Potential operators, maximal functions, and generalizations of A∞

Potential analysis, 19 (2003), 1-33. Potential operators, maximal functions, and generalizations of A∞ Carlos P´ erez§and Richard L. Wheeden 1 Introduction The purpose of this paper is to prove two kinds of weighted norm inequalities for integral operators of potential type and their associated maximal operators in spaces of homogeneous type. The first kind of result is an extension of a result of Muckenhoupt and Wheeden, which showed that for A∞weights in Euclidean space, a weighted Lpnorm of the classical Riesz fractional integral operator Iαfof a function fis equivalent to the same norm of the fractional maximal function Mαf. Our extension of this result involves fairly general integral operators of potential type and their associated maximal operators in spaces of homogeneous type, and classes of weights which are more general than A∞. The second kind of inequality that we will prove gives two-weight Lp, Lqnorm estimates for such maximal operators, assuming again an appropriate (but substantially weakened) version of A∞. An important point here is that we completely avoid using the “good-lambda inequality” technique of Burkholder and Gundy. Finally, we can combine these two results to obtain two-weight Lp, Lqnorm estimates for potential operators that improve similar ones derived by Sawyer and Wheeden. In the usual n-dimensional Euclidean space Rn, if 0 < α < n, let Iαf(x) = ZRn f(y)1 |x−y|n−αdy §Partially supported by DGICYT grant PB940192, Spain AMS Subject Classification 2000: 42B25, 44A15 denote the Riesz fractional integral of fof order α, and let Mαf(x) = sup B:x∈B r(B)α−nZB |f(y)|dy be the corresponding fractional maximal function of f, where Bdenotes a Euclidean ball and r(B) is the radius of B. The pointwise inequality Mαf(x)≤C Iαf(x), with Cindependent of xand f, follows easily from the definitions. On the other hand, the reverse pointwise inequality is false, but it is shown in [MW] that Iαfand Mαfare often comparable in norm. To be more precise, if w∈A∞and 0 < p < ∞, then it is shown in [MW] that ZRn |Iαf(x)|pw(x)dx ≤cZRn Mαf(x)pw(x)dx (1) with cindependent of f. Here A∞denotes the collection of weights won Rnwith the property that there exist constants C, δ > 0 such that if Eis a Lebesgue measurable subset of a ball B, then w(E) w(B)≤C|E| |B|δ , where w(E) = REw(x)dx and |E|is the Lebesgue measure of E. This estimate has had applications in potential theory, such as in the proof of the Hedberg–Wolff theorem [AH] concerning nonlinear potentials. The corresponding theorem for potential operators of convolution type plays a role in the proof of the characterization by Kerman and Sawyer of trace type inequalities, which in turn have applications to eigenvalues estimates for Schr¨odinger operators (see [KS]). Inequality (1) is also related to the fact that the positive cone for the Triebel-Lizorkin space Fα,q p,α < 0, is independent of q; see [JPW] and [AH]. 2 An example of the first kind of result that we will prove is an estimate similar to (1) but with Rnreplaced by a space Sof homogeneous type, Iαreplaced by a potential operator T=TKof the form Tf(x) = T(fdµ)(x) = ZS f(y)K(x, y)dµ(y) (2) where µis the underlying doubling measure on S(see §3 for the exact definitions of a space of homogeneous type and a doubling measure), and with Mαfreplaced by Mϕf(x) = Mϕ(fdµ)(x) = sup B:x∈B ϕ(B)ZB |f(y)|dµ(y) (3) where ϕ(B) = ϕK(B) is a functional which acts on balls and is defined by ϕ(B) = sup x,y∈B d(x,y)≥cr(B) K(x, y) (4) for a sufficiently small positive geometric constant c(see [SW1]). Here d(x, y) denotes the quasimetric associated with S. For example, in the case of the Riesz potential we have K(x, y) = |x−y|α−n,0< α < n, so that ϕ(B)≈r(B)α−n, and then Mϕreduces to the fractional maximal operator Mα. Other examples of operators of the forms (2) or (3) arise from important differential operators; see the next section for more details. In our generalization of (1), we will assume that the weight satisfies a condition that is analogous to the A∞condition described above but with Lebesgue measure replaced by a notion of content, such as Hausdorff content, on the right-hand side. An example of the second kind of result that we will prove is the two-weight estimate ZS {Mϕ(fdµ)w}qdµ1/q ≤CZS (|f|v)pdµ1/p with 1 < p ≤q < ∞and Cindependent of f. Such estimates have been studied extensively, but the nature of the condition that we shall impose on the weights is different from elsewhere. In particular, in addition to the necessary condition ϕ(B)ZB wqdµ1/q ZB v−p0dµ1/p0 ≤c, 3 p0=p/(p−1), for all balls B, we shall assume that v−p0satisfies an appropriate A∞condition of the content type. For comparison purposes, we note that the two-weight boundedness of Mϕis studied in [PW] under a different kind of strengthening of the necessary condition, such as a Fefferman–Phong condition of the type ϕ(B)µ(B)1/p0ZB wqdµ1/q 1 µ(B)ZB v−rp0dµ1/rp0 ≤c for some r > 1. For example, weights for which just reverse doubling conditions are valid may satisfy an A∞condition of content type but not a Fefferman–Phong condition. As mentioned earlier, it is possible to combine the two kinds of results that we will prove. In this way, we obtain two-weight norm estimates for potential operators Tf assuming that both weights satisfy content conditions. 2 Statements of the main results Following [SW1], we consider potential operators of the form T(fdσ)(x) = ZS f(y)K(x, y)dσ(y),(5) where Sis a space of homogeneous type with underlying doubling measure µ, and σis any Borel measure on S. This definition agrees with (2) in case σ=µ. The exact definition of a space of homogeneous type is given is §3; by a doubling measure, we mean a Borel measure µ with the property that there is a constant Csuch that for every “ball” B⊂ S, µ(2B)≤Cµ(B). As usual, 2Bdenotes the ball with the same center as Bbut twice the radius. If d(x, y) is the corresponding quasimetric in S, we will always assume that the kernel K(x, y) is nonnegative 4 and satisfies the following growth conditions: there exist constants C1, C2strictly larger than 1 such that K(x, y)≤C1K(x0, y) if d(x0, y)≤C2d(x, y),(6) K(x, y)≤C1K(x, y0) if d(x, y0)≤C2d(x, y). The main classical examples of such operators are the Riesz integrals Iαfmentioned in the introduction. An important class of examples for metrics other than the usual Euclidean metric consists of potential operators related to the regularity of subelliptic differential equations. In particular, vector fields of H¨ormander type ([H]) as well as the classes of nonsmooth vector fields studied in [FL] and [SW2] lead to integral operators of the type we will study. In addition, the differential operators of Grushin type considered in [FGuW] (at least in the simplest case of Lebesgue measure) are related to integrals of type (2). In fact, for all these examples the associated potential operator has the form Tf(x) = ZS f(y)d(x, y) µ(B(x, d(x, y))) dµ(y) (7) where d(x, y) is a distance function that is naturally related to the vector fields and B(x, r) denotes the corresponding ball with center xand radius r. Associated with Kis the functional ϕ=ϕKdefined in (4) by ϕ(B) = sup x,y∈B d(x,y)≥cr(B) K(x, y) for a sufficiently small positive geometric constant c. As mentioned earlier, ϕ(B)≈r(B)α−n in the case of the Riesz fractional integral Iα. In the subelliptic case (7), note that ϕ(B)≈r(B)/µ(B). The conditions (6) on Klead to useful growth properties of ϕ. If Bis a ball and θ > 0, let θB denote the ball concentric with B whose radius is θr(B). It is shown in [SWZ, (4.2) and 5 (4.3)] that if θ > 1, then there is a constant Cdepending only on θ, C1, C2, the constant cin (4) and geometric properties of Sso that ϕ(B)≤Cϕ(θB) for all balls B⊂ S.(8) Also, for such a constant C(but now one that is independent of θ), ϕ(B)≤Cϕ(B0) for all pairs of balls B0⊂B. (9) We shall assume in some of our results that ϕsatisfies the following additional condition for some  > 0: ϕ(B1)µ(B1)≤Cr(B1) r(B2) ϕ(B2)µ(B2) if B1⊂B2.(10) For example, in the case of the fractional integrals Iα, we can pick =αin (10); for the operator in (7), we can choose = 1. For any Borel measure σ, we define the maximal operator Mϕ(fdσ)(x) = sup B:x∈B ϕ(B)ZB |f|dσ. (11) Note that if σ=µ, this definition agrees with (3). The pointwise estimate T(fdσ)(x)≥cMϕ(fdσ)(x) is easy to show by using the assumptions on K. Our first main theorem will show that the opposite inequality often holds in norm. Thus the theorem generalizes (1) for classical fractional integrals. In fact, the result improves (1) not only by extending it to spaces of homogeneous type but also by allowing a larger class of weights even in the usual Euclidean case. In order to state the result, we first define a suitable class of measures in a space of homogeneous type. The definition is motivated by a similar one in [SW1] for the usual Euclidean case, and it is most conveniently phrased in terms of a grid Dmof dyadic cubes Q. Here mis a large negative integer which indexes the edgelengths `(Q) of the smallest cubes Q∈ Dm, namely, the smallest edgelengths 6 are ρmfor an appropriate geometric constant ρ > 1, and each cube in the grid has edgelength ρkfor some k≥m(see §3 for details about the grid). The cubes in Dmhave the important nestedness property that if Q1, Q2∈ Dmand Q1∩Q26=∅, then either Q1⊂Q2or Q2⊂Q1. Also, each cube Q∈ Dmis contained in a ball B(Q) whose radius is comparable to the edgelength of Q;B(Q) is called the containing ball of Q. Given an integer mand a nonnegative functional τ(B) of balls B, we define the notion of the τcontent of a set E,||E||τ,m, as follows. If E⊂ S, let kEkτ,m = inf{X i τ(B(Qi)) : Qi∈ Dm, E ⊂ ∪Qi}. Thus kEkτ,m is a sort of Hausdorff content of Eassociated with τand the grid Dm. Typical choices of τare τ(B) = r(B)βfor some β > 0, and also τ(B) = µ(B). The first choice corresponds to Hausdorff content of Eof dimension β. By the nestedness property of dyadic cubes, if E⊂Q∈ Dmthen in the definition of kEkτ,m we may assume that all Qi⊂Q. We say that a measure ν∈Ady ∞(τ) if there are positive constants C, δ independent of E, Q and msuch that ν(E) ν(Q)≤CkEkτ,m τ(B(Q))δ if E⊂Q∈ Dm,(12) where ν(E) denotes the ν-measure of E. In case νis absolutely continuous with respect to µ, say dν =w dµ, we say that w∈Ady ∞(τ) if w dµ ∈Ady ∞(τ). We also use the notation w(E) for the w dµ-measure of E: w(E) = REw dµ. If dν =w dx on Rn, it is not hard to see that when τis chosen so that τ(B) = r(B)n(≈ |B|), then w∈Ady ∞(τ) is the same as w∈A∞if wis a doubling weight. If we choose τ(B) = r(B)β,β > 0, for all balls B, then the class Ady ∞(τ) was defined in [SW1]. In this case, we will use the notation Aβ ∞instead of Ady ∞(τ), i.e., Aβ ∞=Ady ∞(τ) when τ(B) = r(B)β. 7 Related notions based on balls instead of dyadic cubes can be obtained by defining Hτ,B(E) = inf{X i τ(Bi) : E⊂ ∪Bi⊂B}if E⊂B and replacing (12) by the assumption ν(E) ν(B)≤CHτ,B(E) τ(B)δ if E⊂B. (13) If νis a doubling measure, these two notions turn out to be the same for many functionals τ; see §3 for details. Recall that we always assume that the underlying measure µis a doubling measure. We say that a Borel measure νsatisfies the reverse doubling condition of order β,β > 0, if there is a positive constant csuch that (14) We write ν∈RDβfor such a measure ν. Any doubling measure belongs to RDβfor some β > 0 by [W, p.269]. We will show in §3 that if νis a doubling measure and ν∈RDβ, then ν∈Aβ ∞for the same value of β. In fact, the assumption that νis a doubling measure is not needed in order to conclude that ν∈Aβ ∞if νsatisfies the following dyadic reverse doubling condition uniformly in m: ν(Q2)≥c`(Q2) `(Q1)β ν(Q1) if Q1, Q2∈ Dm, Q1⊂Q2.(15) We write ν∈RDdy βfor such ν. We can now state our first main theorem. In it, we assume that τ(B) is a functional for which there are constants c,  > 0 with a)τ(B1)≤cτ(B2) if B1⊂B2 b)τ(2B)≤cτ(B) for all balls B(16) c)ϕ(B1)τ(B1)≤cr(B1) r(B2) ϕ(B2)τ(B2) if B1⊂B2. 8 Theorem 2.1 Let σand ωbe Borel measures on S,T(fdσ)be defined by (5) for a kernel which satisfies (6), and Mϕ(fdσ)be defined by (11) with ϕas in (4). Let τbe a nonnegative functional which satisfies (16). If ω∈Ady ∞(τ)and 1≤p < ∞, then ZS |T(fdσ)|pdω ≤CZS (Mϕ(fdσ))pdω (17) with Cindependent of f. In particular, if τsatisfies (16) and w∈Ady ∞(τ), then ZS |T(fdµ)|pw dµ ≤CZS (Mϕ(fdµ))pw dµ. (18) In the important special case when τ(B) = r(B)βfor β > 0, Theorem 2.1 leads to the next corollary assuming that µsatisfies the doubling condition of order D, i.e., that µ(B)≤Cr(B) r(˜ B)D µ(˜ B) for all balls ˜ B⊂B. Any doubling measure satisfies the doubling condition of order Dfor some D > 0. Corollary 2.2 Let 1≤p < ∞,µbe a doubling measure of order D, and Kbe a kernel such that ϕsatisfies (10) for some  > 0. If ωand σare Borel measures on Sand ω∈Aβ ∞with β+>D, then ZS |T(fdσ)|pdω ≤CZS |Mϕ(fdσ)|pdω. (19) In particular, (19) holds if either ω∈RDdy βwith β+ > D, or if ωis a doubling measure and ω∈RDβwith β+ > D. For example, in the usual n-dimensional Euclidean situation with dµ =dx (so that D=n), Corollary 2.2 includes the result (1) from [MW] by choosing dω =w(x)dx and β=n since A∞⊂An ∞. In fact, Corollary 2.2 improves the result in (1) by showing that (1) is valid if w∈Aβ ∞for any β > n −α, since the value of in (10) for the Riesz operator Iαis α. Corollary 2.2 thus extends the result of [R] showing that (1) is valid for any doubling weight wwhich satisfies the reverse doubling condition of order βfor some β > n −α. 9 which implies (32). We will use a grid of dyadic sets in Swhich are “almost balls”, as constructed in [SW1]. In fact, the following has been proved there: If ρ= 8κ5, then for any (large negative) integer m, there are points {xk j}and a family Dm={Ek j}of sets for k=m, m + 1,· · · and j= 1,2,· · · such that •B(xk j, ρk)⊂ Ek j⊂B(xk j, ρk+1) •For each k=m, m + 1,· · · , the family {Ek j}is pairwise disjoint in j, and S=∪jEk j. •If m≤k < l, then either Ek j∩ El i=∅or Ek j⊂ El i. We call the family D=∪m∈ZDma dyadic cube decomposition of Sand refer to the sets in Das dyadic cubes. A dyadic cube will usually be denoted by Q, and Q∗will denote the containing ball described above with 1 ρQ∗⊂Q⊂Q∗; thus, if Q=Ek jthen Q∗=B(xk j, ρk+1). We set `(Q) = r(Q∗)/ρ and call `(Q) the “sidelength” of Q. We note that while the cubes in each Dmhave the dyadic properties listed above, no nestedness properties of the cubes in Dm1 relative to the cubes in Dm2are assumed if m1and m2are different. For any Borel measure ω, define Mω,mg(x) = sup B:x∈B r(B)≥ρm 1 ω(B)ZB |g|dω, and also the dyadic version Mdy ω,mg(x) = sup Q:x∈Q Q∈Dm 1 ω(Q)ZQ |g|dω. If ωis absolutely continuous with respect to µ, i.e., dω =w dµ, then we write Mw,mgand Mdy w,mginstead of Mwdµ,mgand Mdy wdµ,mg. As usual, we say that wis a weight if w(x) is a 16 nonnegative locally integrable function with respect to µ, and for a measurable set E, we write w(E) = REw(x)dµ(x). Let us show, as mentioned in §2, that (12) and (13) are identical notions for many functionals τif νis a doubling measure. In fact, we will show this is the case if τjust satisfies (16)(a),(b). Let νbe a doubling measure. First, suppose that (13) holds for ν, and let E⊂Q∈ Dm. Given η > 0, select cubes {Qi}in Dmwith E⊂ ∪Qi⊂Qand Pτ(B(Qi)) <kEkτ,m +η, where B(Qi) denotes the containing ball of Qi. Note that E⊂ ∪B(Qi)⊂cB(Q) for some geometric constant c which is independent of m, E, Q, and {Qi}. By (13), ν(E) ν(cB(Q)) ≤CHτ,cB(Q)(E) τ(cB(Q)) δ ≤CPτ(B(Qi)) τ(cB(Q)) δ . By the properties of containing balls, and since νis doubling and (16)(a) holds, we obtain ν(E) ν(Q)≤CPτ(B(Qi)) τ(B(Q)) δ ≤CkEkτ,m +η τ(B(Q)) δ , and (12) follows by letting η→0. Conversely, suppose that (12) holds for a doubling measure ν, and let E⊂ ∪Bi⊂B. We want to show that ν(E) ν(B)≤CPτ(Bi) τ(B)δ for some δ > 0, i.e., that (13) holds. Let mbe so large that ρm<< r(B). Cover Bby a finite number N1of disjoint dyadic cubes Q∈ Dmwith `(Q)≈r(B), where N1and the constants of equivalence are independent of Band the grid Dm. By the doubling of ν,ν(B)≈ν(Q) for each such Q, with similar constants of equivalence, assuming as we may that each Qtouches B. By considering the sets E∩Qindividually and adding, we may assume that E⊂Qfor one such Q. Also, by considering those Biwith r(Bi)≥ρm, and eventually letting m→ −∞, 17 we may assume that all r(Bi)≥ρm. Cover each Biby a finite number N2(independent of i and m) of cubes {Qij}N2 j=1 in Dmwith r(Bi)≈`(Qij) uniformly in i, j, m. It follows easily from the properties of the quasimetric that Qij ⊂cBifor a uniform positive constant c. Since E⊂ ∪i,jQij and we may discard any Qij not contained in Q, it follows from (12) that ν(E) ν(Q)≤CPi,j τ(B(Qij)) τ(B(Q)) δ =CPiPjτ(B(Qij)) τ(B(Q)) δ ≤CN2Piτ(Bi) τ(B)δ because the number of j’s is at most N2and (16)(a),(b) hold. Since ν(B)≈ν(Q), a similar estimate holds with ν(Q) replaced by ν(B) in the denominator on the left, which gives the desired inequality. As usual, we say that a measure νbelongs to A∞(µ) if there are positive constants Cand ηsuch that ν(E)≤Cµ(E) µ(B)η ν(B) for every ball Band every measurable set E⊂B. It is not difficult to show that if ν∈A∞(µ) then ν∈Ady ∞(µ) (in the sense of definition (12)). In fact, let Ebe a set with E⊂ ∪Qi⊂Qfor Qi, Q ∈ Dm. If ν∈A∞(µ), it follows easily that νis doubling, and then ν(E) ν(Q)≤cµ(E) µ(Q)η for some η > 0 since Qcan be included in a ball of comparable measure. Therefore, since E⊂ ∪B(Qi), ν(E) ν(Q)≤cXµ(B(Qi)) µ(Q)η ≤cXµ(B(Qi)) µ(B(Q)) η since µis doubling. Consequently, ν∈Ady ∞(µ). Let us now show that any measure νwhich satisfies the dyadic reverse doubling condition RDdy β(see (15)) belongs to Aβ ∞for the same value of β. A similar fact was shown in [SW1] in 18 the Euclidean case. If E⊂ ∪Qi⊂Qwith Qi, Q ∈ Dm, then ν(E)≤X i ν(Qi)≤cX i`(Qi) `(Q)β ν(Q) since ν∈RDdy β. Defining τby τ(B) = r(B)βfor all B, it follows that ν(E)≤cX i τ(B(Qi)) τ(B(Q)) ν(Q), where B(Qi) and B(Q) are the containing balls of Qiand Q. Taking the infimum over all such coverings {Qi}of E, we obtain that ν(E)≤ckEkτ,m τ(B(Q)) ν(Q), so that (12) holds with δ= 1 and thus ν∈Aβ ∞. The conclusion that ν∈Aβ ∞also holds if the hypothesis that ν∈RDdy βis replaced by the assumptions that ν∈RDβand νis a doubling measure; in fact, these assumptions are easily seen to imply that ν∈RDdy β. In passing, we note that if the underlying measure µ∈Ady ∞(τ) and if ν∈A∞(µ), namely, for some η > 0, ν(E) ν(B)≤Cµ(E) µ(B)η if E⊂B, then also ν∈Ady ∞(τ) for the same τ(although the value of δfor νequals ηtimes the value of δfor µ). This follows easily from the definitions since µand νare doubling measures. 4 Orlicz spaces and Orlicz maximal functions To prove Theorem 2.3, we will use some facts about Orlicz spaces which we recall here, referring to [RR] and [BS] for a complete account. A function Φ : [0,∞)→[0,∞) is called a Young function if it is continuous, convex, increasing and satisfies Φ(0) = 0 and Φ(t)→ ∞ as t→ ∞. It follows that Φ(t)/t is increasing, 19 and in particular that Φ(γt)≥γΦ(t) if γ≥1 and t≥0. Sometimes we will also assume that Φ satisfies the doubling condition Φ(2t)≤CΦ(t). For Orlicz norms we are usually only concerned about the behavior of Young functions for tlarge. By definition, the Orlicz space LΦconsists of all measurable functions fsuch that ZS Φ|f| λdµ < ∞ for some positive λ. Note that if 0 < λ1< λ2, then Φ|f| λ2≤λ1 λ2 Φ|f| λ1, so that lim λ→∞ ZS Φ|f| λdµ = 0 if f∈LΦ. The space LΦis a Banach function space with the Luxemburg norm kfkΦ=kfkΦ,µ = inf{λ > 0 : ZS Φ(|f| λ)dµ ≤1}. Each Young function Φ has an associated complementary Young function ¯ Φ satisfying t≤Φ−1(t)¯ Φ−1(t)≤2t(33) for all t > 0, where Φ−1stands for the inverse function of Φ. The function ¯ Φ is called the conjugate of Φ, and the space L¯ Φis called the conjugate space of LΦ. For example, if Φ(t) = tpfor 1 < p < ∞then ¯ Φ(t) = tp0, p0=p/(p−1), and the conjugate space of Lp(µ) is Lp0(µ). An example that we will need is Φ(t)≈tp(log t)−1−for large t, 1 < p < ∞, > 0, with complementary function ¯ Φ(t)≈tp0(log t)(p0−1)(1+)for large t(cf. [O], p.275). A very important property of Orlicz spaces is the generalized H¨older inequality ZS |fg|dµ ≤ kfkΦkgk¯ Φ.(34) 20 In order to define another maximal function which will play a role in the proof of Theorem 2.3, we need local versions of Orlicz norms. If Φ is a Young function and Bis a ball, let kfkΦ,B =kfkΦ,B,µ = inf{λ > 0 : 1 µ(B)ZB Φ(|f| λ)dµ ≤1}. For this norm, the local version of the generalized H¨older inequality (34) is 1 µ(B)ZB fg dµ ≤ kfkΦ,Bkgk¯ Φ,B.(35) Define a maximal function corresponding to Φ by MΦf(x) = sup B:x∈B kfkΦ,B.(36) This maximal function has been used in the usual Euclidean context in [P1] and also in the case Φ(t)≈tlog tin the work of T. Iwaniec and Greco [GI] and in [WW]. The norm behavior of MΦfis closely related to the next definition. Definition 4.1 Let 1< p < ∞. A nonnegative function Φ(t), t > 0,satisfies the Bpcondition if there is a constant c > 0such that Z∞ c Φ(t) tp dt t<∞.(37) Simple examples of functions which satisfy Bpare tp−δand tp(log(1 + t))−1−δ, both when δ > 0. The relevance of condition Bpstems from its relationship to the boundedness of MΦas stated in the next theorem from [PW]. Theorem 4.2 Let 1< p < ∞and Φbe a doubling Young function. Then the following statements are equivalent. i) Φ∈Bp, i.e., there is a constant c > 0such that Z∞ c Φ(t) tp dt t<∞.(38) 21 ii) There is a constant C > 0such that ZS MΦf(x)pdµ(x)≤CZS f(x)pdµ(x) (39) for all nonnegative f. iii) There is a constant C > 0such that ZS MΦf(x)pw(x)dµ(x)≤CZS f(x)pMw(x)dµ(x) (40) for all nonnegative fand w, where Mw is the Hardy–Littlewood maximal function defined in (20). For example, in the standard case when Φ(t) = trwith r≥1, so that kfkΦ,B =µ(B)−1RB|f|rdµ1/r, the equivalence of (38) and (39) reduces to the well-known fact that the mapping f→sup B:x∈B1 µ(B)ZB |f|rdµ1/r is bounded on Lp(S, µ) if and only if p>r. The characterization of Bpgiven above was proved in the Euclidean context in [P3] and used to derive sharp two weight estimates for the classical Hardy–Littlewood maximal function. In the general case, the characterization of Bp plays a main role in some of the results in [PW]. For other applications to different operators from harmonic analysis, see [P1], [P4] [P5], [P6], [CP1] and [CP2]. 5 Proofs of Theorem 2.4 and Corollary 2.5 Recall that Mψ(fdσ) is defined by Mψ(fdσ)(x) = sup B:x∈B ψ(B)ZB |f(y)|dσ(y) for any measure σand any measurable f, where ψ(B) is assumed to be nonnegative and to satisfy (23) and also the doubling condition. To prove Theorem 2.4, we may assume that fis 22 nonnegative, bounded and has bounded support. We may also assume by a limiting argument that Mψ(fdσ)(x) is formed by taking the supremum only over balls containing xof radius at least ρmfor fixed m, where ρis the constant used in §3 to construct the dyadic grid Dm. For each integer kwe let Ωk={x∈S:Mψ(fdσ)(x)> γk} where γ > 1 is a constant to be chosen. For each x∈Ωk, there is a ball Bxcontaining xwith ψ(Bx)ZBx f dσ > γk. Now we claim the following: To each Bx, there corresponds a dyadic cube Qx∈ Dm(Qxmay not contain x, although Bxdoes) of size comparable to Bxwith Qx∩Bx6=∅and ψ(B(Qx)) ZQx f dσ > γk c, where cis a geometric constant. Recall here that if Qis any dyadic cube, then B(Q) denotes the containing ball of Q; the radius of B(Q) is comparable to the edgelength of Q. To prove the claim, first note that we can cover Bxby a fixed number N(independent of x, γ, k) of disjoint dyadic cubes Qof size comparable to Bx. Indeed, let k0be the integer such that ρk0≤r(Bx)< ρk0+1, and consider any mwith m<k0(here ρis the constant used in §3) to construct the dyadic grid Dm. We make the following subclaim: In Dm, there are at most Ncubes {Ek0 j}jmeeting Bx(i.e., there are at most N cubes of sidelength ρk0meeting Bx), where Nis a structural constant which is independent of Bxand m, provided that m<k0. 23 To prove the subclaim, fix m<k0and denote those cubes {Ek0 j}jin Dmwhich have nonempty intersection with Bxby Qj,j= 1,· · · , N. The {Qj}are disjoint, and they are contained in some fixed enlargement of Bxsince they touch Bxand their radii are comparable to r(Bx). Thus, the sum of the µ(Qj) is at most cµ(Bx). Since the size of each Qjis comparable to the size of Bx, each µ(Qj) exceeds a fixed multiple of µ(Bx) by doubling (in fact, the measures are comparable). Therefore, the number Nof Qj’s must be at most a fixed geometric constant, which proves the subclaim. Consequently, for any fixed m<k0,if Qj, j = 1, . . . , N, are the cubes in Dmmentioned above, then χBx≤PN j=1 χQj, and so γk< ψ(Bx)ZBx f dσ ≤ψ(Bx) N X j=1 ZQj f dσ. Thus for some j0, γk< N ψ(Bx)ZQj0 f dσ. Pick Qxto be Qj0. Since the sizes of Qx,Bxand B(Qx) are comparable, the claim follows by using Qx∩Bx6=∅and the properties of ψ. Choosing γ > c we have ψ(B(Qx)) ZQx f dσ > γk−1.(41) Define e Ωk=x: sup Q∈Dm:x∈Q ψ(B(Q)) ZQ f dσ > γk−1 and let {Qk j}jbe the maximal cubes in Dmwith γk−1< ψ(B(Qk j)) ZQk j f dσ. Then e Ωk=∪jQk j. Observe that if Qxis the dyadic cube from (41) then Qx⊂Qk jfor some j, and hence since there exists c0>1 such that x∈c0B(Qk j), we obtain that Ωk⊂ ∪jc0B(Qk j). 24 Now if ˜ Qk jis the next largest dyadic cube containing Qk j, then ψ(B(˜ Qk j)) Z˜ Qk j f dσ ≤γk−1 and therefore, by using the doubling property of ψ, γk−1< ψ(B(Qk j)) ZQk j f dσ ≤cψγk−1.(42) Let Mdy σ,m be defined as in Section 3. To prove the theorem it will enough to show that ZS Mψ(fdσ)qdω1/q ≤cZS (Mdy σ,mf)pdσ1/p (43) with cindependent of mand f, since Mdy σ,m is bounded on Lp(dσ) uniformly in mfor 1< p < ∞(see for example [W], Lemma 3.8). Now, we start with ZS Mψ(fdσ)qdω =X kZΩk\Ωk+1 Mψ(fdσ)qdω ≤X k γ(k+1)qω(Ωk)≤γ2qX k γ(k−1)qX j ω(c0B(Qk j)) ≤γ2qX k,j ψ(B(Qk j)) ZQk j f dσ!q ω(c0B(Qk j)). Since ψ(B(Qk j)) ≤ψ(c0B(Qk j)) by (23)(a), if we use (25) for c0B(Qk j), we can continue with ≤CX k,j σ(B(Qk j))−q/p0 ZQk j f dσ!q ≤CX k,j σ(Qk j)−q/p0 ZQk j f dσ!q , ≤C X k,j σ(Qk j)1−p ZQk j f dσ!p!q/p , where we have used the facts that Qk j⊂B(Qk j) and that p≤q. Recall that e Ωk=∪jQk j. Now let Ek j=Qk j\e Ωk+1. Then Ek j⊂e Ωk\e Ωk+1 and the sets Ek j are disjoint in both j, k. We wish to show that σ(Qk j)≤c σ(Ek j) if γis sufficiently large. It is 25 by the maximality of Ql i. Then, since the Ql iare disjoint in ifor fixed l, and since ω∈Ady ∞(τ), X i:Ql i⊂Qk j ω(Ql i) = ω(∪i:Ql i⊂Qk jQl i)≤c    ∪i:Ql i⊂Qk jQl i  τ,m τ(B(Qk j))    δ ω(Qk j) ≤c Pi:Ql i⊂Qk jτ(B(Ql i)) τ(B(Qk j)) !δ ω(Qk j)≤cγ(k−l)δω(Qk j).(50) Now let Ωk={x: sup Q∈Dm:x∈Q 1 τ(B(Q)) ZQ g dω > γk}, and set Ek j=Qk j\Ωk+1. Then Ek j⊂Ωk\Ωk+1 and the sets Ek jare disjoint in both j, k. Note that Ωk=∪jQk j. We wish to show that ω(Qk j)≤cω(Ek j) if γis sufficiently large. It is enough to show that ω(Qk j∩Ωk+1)<1 2ω(Qk j). But by the dyadic structure and our earlier observations, ω(Qk j∩Ωk+1) = X i:Qk j∩Qk+1 i6=∅ ω(Qk j∩Qk+1 i) = X i:Qk+1 i⊂Qk j ω(Qk+1 i) ≤cγ−δω(Qk j) by (50) with l=k+ 1 <1 2ω(Qk j) if γis large. Next, we rewrite the sum on the right side of (49) as X j,k ϕ(B(Qk j)) Zκ(2κ+1)B(Qk j) f dσ ω(Qk j)1 ω(Qk j)ZQk j g dω and apply H¨older’s inequality to obtain ZS Tm(fdσ)g dω ≤ c X j,k "ϕ(B(Qk j)) Zκ(2κ+1)B(Qk j) f dσ#p ω(Qk j)!1/p  X j,k "1 ω(Qk j)ZQk j g dω#p0 ω(Qk j)  1/p0 . 32 We may replace ω(Qk j) by ω(Ek j) in the numerators of both of these sums and then use the disjointness of the Ek jto majorize the last expression by cZS Mϕ(fdσ)pdω1/p ZS (Mdy ω,mg)p0dω1/p0 , where Mdy ω,mgis the dyadic maximal function defined in §3. As mentioned in §5, Mdy ω,m is bounded on Lp(dω) uniformly in mfor 1 < p < ∞. Thus, we obtain that ZS Tm(fdσ)g dω ≤ckMϕ(fdσ)kLp(dω)kgkLp0(dω), with cindependent of m, f and g, and consequently Theorem 2.1 follows from duality by letting m→ −∞. 2 Let us now prove Corollary 2.2. Let τbe defined by τ(B) = r(B)βfor all B, with β > 0 to be chosen. By hypothesis, there exists  > 0 such that ϕ(B1)µ(B1)≤cr(B1) r(B2) ϕ(B2)µ(B2) if B1⊂B2. Therefore, since µis assumed to satisfy the doubling condition of order D, ϕ(B1)≤cr(B1) r(B2)−D ϕ(B2) if B1⊂B2 =cr(B1) r(B2)(β+−D)−β ϕ(B2) =cr(B1) r(B2)β+−D ϕ(B2)τ(B2) τ(B1). It follows that (16)(c) holds with there taken to be β+−D. Clearly, (16)(a), (b) also hold for any β > 0. Thus, if β+−D > 0, by applying Theorem 2.1 (with τ(B) = r(B)βas above), the conclusion of Corollary 2.2 follows immediately. 2 33 7 Proof of Theorem 2.3 We start with the case p= 1, where the proof will be a variant of that of Theorem 2.1. The cases p > 1 and p < 1 will follow from the case p= 1 using extrapolation ideas and duality between Lpspaces. 7.1 The case p=1 By a limiting argument we may assume that wis bounded with compact support. For f≥0, we start with inequality (49) with dω replaced by w dµ and dσ replaced by dµ: ZS Tm(fdµ)w dµ ≤cγ X k,j ϕ(B(Qk j)) Zκ(2κ+1)B(Qk j) f dµ ZQk j w dµ, where the dyadic cubes Qk jare now the maximal cubes in Dmwith γk<1 µ(Qk j)ZQk j w dµ and satisfy γk<1 µ(Qk j)ZQk j w dµ ≤cµγk. Define Ωk=x: sup Q∈Dm:x∈Q 1 µ(Q)ZQ w dµ > γk, and let Ek j=Qk j\Ωk+1. Note that Ωk=∪jQk j. As usual, if Qk j∩Qk+1 i6=∅, then Qk+1 i⊂Qk j. If γis large enough, then µ(Qk j)≤2µ(Ek j) since µ(Qk j∩Ωk+1) = X i:Qk j∩Qk+1 i6=∅ µ(Qk j∩Qk+1 i) =X i:Qk+1 i⊂Qk j µ(Qk+1 i)≤γ−k−1X i:Qk+1 i⊂Qk jZQk+1 i w dµ ≤γ−k−1ZQk j w dµ ≤γ−k−1cµγkµ(Qk j) 34 =cµ γµ(Qk j)<1 2µ(Qk j) if γis large. Thus there are sets Ek jthat are pairwise disjoint in both jand kwith Ek j⊂Qk jand µ(Qk j)≤cµ(Ek j) for a universal constant c. If we denote ˜ Qk j=κ(2κ+ 1)B(Qk j), then ZS Tm(fdµ)w dµ ≤cX k,j ϕ(B(Qk j)) Z˜ Qk j f dµ 1 µ(Qk j)ZQk j w dµ!µ(Qk j) ≤cX k,j ϕ(˜ Qk j)Z˜ Qk j f dµ 1 µ(Qk j)ZQk j w dµ!µ(Ek j) ≤cX k,j ZEk j Mϕ(fdµ)Mw dµ ≤cZS Mϕ(fdµ)Mw dµ. This concludes the proof of (22) when p= 1. 7.2 The case p > 1 Now let p > 1. As we mentioned above, our argument will be based on duality and the case p= 1. In fact we will prove something sharper than (21): if δ > 0, there is a constant C such that for any weight wand all f, ZS |T(fdµ)|pw dµ ≤CZS (Mϕ(fdµ))pML(log L)p−1+δ(w)dµ, (51) where ML(log L)p−1+δdenotes the maximal function MΦwith Φ(t) = t(1 + log+t)p−1+δ, t > 0. The fact that this estimate is sharper than (21) will be shown later. By the case p= 1, there is a constant cso that for all f, g ≥0 and all m, ZS Tm(fdµ)g w1/p dµ ≤cZS Mϕ(fdµ)M(g w1/p)dµ. 35 Now by the generalized H¨older inequality (35) for an appropriate Young function Ψ that will be chosen soon, we can continue with ≤cZS Mϕ(fdµ)MΨgM¯ Ψ(w1/p)dµ ≤cZS Mϕ(fdµ)p(M¯ Ψ(w1/p))pdµ1/p ZS (MΨg)p0dµ1/p0 . To conclude the proof of (51), we use Theorem 4.2 for an appropriate Ψ ∈Bp0. Indeed, as mentioned in §4, we can choose Ψ(t)≈tp0(log t)−1−, > 0 and tlarge, with complementary function ¯ Ψ(t)≈tp(log t)(p−1)(1+)for large t. Then by combining estimates, we obtain ZS Tm(fdµ)g w1/p dµ ≤cZS Mϕ(fdµ)pML(log L)(p−1)(1+)(w)dµ1/p ZS gp0dµ1/p0 . Since the constant cis independent of m, (51) follows by duality and letting m→ −∞. To show that (51) implies (21), we recall the following lemma from [PW] (see also [P1], [GI] and [WW] in the usual Euclidean case). Lemma 7.1 Let k= 1,2,· · · .Then there is a positive constant csuch that for any measurable function w, kwkL(logL)k,B ≤c µ(B)ZB Mkw dµ, (52) where Mkwdenotes the k-fold iterate of the Hardy–Littlewood maximal function defined in (20) and k·kL(logL)k,B denotes k·kΦ,B with Φ(t) = t(1 + (log+t)k). For k= 1,2,· · · , (52) implies that kwkL(logL)k,B ≤c µ(B)ZB Mkw dµ. Thus, if we choose =[p] p−1−1>0 in the proof of (51), we obtain M¯ Ψ(w1/p)(x)p≈ ML(log L)[p](w)(x)≤c M[p]+1(w)(x), and part i) of Theorem 2.3 follows. 36 7.3 The case p < 1 Assume now that 0 < p < 1. We will prove the inequality ZS |T(fdµ)|pw dµ ≤CZS (Mϕ(fdµ))pMw dµ (53) by an extrapolation argument, after first proving a strengthened version of the case p= 1. We begin with a definition. Definition 7.2 A weight vsatisfies the RH∞(µ)condition (i.e., the reverse H¨older condition of infinite order) if there is a constant c > 0such that for each ball B, ess supBv≤c µ(B)ZB v dµ It is easy to check that RH∞(µ)⊂A∞(µ); in fact we can take η= 1 in the definition of A∞(µ). We will prove the following version of the case p= 1: Lemma 7.3 Let va weight satisfying the RH∞(µ)condition. Then there is a constant C such that for any weight wand all f, ZS |T(fdµ)|v w dµ ≤CZS Mϕ(fdµ)v Mw dµ. (54) For the proof of this lemma we proceed as in the case v= 1: ZS Tm(fdµ)v w dµ ≤cX k,j ϕ(˜ Qk j)Z˜ Qk j f dµ ZQk j v w dµ ≤cX k,j ϕ(˜ Qk j)Z˜ Qk j f dµ ZQk j w dµ (ess supQk jv) ≤cX k,j ϕ(˜ Qk j)Z˜ Qk j f dµ 1 µ(Qk j)ZQk j w dµ!ZQk j v dµ since v∈RH∞(µ) 37 ≤cX k,j ϕ(˜ Qk j)Z˜ Qk j f dµ 1 µ(Qk j)ZQk j w dµ!ZEk j v dµ since v∈A∞(µ) ≤cX k,j ZEk j Mϕ(fdµ)Mw v dµ ≤cZS Mϕ(fdµ)v Mw dµ. 2 Lemma 7.4 Let α > 0and gbe any function such that Mg is finite a.e. Then (Mg)−α∈RH∞(µ). This observation is due to C. Neugebauer [N] where he showed something better: w∈RH∞(µ)⇔w≈(Mg)−αfor some gand some α > 0. As usual we will denote wB=1 µ(B)RBw dµ. To prove (53), we first observe that w−1∈RH∞(µ) if w∈A1(µ), i.e., if wB≤C ess infBwfor all B, since then ess supBw−1= (ess infBw)−1≤c(wB)−1≤c(w−1)B, where in the last inequality we have used H¨older’s inequality. The constant cis in fact the inverse of the constant Cin the definition of A1(µ). The second observation we need is that if w∈RH∞(µ) then wλ∈RH∞(µ) when λ > 1: ess supBwλ= (ess supBw)λ≤cλ(wB)λ≤cλ(wλ)B. Actually this is also true when 0 < λ < 1 but is a bit harder and we don’t need it; see [N]. Let us now prove (53) for 0 < p < 1. We will use an appropriate duality for the spaces Lp(dν) when p < 1 and νis a measure: if f≥0 then kfkLp(dν)= inf{Zf u−1dν :ku−1kLp0(dν)= 1}=Zf u−1dν for some u≥0 such that ku−1kLp0(dν)= 1, where p0=p p−1<0. This follows from the “reverse” H¨older inequality Zf g dν ≥ kfkLp(dν)kgkLp0(dν), 38 which is a consequence of the usual H¨older inequality. Combining this with the Lebesgue differentiation theorem and both Lemmas 7.3 and 7.4 for the weight (M(gδ))−1/δ ∈RH∞(µ), δ > 0 and g≥0, we have ZS Mϕ(fdµ)Mw gdµ ≥ZS Mϕ(fdµ)Mw (M(gδ))1/δ dµ ≥cZS T(fdµ)w (M(gδ))1/δ dµ ≥ckT(fdµ)kLp(wdµ) (M(gδ))−1/δ Lp0(wdµ). However, since the integral on the right side of (53) when raised to the power 1/p equals ZS Mϕ(fdµ)Mw gdµ for some g, everything is reduced to proving that  (M(gδ))−1/δ Lp0(wdµ)≥ckg−1kLp0(M(w)dµ). Since p0<0, this is equivalent to saying that ZS (M(gδ))−p0/δ w dµ ≤cZS g−p0Mw dµ. But if we choose δso that 0 < δ < −p0, then −p0/δ > 1 and the last estimate follows from the known (see [FS]) weighted norm inequality ZS (Mf)qw dµ ≤cZS |f|qMw dµ, q > 1. 2 References [AH] D.R. Adams, L.I. 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