WEIGHTED INEQUALITIES FOR COMMUTATORS OF FRACTIONAL AND SINGULAR INTEGRALS

We dedicate this paper to the memory of José Luis Rubio de Francia, who developed the theory of extrapolation and gave beautiful applications of vectorial methods in harmonic analysis . Through this paper we shall work on Rn, endowed with the Lebesgue measure . Given a Banach space E we shall denote by LE(Rn) or LE the BochnerLebesgue space of E-valued strongly measurable functions such that


Introduction
We dedicate this paper to the memory of José Luis Rubio de Francia, who developed the theory of extrapolation and gave beautiful applications of vectorial methods in harmonic analysis.
Through this paper we shall work on Rn, endowed with the Lebesgue measure.Given a Banach space E we shall denote by LE(Rn) or L E the Bochner-Lebesgue space of E-valued strongly measurable functions such that where lif(x)1IE <-Foo .
Given a positive measurable function ce(x) we shall denote by LÉ(a) the space of E-valued strongly measurable functions such that f 11f(x)IJÉ-(x)dx < o0 and we shall denote by BMOE(a) the space of strongly measurable functions b such that ~Q 11 b(x) -bQ11 E dx < C ~Q a(x)dx, bQ = jQ1-1 IQ b(x)dx .
Given two Banach spaces E and F, we shall denote by .C(E, F) the Banach space of all continuous linear operators from E into F.By a Banach lattice we mean a partially ordered Banach space F over the reals such that (i) x < y implies x + z < y -}z for every x, y, z E F, (ii) ax>0foreveryx>0inFanda>0inR .(iii) for every x, y E F there exists a least upper bound (l.u.b.) and a greatest lower bound (g.1.b.), and (iv) if Ix1 is defined as ,xl = l.u.b. (x, -x) then IIxjj < llyl) whenever Ix1 < jyj .
We shall say that a positive function a belongs to A(p, q) if (1 a-P'(,)d,)1IP'( 1 aq(x)dx) l I9 < C, IQl Q IQI e holds for any cube Q C Rn and p' + p = p'p, the constant C not depending on Q .
Observe that if we denote by Ap the Muckenhoupt's class, then, for p > 1, w E A(p, p) if and only if wP E Ap .
Finally we shall say that a Banach space E is U.M.D. if the Hilbert transform is bounded from LÉ into LÉ, see [2] .
The paper is organized as follows: in section 1 we state and prove the extrapolation results, in section 2 we state the commutator theorems, these theorems are proved in section 4, we give several applications in section 3.

Two extrapolation results
Let v > 0 be a measurable function , 1 < p < q < oo, 1 < A < oo and r v = 1 .We shall say that a weight w belongs to the class A(°)(p, q) if for every w such that w -1 E A 1 and (vw) -1 E A 1 .Then holds for every w E AP°) and p > 1 .
Let h = g"P .Then (1 .4) is equivalent to Since w E A ( ' ) (p, q) we have w-P' E A(' ,lg ; setting r = 1 -fg , we can apply lemma (1 .1),observing that r' _ -~ to obtain a function H > h such that (1 .6)J Hgl a'w -P'dx < c and Hw -P' E A( °-A') Therefore the weight v = H -1 h'wP'l a ' is such that v -A ' E Al and (vv)" E A1 .
Note .The theorems of this section are heavily inspired in [10].(2 .2) Definition : We shall say that a positive function a belongs to A(p, q) ( 1 ¡ a -P'(x)dx) 1 IP ' ( 1 ho1ds for any cubo Q C Rn and p' -fp = p'p, the constant C not depending on Q .
Now we state the theorems of this section .
(2 .9)Remark .If v2 E A2 then b E BMO(v) if and only if To see this it_is enough to observe that if v2 E A2 then v satisfies a reverse Holder condition with exponent 2, see [9] .
(2 .10)Remark .The theory of vector-valued Calderón-Zygmund operators, see [5], and potential operators, see [61, can be applied in both theorems despite of the fact that smoothness is required only on the first variable of the kernel .Thus the operator T (respectively V) tucos out to be a bounded operator from LÉ(ceP) into LF(a9) (respectively from LP(cJ) into LF(a9» for a E A(p, q), P _ q _n .
(2 .11)Remark .Let v2 E A2, and a Q such that a# -' = v2 .It is easy to check that if 5 = a 1/2p1/2, then b-1 belongs to A1 if a-1 and fi -1 belongs to A1 and 5 E A(p, q) if a and f belong to A(p, q).
In particular, the commutator of any Calderón-Zygmund operator with standard kernel will be bounded from LP(a) into LP(fi) for a, f E AP and al -1 = vP .Also the commutator of the fractional integral of order -y will be bounded from LP(aP) into L9(l9), ñ -1 = ñ, a, f E A(p, q) and af -1 = v.Analogous results are true for the second commutator assuming v = p, 2 .For the case of the Hilbert transform see [1], for the case of singular integrals with unbounded kernel see [7], and for .the case of fractional integrals, see [3] for an unweighted version .
B. Let T, k, ca, f3, v, a and b as in application A ; and assume that in addition k satisfies We define and 3) for any pair a, f3 E A(p, q), v = af -1 , the operator C6 is bounded from LP(aP) into L9(pq), and the operator exists a.e. for f E LP(aP) and it is bounded from LP(aP) into (3.4) for any pair ce, ,3 E A(p, q), v 2 = a,0 -1 , the operator from LP(aP) into L9(ag), and the operator exists a.e. for f E LP(aP) and it is bounded from LP(cJ) into L9(fl9) .
The proof of (3.3) in the case p = q can be found in [8] ; here we shall give a sketch for the case (3.4) .
It is clear that choosing 0 as above and such that X[-, , ,] < we can deduce that the operators To see that (b) =* (c) observe that with this election of q o we have pá = qo and _ ( v j1 v1 )1/qo E A(p,q), a = (vovl 1 ) 1 /qo E A(p,q), and afl -1 = v .Now we prove (c 4 .Proofs of the commutator theorems (4.1) De$nition .Let 1 <_ s < oo, E be a Banach space, v E A2, a, 0 positive functions, a and b functions belonging to BMOr-(E,E)(v) and f be an E-valued function.We define the following maximal functions .We postpone the proofs of these Propositions .Now we state and prove the following Corollaries.For the proof of these Corollaries it is enough to observe that for a sublinear operator S, the inequality IISfJILP(0) < CIIfJILP(«) a,fl E AP and a0 -1 = vP is equivalent to the inequality IISf ilL9(p9) _< CII f II LP(aP) , a, 9 E A(p, q) and a/3 -1 = v , is equivalent to the inequality IIU( 9)IILP( .)< CII9IILP(w), w E AP°) , U being the operator U(g) = S(gv-1) .
In fact, we shall give only the proof of (2.5) assuming that (2.4) is true.The proof of (2.4) is similar using remark (2.10) .
(4 .38)Lemma.Let E be a Banach space.Let Q be a cabe and Q k = 2''Q .
(4 .41)Lemma.Let E be a Banach space, if b E BMOE(v) and vi _ o,fl -I , a-t E Al , fl -t E Al then there exists e > 0 such that The proof of there lemmas can be found in Now if we choose t such that st < n n -y (1 + E) and st' < 7, where E is the one which appears in lemma (4.41), we get that the last product is less than CIIfaII 7 I QI -7/n2n fzEQ~-1(x) .
To show (4.19), choose r such that nny < r < ( (1 + E) , r' < 1 and car E A1 then by Since 2 < nn7 then p-1/2 E A1 and then we get the desired result .
Let Q be a cube in Rn with center ar xo.Given a function f with compact support, we define  We shall estimate (Cbf)*(xo) in terms of the a l (x) .Obviously 1 j Uj (x)dx < M,(Tf)(xo) .

IQ I JQ -
Now, for U2(x) choose r such that 1 -= ñ and s < (1 + e).Then using T, the boundedness properties of T, we have, On the other hand, by using hypotheses (K .1)and (K.2), we have, In order to prove (4.37), given a cube Q and a positive compactly supported function f, we decompose f into fl and f2 as before and we consider wQ = f IaQ -a(y)IIbq -b(y)IW(xo,y)f2(y)dy .