rlonempty interior) . Our aim is to characterize the Orlicz spaces L*, associated to N-functions 0, for which the operator defined by (1 .3) is bounded . Such Orlicz spaces are symmetric spaces, but, for the case n = 1, the proof given in here represents a direct and diferent proof from that given in [2] . Moreover, our characterizing condition is more manageable than (1 .2) . Except in basic questions, this paper is intended to be seNcontained . Now, we shall present the basic definitions and results concerning to Nfunctions and Orlicz spaces which will be used in this paper . The prooos of most of these results can be found in [1] or in II-13 of [3] . An N-function is a continuous and convex function  : [0, oo) -r R Such that O(s) > 0, s > 0, s-1O(s) -> 0, as s - ; 0, and s -1 «s) -> oo, as s -> oo . As example of N-functions we have : 01(s) = s P , p > 1 ; 02(s) = sP logk(1+s), p > 1 and k > 0 ; 04(8) = es - s -'1 ; 0s (s) _ ( 1 + s) log( 1 + s) - s ; 0s (s) _ = exp s 2 - 1 and 0s(s) = fo p where p : [0, oo) --> [0, oo) is defined by p(0) = 0, p(t) = 2 -n if t E [2 -n , 2 -n + 1 ) and e(t) = 2 n-1 if t E [2n-1, 2n), n a positive integer . An N-function 0 has the representation O(s) = fo cp where cp : [0, oo) -> R is continuous from the right, non decreasing Such that cp(s) > 0, s > 0, cp(0) = 0 and W(s) -> oo for s -+ oo . More precisely cp is the right derivate of 0and will be called the density function of 0 . Associated to co we have the function p : [0, oo)  + ff8 defined by p(t) = sup{s cp(s) < t} which has the same aforementioned properties of cp . We will call p the generalized inverse of 0 . The N-function 0 defined by O(t) = fo p is called the complementary Nfunction of 0 . Thus, if O(s) = p-1sP, p > 1, then O(t) = q-1tq where pq = p+q . Young's inequality asserts that st <_ O(s) + O(t) for s, t >_ 0, equality holding if and only if cp(s-) < t < cp(s) or else p(t-) < s < p(t) . An N-function 0 is said to satisfy the 0 2 -condition in [0, oo) (or merely the '~12-condition) if sup 9>o 0(2s) f s) < oo . If cp is the density function of 0, then, 0 satisfies A 2if and only if there exists a constant a > 1 such that sw(s) < ceO(s), s > 0 . The 02-condition for 0 does not transfer necessarily to the complementary N=function ; for example, 0 defined by O(s) = (1 -fs) log(1 + s) - s satisfies the 0 2 -condition but its complementary N-function 0, defined by O(t) = et-t-1, does not . In this paper the 02-condition for the complementary N-function 0 of 0 plays a fundamental role ; precisely, this is the characterizing condition . for the boundeddness of the Ha,rdy-Littlewood maximal operator defined in (1 .3) . For Chis reason it is very interesting to give some characterizations of this condition, which permit to know wheter 0 satisfies 0 2 even if we do not where S,p = {g E L,p : fx0(1g1)dp < l}, and ORLICZ SPACES  26 3 know explicitly the function 0 . First, it is known that 0 satisfies the 02-condition in [0, oo) if and only if there exists a constant a > 1 such that ¢(s) < (2a) -1 O(as), s > 0 . The following characterization, which is used in this paper, appear in the literature with more restrictive hypothesis than the one we shall use and so we shall include its proof . Proposition 1 .4 . The complementary N function of 0 satisfies ¡he 02 - condition in [0, oo) if and only if inf s>o scp(s)10(s) > 1, where ep is the density function of 0 . Proof of (1 .4) : Let p be the generalized-inverse of cp and let 0 be the complementary N-function of 0 . Assume that 0 satisfies 02 ; then, there exists ce > 1 such that tp(t) < ao(t) for every t > 0 . On the other hand, it follows from the equality cases in Youngs's inequality that tp(t) = O(p(t)) -iO(t) and therefore O(p(W(s)))/p(W(s)) < a -1 (a - 1)cp(s), s > 0 . Since p(cp(s)) > s and the function u -> u -1 O(u) increases strictly for u > 0 we obtain that inf 9 >oscp(s)1O(s) > a(a - 1)-1 . Assume now that inf 9 >os~o(s)/¢(s) > 1 ; then, the function s -> s -,3 0(s) increases strictly for exists a > 1 such that O(s) < (2a)-1O(as), s > 0 . finishes the proof . Ilf¡lo = sup{I jfgIdp : gE S,p}, x 1If1j<p) = inf {A > 0 : 10(A -1 1fi)dp ~ l}, x there exists fl > 1 such that s > 0 and, therefore, there Thus, 0 satisfies 0 2and this If (X, .M, p) is a o - -finite measure space we denote by 9 972 the space of Mmeasurable and p-a .e . finite functions from X to R (or to C) . If 0 is an N-function the Orlicz spaacs LoLO(X^p) and L* - L*(X, M, p) are defined by L,5 = { fE 9N : fx O(I f j)dp < oo} and L* = {fE 9N : f gE L1 (p) for all g E Lp}, where 0 is the complementary N-function of ~ . We have L o C L* and L* coincides with the set of 91T such that A fE Lt for some positive real A . The space L* is alinear space with the usual operations and we may define the following norms in L* : which are called the Orlicz norm and the Luxemburg norm respectively . Both norms are equivalent, actually Ilf 11( q5 ) _< Ilf 11 .p < 2jif 11( 0 ), and they make L* into a Banach space . 264  D . GALLARDO For a measurable set A, with 0 < M(A) < oo, lIXA11m = p(A)0-1(1/m(A)), where XA denotes the characteristic function of A . If O(s) = sp, p > 1, then, L* = Lo = Lp, Ilfll(o) = IIflip and 11911,p = 11911p where pq = p + q . 2 . The main result Theorem 2 .1 . Leí 0 an N-function, 0 the complemeniary N function of 0, L* ¡he Orlicz space associated to 0 and leí M be the Hardy-Littlewood maximal operator defined in (1 .3) . The following conditions are equivalent : (a) There exist positive constante A and b such that e I n O(bM f )dx < A l n 0(j f I )dx  ,  (f E LO) . (b) There exists a positive constant C such that 1IMf1I(0) < Cl lfll(,t)  ,  (f E L*) . (c) There existe a positive constant h' such that IIMXAII(0) < KIIXAII(0)  ,  (¡Al < oc) . (d) 0 satisfies the '~12 - condition in [0, oO) . Proof . It sufhces to prove (a) => (b), (c) => (d) and (d) => (a) . The proof of (a) ==> (b) is easy . In fact, if f E L* there existe .\ > 0 such that .\f E LO ; hence \bM f E Lo and therefore M f E L* . Moreover, we have IR n O(M f /(b -1 max(1, A)11 f 11(0) ))dx < < A(max(1, A))-1 IR n 0(If 1111f II (0))dx < 1, for f :~ 0 and thus we get I IM f 11(0) < b - ' max(1, A)I I f 11(,6) for every f E L* . The proof of (d) =~> (a) follows from the following interpolation result, taking into account that M is of weak type (1,1) and bounded in L . . Theorem 2 .2 . Leí (X, JVI, tC) and (Y, .F, v) be two Q-finite measure spaces, 0 an N function whose complementary N function satisfies the A2-condition and leí T : Lj(h) -k L .(p) --> 91t(Y) be a quasi-additive operator which is simultaneously of weak type (1,1) and of type (oo, oo) . Then, T is defined on L* (M) and there exist positive constante A and m such that (2 .3)  Y 0(ITf I )dv < Al 0(ml f 1)dh  ,  (f E LO(p)) . Proof of Theorem 2 .2 : By hypothesis there exists a constant C such that v{yE Y : ITg(y)I > A} < CA -1 I Igldp, x IIThil . <- CIIhli .  ,  IT(g + h)I <- C(ITgi + IThi) for every g E Ll(h), hE L, .(p) and A > 0 . For f E L* and \ > 0 let fa = f XA(a/2C2) and f' =ffa where A(a) _ {x E X : I f(x)I > a} . We have fa E L1 (p) and f'E L,, > (M) and therefore f O(ITfl )dv = f  p(a)v{y E Y : ITf(y)I > A}da < Y  0 _ 00 <2C 2 lo  A-1 P(A)( f Ifa1dp)da = x 2C'jf(i)j =2C 2 If(x)I(  A -I CP(A)dA)dp(x), L where cp is the density function of ~ . Integrating by parts, we obtain ORLICZSPACES  26 5 (2 .4)  J s A- ] (p(a)lo = s -1 O(s) + J 9 A -2 O(A)dA  ,  (s > 0) . 0  0 Since 0 satisfies 02, it follows from Proposotion 1 .4 that there exists ,Q > 1 such that 00(s)< scp(s), s > 0 ; then, if 0 < A < 1 we have that .\-20(A) <_ and therefore fo  -2 ¢(X)dA is finite . then (2 .4) shows that f0 9  < ~( O -  ,  (s > 0) and thus, we obtain (2 .3) with A = fl(0 - 1) -1 and m = 2C2 . Note . If T is also positively homogenous then it follows from (2 .3) that /Y O(m-1 I T f I )dv <A Jx O(I f ~ )dj,  ,  ( .f E LO(p)) and T applies L* (X, M, M) in L* (Y, T', v) . Proof of (c) =~> (d) : It follows from (c) that there exists a constant K > 0 such that IIMEXAIIm :5 KIIXAII0 for every A with ¡Al < oc,with ME being the maximal operator on balls, defined by MEf (x) = supzCB IBI -1 fB I f I dx, where the supremum is taken over all euclideans balls B containing x and 1111 m is the Orlicz norm . We denote by B(x ; r) the ball with center x and radius r and let a,, be the measure of B(0 ; 1) .  For every pair of reals (v, s), with v > 0 and s > 1, we 266  D . GALLARDO denote by A(v, s) and D(v, s) the halls B(0 ; (a n vs) -I / n ) and B(0 ; (a n v) -l en) respectively . If x 1 A(v, s) then A(v, s) C B(x ; 21 IX 112) and therefore MEXA(v,s)(x) ? (2nanVSIIXII2)-1 . On the other hand, if g = 0 -1 (v)XD(v,8) we have fRn 0(Igj)dx = 1 and consequently IIMEXA(v,s)II0 ~ .0-1(v) 1 MEXA(v,s)dx > D(v,s) > (2nanvs)-10-1(v) (anvs) -11n <1Ix11z<(anv) -1 / n _ (2 n vs) -1, 0 -1 (v) log s . Since IIXA(v,s)IIá = (vs)-10-1(vs) we conclude that there exists K > 0 such that 2-no-1(v) log s < Ko_ 1 (sv)  (v > 0) (s > 1) . Therefore 20 -1 (v) <_ 0 -1 (v exp (K2 n+1», v > 0, which preves that 0(2t) < < exp (K2 n+1 )0(t) for every t > 0 and so 0 satisfies A2 in [0, oo) . Final Remark . It follows from Theorem 2 .1 that the A 2 -condition on the complementary N function of 0cannot be eliminated from the hypothesis of the interpolation theorem 2 .2 since if the complementary N-function of 0 does not satisfy A 2(for example, in the case of 0 defined by O(s) = (1 + s) log(1 +s) - s) the result of the interpolation theorem does not hold for 0 and the Hardy-Littlewood maximal operator in spite of the fact that this operator is of weak type (1, 1) and of type (oo, oc) . (Moreover, observe that in the abovementioned case the N-function 0 satisfies the A2-condition) . R .eferences 1 .  M . A . KRASNOSELSKY AND V . B . RUTITSKY, Convex functions and Orlicz spaces, Noordhoff Groningen (1961) . 2 .  S . G . KREIN, J . I . PETUNIN AND E.M . SEMENOV, Interpolation of Linear Operators, Translations of Mathematical Monographs (1982), Amer . Math . Soc . 3 .  J . MUSIELAK, "Orlicz spaces and modular spaces," Springer-Verlag, 1983 . Departamento de Matemáticas Facultad de Ciencias Universidad de Málaga, SPAIN . Rebut el 25 de Mari de 1988 IIXII2ndx=