WEIGHTED NORM INEQUALITIES FOR GENERAL MAXIMAL OPERATORS

The main purpose of this paper is to use some of the results and techniques in [9] to further investigate weighted norm inequalities for Hardy-Littlewood type maximal operators.


Introduction
In [13] Muckenhoupt proved the fundamental result characterizing all the weights for which the Hardy-Littlewood maximal operator is bounded ; the surprisingly simple necessary and sufficient condition is the so called AP-condition (see below).A different approach to this characterization was found by Jawerth (cf .[9]) .An advantage with this approach is that it generalizes to more general situations ; for instante, to Hardy-Littlewood type maximal operators, obtained by replacing the cubes by any collection of sets in R"°, and to spaces of homogeneous type.For a general introduction, and historical comments we refer to [7].
The main purpose of this paper is to use some of the results and techniques in [9] to further investigate weighted norm inequalities for Hardy-Littlewood type maximal operators .We start by introducing some notation .By a basis !3 in Rn we mean a collection of open sets in R" .We say that w is a weight associated to the basis 13 if w is a non-negative measurable function in Rn such that w(B) = fB w(y) dy < oo for each B in 13.Mr3,w is the corresponding maximal operator defined by Mrt,wf(x) = s u á w(B) Ájf(y)jw(y)dy if x E UBEt3 and MB ,f(x) = 0 otherwise .If w -1, we just write MBf(x).
We say that the weight w belongs to the class A,,j3, 1 < p < oo, if there is a constant c such that IBI ~B w(y) dy) ~BI JB w(J) 1-P' dy) P-1 <_ c for all B E 3. p' will always denote the dual of p, that is C .PÉREZ In the limit case p = 1 we have that w belongs to the class A1, 13 if for all B E 13; this is equivalent to saying almost everywhere x E Rn .For the other limit case, p = oo, we set It follows from these definitions and H&lder's inequality that if1<p<q<00 .
In section (5) we shall use also the following notation .The weight w belongs to the class Ap u (dp), 1 < p < oo, if these is a constant c such that if x E UBEa and Ma ,d p f(x) = 0 otherwise, with (wdF,)(B) = fB w(y) dp(y) .
One of the main results in [9] is the following Theorem .if and only if dlc = w(y)dy, with w E AP 2 .Here Q is the basis of all open cubes in R" .
A key fact concerning Theorem 1.1 is that the proof completely avoids the (difficult) "Reverse H6lder inequality." Acknowledgements .The content of this paper is part of my Washington University Ph.D. Thesis .I would like to express my deepest gratitude to my teacher Bjórn Jawerth for his guidance and all his teaching.I also would like to thank R. Howard and A. R. Schep from the University of South Carolina, for several conversations concerning their work in [8] .The referee has made several useful observations for which I am grateful.Finally, it is a great honour for me to dedicate this work to the memory of José Luis Rubio de Francia .He introduced me to the field of Harmonic Analysis, and, later, always kindly supported and encouraged me.

. One-weight theory
It is a fundamental fact that M2, . is bounded in LP(w) for each 1 < p < oo, if the weight w is doubling (cf .[7] p.144 ).In particular MQ,. is bounded if w is a A,,,2 weight.R. Feferman in [3] and B. Jawerth and A. Torchinsky in [11] (also cf.[7] p .463) proved that the weighted strong maximal operator M-R,,,, that is the weighted maximal operator associated with the basis 13 = R of all rectangles in R'°with sides parallel to the coordinate axes, is also bounded in LP(w) whenever the weight w belongs to the class A,,.-.The proof of this result is based on a geometric covering lemma which goes back to the work of Córdoba (cf.[1]) .In this section Nve show that these results are particular phenomena of a general fact.Theorem 2.1 .Let Ci be a basis.The following statemenis are equivalent.i) For each 1 < p < oo, arad whenever w E A P ,5 (1) Mi3 : LP(w) --, LP(w) ; ü) for each 1 < p < oc, and whenever w E A .,13 (2) Mj3, .: L P (w) -+ LP(w) .
Proof: Assume that the basis 8 satisfies ii) .Assuming now i), we fix 1 < p < oo, and we take w E A. .. Suppose that w E Aq , 1 < q < oo.There are two cases.a)q<p'; b) q > p' .In the first case we have that w E AP , , which means that w' -P E Ap .Hence, by hypothesis, Me : LP1(w) -> LP'(w) M13 : LP(w 1-P) -LP(wl-P) .
since wl-9 ' E Aq, Now by using Theorem 1.1 we get Since we always have that we can interpolate to get Mr3,w : Lq'(w) -> Lq'(w) .
Example 2 .2.Let MR be the Córdóba-Zygmund maximal operator .The basis R defining this maximal operator is formed by these rectangles in R3 with sides parallel to the coordinate axes whose sidelengths are of the form {s, t, st}.It has been shown by R. Fefferman (see for instante [4] ) that MR is bounded in LP(w), for each 1 < p < oo, if and only if w E AP,R .Hence, by Theorem 2.1 MR, , : LP (w) -+ LP(w) .
for each 1 < p < oo, whenever w E A .,R In view of all these important examples we make the following definition.Definition 2 .3 .We say that the basis 13 is a Muckenhoupt basis if for each 1<p<oo,andeverywEA,a M,3 : LP(w) -> LP(w).
for each 1 < p < oo, and whenever w E A,,,s Next result is an extension of Lin's result (cf .[12]) for the strong maximal operator to any maximal operator whose basis is a Muckenhoupt basis .
Corollary 2 .4.Suppose ¡ha¡ 8 is a Muckenhoupt basis.Let 1 < p < oo, and suppose that w E A,,,s, then the following Fefferman-Stein inequality holds (4) n Msf(y) P w(y)dy C c n f(y)PMs w(y)dy .

IR R
Proof. .Suppose that w E A,,s, for some 1 < r < oo .The following pointwise inequality then follows easily from Hólder's inequality and from the A,., scondition ( 5) Since l3 is a Muckenhoupt basis, Theorem 2.1 yields (6 ) Ms,w : LP(w) --> LP(w), and together with (5) we easily conclude that for each measurable E and each 0 < A < oo the following inequality holds : Since also Ms : LP(R") -> LP(R'), we are now in a position where we can proceed as in the proof of Lemma 7.1 in [9], to conclude the proof of the Corollary.
Remark 2.5 .We point out that for (4) to hold, we do not need to assume that w E A,, .,s; (7) for some 0 < A < 1 would be sufficient .
For the sake of completness we observe that Muckenhoupt bases satisfy Jones' factorization theorem .We just state the result and refer the reader to [9], Corollary 6 .1, for the proof under a weaker condition on the basis .Proposition 2.6.Suppose that Ci is a Muckenhoupt basis, and let 1 < p < oo .Suppose that w E AP ,s .Then there are weight.sw1,w2 E Al,s such that w = WJw2-P .

. Vector-valued inequalities
It is well known by now that there is an intimate connection between vector valued inequalities and weigthed norm inequalities (cf.[7] Chapter 5).In this section we shall use the results from the previous one to obtain an extension of the classical Fefferman-Stein vector-valued inequality (cf.[2]) lIq lIq where 1 < p < oo and 1 < q <_ oo, to any maximal operator ML3 whose basis is a Muckenhoupt basis (cf .the definition above) .We would like to point out that (8) has played a fundamental role in the analysis made in the recent works [5] and [10] .We shall assume throughout the section that X3 is a Muckenhoupt basis.
For a fix 1 < p < oc, and motivated by the method introduced by J. L. Rubio de Francia (cf.section 5 .5 in [7] and section 6 in [9] ), we denote by R13 the operator ' , where MI = Id and M,'3 is the ith iterate of the operator M,3 .K is the norm of M13, as an operator on LP(R") .Although RL3 is pointwise larger than ML3, it preserves most of its properties, namely i) u<RL3u IIRi3 UJILP(R^) :~2 IIuJILP(R^)' Furthermore, RB has the property that iii) for each f E LP(R") .The last statement is not true for MB f as the following argument shows .Consider the Hardy-Littlewood maximal operator 117 = MQ, and take any positive function cp E L l (R") .We shall show that DrlW is not even an A,,,, weight .Indeed, suppose that Mcp  Proof. .The first inequality follows from above remarks and (4) .For the second we observe that (RL3 U)-11t = [(R5 u )1 /t(P -1 ) J 1-P E AP,t1, and this, in turn, follows from the fact that if w E A1,8 then wó E A1,r3, if 0 _< 6 < 1, and from the easy part of (2.6) .Finally, (9) follows from the definition of Muckenhoupt basis and from above remarks about RB .Remark 3.2.We point out that for the case B = Q the following inequality holds : (10) IR-Mf(y)P M(u)(y)1/t < cJR-f(y) P uy , for 1 < p < oo and t > 1 1 .The result is false if t = P 1 1 , (cf .[14]) .

P_-
We may think of Rt; as being the right substitute in the general case.Once we have this lemma then the vector-valued inequality for Mt; follows from Theorem 5 .2Chapter 5 in [7].Theorem 3.3.Leí 1 < p < oo, and 1 < q _< oo .If B is a Muckenhoupt basis, then 1/q 1/q ~~I MCifi I ql < CP,q ~~I fi q q-o La i-o Ly Remark 3.4.Although we have mentioned that the theorem follows from Lemma 3.1, it is to be mentioned that we just need the first half of it .Indeed, this and a standar procedure would give the case q < p. Now, since the theorem is obvious for q = p, and also for q = oo, the case 1 < p < q < oo is obtained by interpolation .
In the same spirit we make the following observation about the class A1 ,1;.Lemma 3.5 .Assume that B is a Muckenhoupi oasis.Le¡ 0 < q <_ 1, and let w E UP>1LP(Rn) .Then if and only if there is a positive function g E Up>1LP(Rn), such that for some large enough constant A (12) Since obviously Proof: By iteration Mgw <_ c'w, for some constant c .Then putting A c2 1 / 9 and g = w we get < 2w .(12) follows.
To prove the converse, let G denote the right hand side of (12).Since w ',Z~G, it is enough to deal with G.We first show that G is in UP>1LP(Rn) .Indeed, suppose that g E LP(Rn ), 1 < p < oo .Then for some u E L(Pl9)'(Rn) with unit norm.Therefore, by the above remarks and by iterating (4), the last expression is dominated by 00 C M~(y) ) v Reu(y)dy< g(y)9 RBu(y)dy.

An alternative formulation of Muckenhoupt's theorem
In this section we give a different criterion to decide whether the operator M13 is bounded on LP(w), assuming that the basis is a Muckenhoupt basis.In particular this result applies to the case X3 = Q, providing a different characterization of Muckenhoupt's theorem.This approach is inspired by the results in [8] .Theorem 4.1 .Le¡ 1 < p < oo, and suppose that 13 is a Muckenhoupt basis.Let w be a weight for 13 .Then (13) Mí3 : LP(w) --> LP(w) if and only if (14) Me (w (M6go) P-1) < c w go 1,-1, for some nonnegative measurable function go , with fB go (y) dy < co for each B in 13, and for some positive constant c.
Proof. .Assuming (13) it is standard to see that w E AP L3, and hence o = W1-P' E A P , ,r3 .Since Li is a Muckenhoupt Hence, by using the iteration technique of Rubio de Francia (cf.[7] p. 434, [9] p.392, or the previous work of Gagliardo in [6] ), there exists a nonnegative measurable function h E LP'(Rn) for which or, equivalently, Taking go = w -1/php'lp we obtain (14) .Note that fB g o (y) dy < oo for each B in 13 since To prove the converse, we note that it is enough to show that w satisfies the Ap,g -condition since 8 is a Muckenhoupt basis.For each B E 13 we define the constant A by We denote by RHq B, 1 < q < oc, the class of weights satisfying a reverse H51der inequality of order q uniformly on each B E 13 .That is (15) RHq ,a = {w : ( PBI B w(y) q dy) 1/q < PBI Á w(y) dy, B E r3} for some positive constant c, independent of B E 13.
To illustrate the interest of the reverse H51der classes, consider the simpler operator if and only if where B is an arbitrary fixed set in 13.It readily follows from the next (easy) lemma that for 1 < p < oo mw,B : LP(R") -+ LP(R") w E RHP ,,a .Proof.. ii) follows from i) by taking f = wP'/P, and i) follows from ii) by H51der's inequality : To obtain a result for M6,u we peed a way of measuring how the different mw,B's interfere .This is intimately connected with the geometry of the particular basis 13, and hence to covering properties of families of sets belonging to B. This, in turn, is essentially equivalent to mapping properties of the maximal operators .
Our characterization is the following Theorem 5. Proof. .Let dw denote the measure dw = wdx .Noticing that w E RHP, , a is equivalent to w-1 E AP,g(dw), and writing dx = w-ldw, and wP' = o,-Idw, where o, = w1-P', the following equivalente follows And this concludes the proof of the Theorem .
Here Mp,w is the following weighted fractional maximal operator IQl p~n MO,wf(X) = s p w(Q) f Q f(y) w(y)dy, where the supremum is taken over all cubes.Remark 5.6.There is another proof of the nontrivial part of Corollary 5.4, closer in spirit to the classical proof of Muckenhoupt's theorem .Suppose w E RHp ,,2 .It was discovered by Gehring that w E RH(p_E),,2 for some tiny e > 0 .Then by Lemma 5.  Proof.As in the proof of Corollary 5.4 we just need to check both MR LP(Rn) -+ LP(Rn ) and MR ,,, : LP(wP') -> LP(wP') .The first one is the classical Theorem of Jessen, Marcinkiewicz and Zygmund .Now, since R is a Muckenhoupt basis, (3) yields the boundedness of MR,w,, if we show that wp'EA .7z<!=> wERHp,R .
To prove this, we use Theorem (6 .7)p. 458 in [7], and the proof of the case Li = Q in [17] applies mutatis mutandis to the case fi = 9Z.

Two weight theory
In this section we discuss two weighted norm inequalities for MS.We shall extend the two weights results in [9] to the Lorentz spaces .
We recall that a function f belongs to the Lorentz space L(r, s) if IIfIIL(r,,)(j .)_ [100 (tlu{x E R'1 : I&)I > t}'/r)' dtl 1/s < 00 Theorem 6.1 .Le¡ 1 < p, q < oo, and le¡ (v, u) be a couple of weights .Call v = u' -P' .Assume that the basis 13 satisfaes the condition that for every set G which is a union of sets in Ci the following holds Then for each smooth f We view the sum Ek j Pk jgk , j, as an integral on a measure space (X, tt) built over the set X = {k, j}, assigning to each (k, j) the measure [¿k,j.For A > 0, set Then We can estimate p(F(,\)) as follows f MI; (o-XB,k,i )(J) 9 v(J)dy < _ (k,j)Er(A) Ekj   < cu(G(A))q/p < cv({J E R' : DIB,v(fla)(J) p > ,\})9/p Here we have used the hypothesis on AIB Q in the third inequality.Finally by making a change of variables we obtain As a consequence of this theorem we can deduce the following characterization.
Proof. .By setting f = UXG(a) in (24) we readily get (25) .To prove the converse we use Theorem 6.1, that p < q, and our hypothesis on As a consequence of this result we can obtain Sawyer's characterization of those couple of weights (v, u) for which the Hardy-Littlewood is bounded from LP(u) to Lq(v) .We just state the result since the proof is like the given for the case p = q in [7] p.432, with some obvious modifications .Corollary 6.3 .La 1 < p < q < oo, and let (v, u) be a couple of weights, ando,= u' -P' .Then (26) M : LP(u) -> Lq(v) if and only if \ 1/q (27) (IQ M(-XQ)(y)q v(y)dy) < co(Q)1/P for every cabe Q .

Lemma 5 . 1 .
Let 1 < p < oo .The following statements are equivalenL i) There is a positive constant c, independent of B E 13, such ¡ha¡ for every nonnegative locally integrable function f 1/p w(B) Á f (y) w(y)dy < c (B~Á f (y)P dy) ; w E RHP,,B.
2 .Let 1 < p < oo .Suppose thai 13 is a basis and that w is a