Vector-valued inequalities with weights

Publicacions Matemátiques, Vol 37 (1993), 177-208 . A bstract VECTOR-VALUED INEQUALITIES WITH WEIGHTS LUZ M . FERNÁNDEZ-CABRERA AND JOSÉ L . TORREA( * ) This paper deals with the following problem : Let T be a given operator . Find conditions on v(x) (resp . u(x)) such that f ITf(x)IPu(x)dx < C J If(x)IPv(x)dx is satisfied for some u(x) (resp . v(x)) . Using vector-valued inequalities the problem is solved for : Carleson's maximal operator of Fourier partial sums, LittlewoodPaley square functions, Hilbert transform of functions valued in U .M .D . Banach spaces and operators in the upper-half plane . Introduction This paper deals with the following Problem A . Let T be a given operator . Find conditions on v(x) (resp . u(x)) such that (0 .1)  ¡T f (x) (Pu(x)dx < C f l f (x) ¡Pv(x)dx is satisfied for some u(x) (resp . v(x)) . This problem was studied for different operators in [C, J], [G, G], [H, M, S] . The operators treated were singular integrals, fractional integrals, Hardy-Littlewood maximal operator and fractional maximal operator . In all the cases the method used was constructive . (*)Partially sypported by DGICYT . PB90-187 . 178  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA On the other hand it was hnown that a "good" weighted norm inequality of type (0 .1), for an operator T, gives a vector-valued inequality of type (0 .2) II(E ITfjlP)1/PIIS <CII(E IfjI P ) 1 lPlls . .7  7 For instante, the vector valued inequalities for the Hardy-Littlewood maximal operator, M, were obtained in [F, S] from the estimate f I M .f I Pu < C f I f I PMu,  1 <p< oo . In 1981, José Luis Rubio de Francia showed that weighted norm inequalities and vector valued inequalities were equivalent in some sense, see [R de F, 1] . Using that equivalente, he developed a non constructive method in order to solve the Problem A for come operators . The aim of this paper is to show that a slight generalization of the method of Rubio de Francia allows us to solve the Problem A for a huge family of operators . The generalization is two fold : First we shall consider vector-valued versions of inequality (0 .1) and we shall prove the relation with the corresponding inequality (0 .2), see Theorem (1 .1) . We use this vector-valued version in order to : (a) solve partially Problem A for Carleson's maximal operator of Fourier partial sums, see Theorem (2 .9) . (b) find conditions on v(x) (resp . u(x)) such that R II H f (x) II Éu(x)dx < C fR .II f (x) II Fv(x)dx is satisfied for some u(x) (resp . v(x)), where H is the Hilbert transform and E is a U .M .D . Banach space, see Theorem (2 .24) . In the case that E is a U .M .D . Banach lattice we solve the same problem for the operator M&)  s u p -QI fQ I .f (y) I dy, x E R'' where 1 .1 is the absolute value in E and the supremum is taken in the lattice order, see Theorem (2 .26) (c) solve Problem A for Littlewood-Paley square functions, see Theorem (2 .11) . VECTOR-VALUED INEQUALITIES WITH WEIGHTS  179 The second generalization is to consider abstract measure spaces, instead of Rn . With these ideas we are able to solve Problem A for operators which map functions in Rn x [0, oo) into functions in Rn x (0, oo), see Theorems (2 .16) and (2 .17) . The vector-valued inequalities obtained in this case, see Theorem (3 .16), can be of independent interest when working with operators acting on functions defined on the upper half plane . These operators include as particular cases Poisson integrals, balayages, see (2 .19), and some well known maximal operators, see (2 .23) . Our method gives in all the cases some extra information about the size of the weight that is found . It is an honour for us to use ideas of our friend and advisor José Luis . Throughout this paper we shall work on general measure spaces (Y, dv), (X, dp), where dv and dp are positive measures . Given a Banach space E, we shall denote by L E ' (Y, dv), LÉ(Y) or LÉ(dv) the Bochner space of E-valued strongly measurable functions such that .lY I1f(x)IIÉdv(x) < +oo . Given a positive measurable function w(x), on (Y, dv), we shall denote by LÉ(w(x)dv(x)), or LÉ(w) the space of E-valued strongly measurable functions such that fY lif(x)II Éw(x)dv(x) is finite . Given a Banach space E, we shall denote by PÉ, or &(E) the Banach space 00 {{an} C E :  IIan1IÉ < + 00} . n=1 If E is a Banach lattice we shall denote by E(P') the lattice {{xn} : sup Ixnl E E} with the norm I I {xn } I I E(e-) = II SUP Ixn I II En The organization of the paper is as follows : in Section 1 we state and prove the abstract results that generalize the previous work of Rubio de Rancia, in Section 2 we give (without proof) the applications that solve Problem A for several operators and Section 3 is devoted to the proofs . 180  L . M . FERNÁNDEZ-CABRERA, ,I . L . TORREA 1 . Abstract results We begin this section with a lemma, that it is known for the scalar case, see [GC, R de F, VI .4 .21 . The proof in the vector-valued case is essencially the same, but we include it here for the sake of completeness . (1 .0) Lemma . Let (Y, dv) be a measure space, let F and G be Banach spaces . Assume that 0 < s < p < oo and T is a sublinear operator which satisfies II(E IITfjllF) 1'p iiLs(Y,dv) < G( E IIfj1iG)1'P, j  j Then, if o,  there exists a nonnegative function w(x) with II W II Lo1 (Y,dv) < 1 and such that fY IITfj(x)IIF , w(x)dv(x) < GIIfj1iG The proof of this lemma is based in the following result, see [GC, R de F] . Mini-max Theorem . Let A, B be convex sets in some vector spaces and assume that B is compact for a certain topology . Let 0 be a function, 0 : A x B --> R U {+oo}, which is concave on A and conex and lower semicontinuous on B . Then min sup ~b (a, b) = sup min O(a, b) . bEB aEA  aEA bEB Proof of the lemma (1 .0) : Let A and B given by A={E IITfjIIF : fjEG,~IIfjIIG<-1} B= {b E L°(Y) : b(x) > 0, IIbil, < 1} and we define on A x B the function 0 as O(a, b) = f  ~I T fj (x) II Fb(x)"dv(x), B is convex and weakly compact, A is convex ; 0 is convex and lower semicontinuous in B, see VI .4 .3 in [GC, R de F], and 0 is linear on A, therefore by the MinimaxTheorem we have, min sup O(a, b) = sup min O(a, b) < bEB aEA  aEA bEB < sup II(E IITfjII F)lIpllLs(Y,dv) <_ aEA  Cp . j VECTOR-VALUED INEQUALITIES WITH WEIGHTS  181 Therefore, there exist bo E B such that, for every aE A, we have the proof of the lemma finishes by choosing w(x) = bo(x)-°' . (1 .1) Theorem . Let (Y, dv) be a measure space, F and G be Banach spaces, and {Ak}" be a sequence of disjoint sets in Y such that Uk o Ak = Y . Assume that 0 < s < p < oo and T is a sublinear operator which satisfies f  11Tf ;(x)IIFbo(x)"dv(x) < CP, .7 (1 .2)  II(EIITf ;IIF) I IPIIL-(A,,dv) <Gk(EIIfj1I G )IIP, kE0,1, . . . .7  7 where for each k, Ck is aconstant depending on G, F, p and s . Then there exists a positive function u(x) on Y such that (1 .3)  ( .lY IITf(x)IIFU(x)dv(x)) 1 IP < CIIfJIG holds, where C is a constant depending on G, F, p and s . Moreover, given a double sequence {ak}k á such that and o, = (p~~, u can be found such that ¡¡u -1 xAkIILi (Ak,dv) ~ (ak1Ck)P . Proof . . Given k, we define the operator Tk by Tkf (x) = T f (x) MA, (x),  xE A k . Tk satisfies Ak such that llw -l llL , - 1 (Ak,dv) < 1 and A IITkf(x)IIFwk(x)dv(x) <_ Ckliflic* k k=0 II(E IITkf ; IIFWPII L-(Ak,dv) < Ck(E IIf ; IIG)lIP, 7 ak < +oo then by lemma (1 .0), there exists a nonnegative function wk defined on 182  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA +oo We define u(x) by u(x) = y~akCk p wk (x)XA k (x),  x E Y ; then 0 l IIT f (x) II Fu(x)dv(x) _ akC k p J  IITj(x) II Fw(x)dv(x) <_ r ak ll f II +00  +00 G = 0  Ak  0 =Cll .fllG, and ¡¡u-1XAk IIL - 1(Ak,dv) = (ak 1 Ck)plluwkIlL- -1 (Ak,dv) :5 (ak 1 Ck) p . We introduce first some notation . of weights w(x) in R n . 2 . Applications Given 1 < p < oo and 0 < -y <n we shall define the following classes (2 .1)  Z P , . y = {w :  w(x)(1 + Ixl)('' -n )pdx < +oo} Rn (2 .2)  D p ,_ y = {w : L  w(x) 1p ' ( 1 + IXI)(-y -n)p' dx < +oo} Rn (2 .3)  Dp , , y = {w : sup R('Y -n )p  W 1p (x)dx < +oo}, R>1 fl-I<R when y= 0, we shall write simply Z p , D pand D* p , Remark . It is easy to check that if p < q then D p C Dp C D9, D p ¢ Dq and DP ~¿ D q , in fact the weight v(x) = (1 + IxIn)-1 be longs to DT, 1 < r < oo but v 1 D q for any q, 1 < q < oo . Finally the weight w p (x) = IXI n (p -1 ) (1 + . IxIn) - p(1 + I log IXII)2(p-1) belongs to D p but w p ¢ Dp 1 if pl < p . (2 .5) (2 .6) VECTOR-VALUED INEQUALITIES WITH WEIGHTS  183 Analogoulsy, given 1 < p < oc, 0 <_ -y < nand a measure p on the upper half plane R++ 1 , we shall consider the following classes of weigths w (x, t) in R++ 1 , (2 .4) Zp,7(dp,) = {w :  fR'+1 w(x, t)(1 + t + I x1)( 7-n )Pdp(x, t) < +oo} DP,7(dp) = {w : fR'+1 w(x~ t)1-p'(1 + t + I xI)(- ,-n)P dp(x, t) < +oo} D* , , y (dp) = {w : sup R(-1-n)p' f  w(x,t) 1P dp(x,t) < +oo}, R>1 (xI+t<R when 7 = 0 we shall write Zp (dp,), D p (dp) and D* (dp) . A . Partial sum operators . Consider a homogeneous function of degree 0 in Rn, p(X) = q(x'), which we assume to be of class C( 1 outside the origin and satisfying the cancellation property For each ~ E Rn, we define the kernel k g (y) = e 2 n 2 C'yQ(y')IyI -n , and the corresponding operator Tgf(x) =p .v . f kg (x - y)f(y)dy = = e27r2l .xp .v . f  q((x - y) , )IX - yI -n e -27,i ~ * yf(y)dy, Rn then, we define the operator and consider the inequality SZ(x')dv(x') = 0 . fl-II=l T* f (x) = sup TI f (x) I (2 .7)  f  IT*f(x)IPu(x)dx <_ C f nIf(x)IPv(x)dx . R  R We have the following 184  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA (2 .8) Theorem . Let 1 < p < oo . (i) If uE Z p then (2 .7) holds for some v, such that va E Z p , for n<1 . (ii) If v E D pn Dp l , for some p1 < p, then (2 .7) holds for some u, such that u« E D p , for a < 1 . When n = 1and Q(y) = - 1 sign (y) then To = H (Hilbert transform) and the partial sum operarors of the Fourier Series can be expressed in terms of {Tr},ER : For each interval I = [a, b] SIf(x) = f f(1)e2-i-ld1 =2(Tbf(x) - Taf(x)) . I In this case T* is Carleson's maximal operator, see [C], [H], and we have (2 .9) Theorem . Let 1 <p< oo . (i) uE Z p if and only if (2 .7) holds for some v, (ii) If vE D p n Dp l , for some p1 < p, then (2 .7) holds for some u . Conversely if (2 .7) holds for some u then v E D p . (iii) both (i) and (ii) are trae if we replace T* by S* . B . Littlewood-Paley operators . Let cP E S(Rn) be such that supp(0) C {1 E Rn : 2 <_  <_ 2}, cp( ) = 1 in a neighbourhood of 111 = 1, and We define the following operator S* f (x) = sup ¡Si f (x) j < CT* f (x) I c0(2 k j) = 1 for all l ;  0 . kEZ gf(x) = (E I'Pk * f(x)12)1/2, kEZ where Wk(x) = 2 k n,P(2 k x), and we consider the inequality (2 .10)  ~n .~ IGf (x) I pu(x)dx < C IR .~ I f (x) I p v(x)dx then we have VECTOR-VALUED INEQUALITIES WITH WEIGHTS  185 (2 .11) Theorem . Let 1 < p < oo (i) If u E Z p , then (2 .10) holds for some v, such that v« EZ p for n<1 . (ii) If v E D p , then (2 .10) holds for some u, such that ua E D p for a<1 . C . Operators on the upper half plane . We shall consider the upper half plane R++1 = Rn x [0, 00) . Given a measure dv on R++ 1 , we shall say, as usual, that dv is a Carleson measure if there exists a constant C such that for any cube Q in Rn, v(Q) <_ CIQ1 where Q = {(x, t) : x E Q, 0 <_ t <_ side length of Q} and IQI stands for the Lebesgue measure of Q . Given a measure dp on R++1, we shall consider the following operators : (2 .12) Generalized fractional integrals . Let 0 < 7 <n and K y (x, t, u) = Cn (IXI -}- t + u)?'-n,  xE R n ,  t, u E [0, 00) . Then we define, for compactly supported f on R++ 1 Tp,yf (XI t) = fR'+1 Ky(x - y, t, u)f (y, u)dp(y, u),  x E R n ,  y ? 0 . If f is a compactly supported function on Rn and K . (x, t) = cn(1x1 + t)y - n, xERn, t E [0, 00), we define Ty f (x, t) = I  K y (x - y, t) f (y) dy,  (x, t) E Rn+ l Rn (2 .13) Generalized Poisson integrals and Balayages . Let K be a function K : Rn x Rn x [0, 00) x [0, 00) --> R+ that satisfies K (x, y, t, u) C (I x - y1 + t + u)n'  x, yE R n , t, u E [0, oc), we define the operator T p ,,o f (x, t) = P .V . IR-+ 1 K(x, y, t, u) f (y, u)dp(y, u),  (x, t) E R+ +1 . 192  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA where in the last inequality we have used that v E D P , in particular we have 1 /P  1/P sup EITfí'(x)IP  <C ~Ilfj~~LP v XESk  j  j  £1 ( ) (  ) 1 hence get II  1 : IT .ff IP  IILs(S i,) <C2kn/s  ~IIfiIILPl(v) ( j  j On the other hand, as we said before, T maps L'P(Pl) into weak-Ll therefore we use Cotlar's inequality (see [GC, R de F, V . 2 .8]) and we 1/P  1/P II  E ITf j IP  IILs(S,) < ISkll /S-1 11  E ITf j IP  Leak-L l (SO ( j  ( j < CpISki 1/S-1 111 E Ilf,llél  IILl j ) 1/P  1/P < CpISkl l/S-1  fn ( j  ~ Il .fj(x)~~11  v(x)áx  fxI<2k+1 v(x)1-p,dx R I ) 1/P < Cp2kn/s  II fJ II Le l ( v ) j where in the last inequality we have used that v E D P . This finishes the proof of (i) . In order to prove (ii) we decompose again each function as before f=ft+ffl . Since ~ j K(x, y) II Q- <_ CI X I - n and v E D p , we have analogously as for T, that 1 /P  1/P II  IITf j "IIe-  IIL'(Sk) <_ C2knls  ~IfALP(v) j  j On the other hand T maps LQP(R n ) into Lép(e-)(Rn), 1 < r, p < oo, therefore by Hólder's inequality we have 1/p  1/p 11  EIITfjllIILs(Sk)  ISkls  r¡¡  E IITfjIIllLr(Sk) ( j  ( j <CISkIl rll I I :If j lp  JIL- \ .7 < CIS k I1/S  Ifj l p  v(x)d  v(x)-(P)(-,)~ ~ ~ 11 r. j  IXI<2k+1 Now we choose r such that 2 = p1 and by using the hypothesis v E Dp l , we have VECTOR-VALUED INEQUALITIES WITH WEIGHTS  193 II IITfj1Ie-  p ) IILs(Sk) <_ C2kn/s  ll f JllLp(v)  . j  j this finishes the proof of (ii) . Proofof Theorem (2 .8) : S k Moreover u can be found such that Aswe said above in order to prove (i) it is enough to prove (3 .2) . We observe that by the last Proposition the operator T satisfies (1 .2) with F= R,Ak = Sk, Ck = C2 k, /' and G = LP l (v), then by Theorem (1 .1) there exists a weight u satisfying (3 .2) . < (ak12kn/s)p L (1 + lxI)np' dx < E 2-Knp ~K U(X)q1 dx < L 1 00  Q(  1 7 < 572 -Knp' ( a K12kn/s)p(q-1) (2kn) \ 9-1 K=o with Q = (P)' and ak such that E aP < oo . Therefore if we take q, 1 < q < oo, such that q - 1 < p' - 1, we have 194  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA but -p'+ P (q - 1) +  _a 1 1  ,  p') + (q - 1) < 0, then if we chose e -1 aK = 2K,E, with e small enough, we get that u« E D P with a = P . Analogously in order to prove (ii) we observe that by using Proposition (3 .3) (ii) T satisfies (1 .2) with F= Q°°, Ak = Sk, C k = C2 kn / s and G = LP(v), then Theorem (1 .1) can be applied as before . Proof of Theorem (2 .9) : The sufficient conditions on (i) and (ii) have been proved in Theorem (2 .8) . In order to obtain the necessary conditions we observe that T * f(x) ? IHf(x)I, where H is the Hilbert transform, then the conditions in order to have (2 .7) are also necessary in order to have (3 .4)  ~R IHf(x)IPu(x)dx < CJR ~f(x)IPv(x)dx, but it is well known that in order to have (3 .4) for some u (resp . some v) it is necessary that v E D P (resp . u E Z P ), see [GC, R de F] . Finally to prove (iii) we observe that as S* f (x) <CT* f (x) we have that the sufficient conditions for weigths in order to have (2 .7) are also sufficient for (3 .5)  ~S* f (x) ¡Pu(x)dx < C l I f (x) IPv(x)dx . IR  R On the other hand observe that an inequality of the type (3 .5) implies that for any interval I C R, the inequality l R ISI f (x) I Pu(x)dx <C fR I f (x) I P v(x)dx holds with C independent of I and hence (3 .4) holds and the necessary conditions again are the some that they are for the Hilbert transform . B . Littlewood-Paley operators . Our idea is to prove Theorem (2 .11) following the lines of the proof of Theorem (2 .8) . We consider the PZ-valued operator Tf(x) = {Wk * f(x)}kEZ VECTOR-VALUED INEQUALITIES WITH WEIGHTS  195 where Wk are the functions defined in section 2 part B . It is clear that G f (x) = II T f (x) IIQ2 and then we are going to deal with the following inequality (3 .6)  IR . IITf(x)IIé2u(x)dx < CL .n If(x)Ipv(x)dx, in instead of (2 .10) . It is known that T is given by the 2 2 -valued kernel K(x, y) = {W k (x - y)}kEZ that satisfies IIK(x,y)11£2 <CIxy l - n . Moreover the operator defined by (f j ) j -> (Tfi ) j is bounded from L .1P(R n ) into weak -L,lP(e2)(Rn ) , 1 < p < oo, see [R de F, R, T] . We can consider also the adjoint operator T, acting on 2 2 -valued functions, f (X) = (fk(x))k, and defined by Tf (x) =T ((fk)k)(x)  Wk * f (x), k T is defined by the 0= G(P 2 , C)-valued kernel K(x, y) = {'Pk(y - x)Ík and again the operator T((f j ) j ) is bounded from LQP(e2) into weak - LlP, 1<p<oo . Therefore (i) and (ii) of Theorem (2 .11) are equivalent statements and it is enough to prove (ii) . In order to prove (ii) we need the following (3 .7) Proposition . Let v E D p , 1 < p < oo and let s < 1 < p . Then we have 11(1 : IITfj1Ie2WpIILs(Sx) :5 Cs,p 2xn/ s(E IIf II L P(v) )l ip 7 K = 0, 1, 2, . . ., where SK are th sets defined in (3 .3) . The proof of this Proposition follows the pattern of the proof of Proposition (3 .3) . Once we know Proposition (3 .7) the proof of Theorem (2 .11) can be built as the one of Theorem (2 .8) . 196  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA C . Operators on the upper half plane . Our goal is to establish inequality (1 .2) in this context . We shall denote by F(x) the cone of aperture one whose vertex is x, xE Rn, Le . F(x) = {(Y, t) E R++i :  IX _ y I < t} . (3 .8) Definition . Given a positive measure dp on R++ 1 , we define Al, f (x) = f  f (y, t) dp(n, t)  xE R n . r(x) t (3 .9) Proposition . Let 1 < p <_ oc and dp be a measure on R++1 . Then A~ is a bounded linear operator from L',(R++1, dp) into Lép (Rn, dx) . Proof . A P , is a positive linear operator, then it is enough to prove that A wmaps L' (R+'-'-', dp) roto L l (Rn, dx), but JI A Wf 11L1(d5) =  I  f (y, t) dp(n, t) jdx fR r(-) t fRn+1 CfR~ X , ( .) (y, t) ¡f (y, t) I dx) d~ty' t) -  n 1 < fR++1 (tn fg(y,t) dx) l f (y, t)idtt(y, t) < CnilfilL'(R++1,dtt) (3 .10) Remark . Given a positive measure dp on R+ +1 , we can define the operator Alf(x) = f  lf(ylt)l d~tn~ t) , x E Rn . The operator Al is related with "tent spaces", see [C, M, S] and [R, T2] . It can be showed that A1 maps Ls(R++1, dp) into LS(Rn, dx),1 < s < oo, if only if dp is a Carleson measure, see [R, T2] . Therefore, since M j f 1) (x) = Al (f) (x), we have that A w maps L', (R++1, dp) into LQP (R n , dx),1 < s < oo,1 < p < oo if and only if dp is a Carleson measure . VECTOR-VALUED INEQUALITIES WITH WEIGHTS  197 (3.11) Proposition . Let du be a measure on R++ 1 . The following inequalities hold : Where M y , T, y and P are the operators defined on (2 .14), (2 .12) and (2 .18) . By C n we denote a constant no necessarily the same at each ocurrente . Proof . Let B = B(xo, r), xo E R n, r > 0, be a ball in Rn . If (x, t) E B then I x - xo I + t < r, in particular we have IBI¡ n-1JB If(y,u)Idw(y,u)=enIBI n-1 f B If(y,u)I (~ J// B(y,u)dz)dp(y,u) = en I BI n -1  RTy' 1 1 R XB (y, u)XB(Y,u) (z) l f (y, u) l dzd'(y, u) < CniBIn-1 fBAl,(If1)(z)dz < cnM,(Al,f)(x,t) . In order to prove (3 .13), we observe that Tw,7f (x, t) I = ¡en  f (y, u)  1  dz  dM(y, u) I fR7+, (Ix-yI +t+u)n-7 (un fB(Y,U)  ) < Cn  I f (y, u) I  n-7 XB(y,u) (z) dzda(y, u) fRn +1 IRA (IX - yI + t + u)  ,un but if z E B(y, u), then I x - zI + t < I x - yI +u+ t and we have Ti .,7f (x, t) I :5 : cn f R :  f R- (Ix I zI y + )n-~ X B(y,u) (z) dzdU n u) + -c nR~ (IxA'zll_  f+)t)) -~dz=enTy(A~If1)(x,t) . The proof of (3 .14) es analogous . The following Theorem can be found in [R, T1] . (3 .12) M,,,-,f (x, t) <_ C .My (AI, I f I) (x, t), (x, t) E R+ +1 0<y<n (3 .13) IT ., .yf(x,t)I <-C .T7(A, .IfI)(x,t), (x,t)ER++1 0<y<n (3 .14) I T,,of (x,« I G CnP(A W I f I) (x, t), (x, t) E R++ 1 . 198  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA (3 .15) Theorem . Let 1 < p < oo, and dv be a Carleson measure on R+ +1 , then the operators ,M .,, with 0 < " y < n, T y , with 0 < -y < n and P are bounded from LPp (R n , dx) finto weak-Lé,7(R++ 1 , dv) and from LQ P (Rn, dx) finto Lép (R + + i, dv), n n~ , < S < 00, q =- l + T . Taking into account this theorem and Proposition (3 .11) we shall be able to prove the following (3 .16) Theorem . Let 1 < p < oo, 0 < ,y <n and dv be a Carleson measure in R++ 1 . Given a measure dp in R++ 1 then the operators Mand T,,,,-, are bounded from L' P (R++i, d[t) finto weak L,p " (R++ 1 , dv) . Moreover if dp is a Carleson measure, then the operators M,,,7 and TI ., ., are bounded from LQp (R++1, d~) finto L' P (R++1, dv ) , q = ñ+ r . Proof .. The first part of the theorem is a direct consequence of (3 .9), (3 .11) and (3 .15) . The second part is a consequence of (3 .10), (3 .11) and (3 .15) . (3 .17) Remark . The proof of Theorm (3 .15) is based in the theory of vector-valued Calderón-Zygmund Kernels . That proof is not avalaible for the case of Theorem (3 .16) . Now we can prove inequality (1 .2) for these operators . (3 .18) Proposition . Let 0 < s < 1 < p < oo, 0 < ,y <n and let dv a Carleson measure in R++ 1 . Let SK, K = 0, 1, 2, . . ., be the sets in R++ 1 defined by So ={(x,t)ER+ 1,  Ixl+t<1} SK = {(x, t) E R++1,  2 K-1 < I XI + t < 2 K }, K = 1, 2, ... (i) If v E DP,- y (dp) and G = LP(R+1, vdp) then II(1 ITp,7fif l' P II L-(SK,dv) < C2K(E IIf ;IIG)üP . 7  7 (ii) If v E DP ,, y (dp) and G = LP(R+ +1 , vdp) then II(1 :IMw7fjl p ) 1 ' p ilLs(SK,dv) <c 2~s (EIIf ;II G)"P . VECTOR-VALUED INEQUALITIES WITH WEIGHTS  199 Proof . Given K >_ 0, we decompose each function f = f' + f" where f=fXB K ,f"=f-f'and If ¡y¡ + u > 2(Ixl + t) then Iyl +u < Iy¡ +u+ 4t < Iy1+u+2t+Iy¡+u-2Ixi < 2(Ix-y¡+t+ u) . Therefore if (x, t) E SK we have ITil,7f " (x>t)l = 1 /,y¡  K7 (x - y, t, u)f(y,u)dp(y,u)I +u>zK+ 1 >2(Ixl+t) lyl+u>2K+ 1 flyl+u>2K+ 1 BK = {(x, t) :  Ixi +t < 2K+I} . 1 .f (y, u)  dp(y, u) (Iy¡+u) n-7 < Cn (fRn+1 l .f (y, u) I P v(y, u)dp(y, u) v(y,U)1-P  ) 1/p < (¡y¡ + u)(n-7)P' dl~(y, u  .IIf ~~G Thus  sup  (E m .,7fi(x,t)IP)1/P < C(57 IIfjIIc)1/P and then, (a,t)ESK 7  .7 II(1ITp,7fii1/PIILs(SK,dv) < CV(5K)1/s(1 : IIf ;IIG)1/P < < C2ns (1 : lif ;IIG) 1 /P . On the other hand, as s < nn7, we use Cotlar's inequality (see [GC, R 200  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA de F, V .2 .8]) and Theorem (3 .16) to get therefore 11 (1 : I Tw,y .fj ip) 1 /plI LI (SK,dv) j <CV(SK  n )1/s_  II(Y :ITw,y .fjlp)1/pIILn,ny (Rn+1,dv) GC2 K s ["-(n-y)] ¡¡(Y : 1 fi j =C2 K s  (~ Lfj (y, u) I p)1/p dp(y, u) ~B K  2K(n - y) j <C2 K s  Lfj (y, u) I p v(y, u)dM(y, u) j vl-p, (y, u  11/p  n +u+ 1))-ydu (Y,u)J  < C2 K!' (1 II .filiG)1/p . This completes the proof of (i) . In order to prove (ii), we observe that if (x, t) E SK, then (x, t) E QK where QK is the cube QK= {y E R n "  y = (Y1, . . ., yn), I NI :~ 2K,  i = 1, . . ., n}, 1 IQxIn-1 ~GlK I f l ~(y,u)Idl~(y, u) \1/p , < C n II .f II G 2K(7-n) ( f  v ( y u)1-p dp(y, u) I B/x 1/p' < C n IIf IIG s11P CR(y -n )p'  v (y, u)1-p dM(y, u) I  < R>1 fI .J+U<R < CnII,f IIGNow the rest of the proof follows as in (i) . Proof of Theorem (2 .16) : If v E Dp,y(dp), then, by the last Proposition, inequality (1 .2) is satisfied for T p ,, . y with AK= SK, G= Lp(vdp), F = R, cK = 2K' . Therefore by Theorem (1 .1) there exists u satisfying (2 .15) for T p ,, y . Moreover u is such that IIu -I XSK IIL1 (AK,dv) < (aK12 sn )p VECTOR-VALUED INEQUALITIES WITH WEIGHTS  201 with u = (P )r and ~aP < +oo, then U(X) 1-a  °° n-7)P~ dv(x, t) < ~ 2-K(--7)p' (aK12 '-)P(a-1) ,f R n+1 (1 + t + ~x~)(  K=O but as (2)' < p' we have -(n -- y)p' + sp(a - 1) < 0 . Therefore it is enough to choose aK = 2-K, with E small enough and then u« E DP,7 with cx - 0 - 1 . p This finishes the proof of (i) . The proof of the suffciency of condition DP , , y (d¡z) in (2,16) is obtained in a similar way using (3 .18) (ii) . For the necessity observe that for any ball B BC {(x,t) : Mw,7f(x,t) > IBI~-1 then (2 .15) for T = M,,,7 implies that therefore for f = XBv 1p we get the result . 1f (y, u) I dw(y, u) }, Á u(x,t)dp(x,t) < < C (Á I f (y, u) I dp(y, u» -P I BI (!-1)P f R ~ +1 f (y, u) I P v(y, u)dit(y, u), Proof of Theorem (2 .17) : Since T N ,, . y is essentially self-adjoint, a simple duality argument shows that the pair (u (x, t), v (x, t)) satisfies (2 .15) for the exponent p if and only if the pair (v(x, t) 1P , , u(x, t) 1-P ) satisfies the some inequality with exponent p' . Thus (i) is actually equivalent to (2 .16) (i) . The necessity of (ii) is obtained as in (2 .16) (ii) . For the sufficiency we consider the Q°°-valued operator Tw,7f (x, t) = { X QT ( x ' t) IQr I1-7/n .fQT f (y, u)dp,(y, u»IER where Qr is the cube centered at origine and with side length r . It is clear that M m ,, f (x, t)  =  JIT N ,, y f (x, t) I1E_ .  Therefore (u (x, t), v (x, t)) satisfies (2 .15) for M N ,,, y if and only if satisfies (3 .19) a pair L .+1 IIT~,7f (x, t) IIé-u(x, t)dv(x, t) < C IRn+1 I f (x, t) I Pv(x,t)dp(x, t) . 208  L . M . FERNÁNDEZ-CABRERA, J . L . TORREA [R de F, R, T] RUBIO DE FRANCIA, J . L ., RUIZ,F . J . AND TORREA, J . L ., Calderón-Zygmund theory for vector valued functions, Adv . i n Math . 62 (1986), 7-48 . [R, T1] Ruiz, F . J . AND TORREA, J . L ., Weigthed and vector-valued inequalities for Potential operators, 7'rans . Amer . Math . Soc . 295 (1986),213-232 . [R, T2] Ruiz, F . J . AND ToRREA, J . L ., Vector-valued CalderonZygmund theory applied to tent spaces, Colloquium Math . 62 (1991),265-277 . [S] SAWYER, E ., A characterization of Two Weight Norm inequalities for Fractional and Poisson integrals, 73 - ans . Amer . Math . Soc . 30 8 (1988),533-545 . Luz M . Fernández-Cabrera : Escuela Universitaria de Estadística Universidad Complutense de Madrid Madrid SPAIN Rebut el 4 de Setembre de 1992 José L . Torrea : Departamento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid SPAIN