Uncertainty principle estimates for vector fields

JOURNAL OF FUNCTIONAL ANALYSIS 181 (2001), 146–188 Uncertainty principle estimates for vector fields Carlos P´ erez§and Richard L. Wheeden ¶ 1 Introduction One form of Hardy’s inequality is the estimate ZRn |f(x)|pdx |x|p≤CZRn |∇f(x)|pdx, for 1 ≤p<nand any smooth fwith compact support, where the constant Cis independent of f. This inequality, which can be found in [HLP], has had many important applications. For instance, in Mathematical Physics, it is related (in fact, equivalent) in case p= 2 to the Uncertainty Principle of Heisenberg ([RS], vol. II, p. 169). In this paper, we will derive norm estimates for a wide class of integral operators of potential type. These estimates can be used to obtain inequalities like the one above. In fact, if Tis the integral operator defined by Tf(x) = ZRn f(y)1 |x−y|n−1dy, then by using the well-known pointwise inequality |f(x)| ≤ cnT(|∇f|)(x) for any smooth fwith compact support, one can deduce the Hardy estimate above from the corresponding (weighted) Lpnorm estimate for T. The same method can be used for more §Partially supported by DGICYT grant PB940192, Spain ¶Partially supported by NSF grant DMS 95-00799. general differential operators Xf often called generalized gradients; that is, one can bound norms of fby norms of Xf provided there is an integral operator Twhich is bounded on appropriate weighted Lpspaces and for which the pointwise estimate |f| ≤ cT(|Xf|) is valid. For example, this pointwise estimate is known to hold for vector fields of H¨ormander type when Tis given by Tf(x) = ZRn f(y)d(x, y) |B(x, d(x, y))|dy, where d(x, y) is the associated Carnot–Carath´eodory metric and B(x, r) denotes the metric ball with center xand radius r. In addition to norm estimates for integral operators, we will also study norm estimates for maximal operators that are closely associated with the integral operators. Our main theorems generalize and sharpen some of the principal results obtained in [SW], [P1] and [P2]. We improve these results in several ways, such as by considering spaces of homogeneous type without any group structure, and by enlarging the classes of weight functions for which some of the results hold. In particular, with regard to weight functions for integral operators, we are able to avoid assuming the doubling conditions that are imposed in some of the results in [SW], as well as improve the results there which deal with generalizations of the Fefferman–Phong “r–bump” condition (see below). We will show that this sort of condition can be replaced by weaker ones like those considered in [P1] and [P2], and which are closely related to estimates derived in [CWW] and [ChW2] for Schr¨odinger operators. In order to obtain weighted results, the kinds of conditions that we will impose on the weights are in the spirit of simple sufficient conditions which are close to necessary. For example, in n-dimensional Euclidean space Rnwith the usual metric, the classical Riesz fractional integral operator Iαf(x) = ZRn f(y)1 |x−y|n−αdy, 0< α < n, 2 is known to satisfy the norm inequality ZRn {|Iαf(x)|w(x)}qdx1/q ≤cZRn {|f(x)|v(x)}pdx1/p ,(1) 1< p ≤q < ∞, if for some r > 1 and all balls B, the weights satisfy r(B)α−n|B|1 q+1 p01 |B|ZB w(x)rqdx1/rq 1 |B|ZB v(x)−rp0dx1/rp0 ≤C, (2) where p0denotes the conjugate index of p, that is, 1 p+1 p0= 1, and r(B) the radius of B. On the other hand, the same condition with r= 1, i.e., the condition r(B)α−nZB w(x)qdx1/q ZB v(x)−p0dx1/p0 ≤C, is necessary but not sufficient for the norm inequality (this is exactly the Aα p,q condition of [SW], but with a different normalization). These facts are proved in [SW]. Extensions to spaces of homogeneous type are also proved there, but with extra restrictions on either the weights or the space, such as doubling conditions on wrq and v−rp0, or a group structure for the space. One of our goals is to remove these extra restrictions. We refer to (2) as a Fefferman–Phong “r–bump” condition. It is simpler in nature than two other kinds of conditions known to be both necessary and sufficient for such norm estimates. To put our results for potential operators in perspective, it may help to briefly recall these other conditions, even though they play no role in the paper. One of them involves “testing” conditions of the type found first in [S] in the usual Euclidean situation, and then generalized and sharpened in [SW], [SWZ], [WZ] and [VW]. Testing conditions are phrased in terms of norm estimates for the integral operator when it is restricted to acting on the weight functions themselves. The second sort of necessary and sufficient condition involves integrals with “tails”, i.e., integrals extended over the entire space of products of weights times suitably truncated powers of the kernel which appears in the integral operator. Such results are known for 1 < p < q < ∞but not for q=p: see [GK], [SW], [SWZ], [GGK]. Compared with 3 conditions of these two types, those of the Fefferman–Phong sort have the disadvantage of not being necessary, but they have the advantage of being relatively simple and close to necessary. No simple method is known for proving that Fefferman–Phong conditions imply either of these other two types of conditions. The r–bump requirement (2) was weakened in [P1] within the Euclidean framework; more general potential operators of convolution form were also studied in [P1]. In the case of the Riesz fractional integral, these weaker assumptions can be described as follows: let Ψ and Φ be doubling Young functions such that both Z∞ ctq Ψ(t)q0−1dt t<∞and Z∞ ctp0 Φ(t)p−1dt t<∞(3) for some positive constant c. Some examples of such Φ(t) are, for large t, Φ(t) = tp0(log t)p0−1+βand Φ(t) = tp0(log t)p0−1(log log t)p0−1+β, where β > 0. Then (1) holds if, for all balls B, the weights satisfy the condition r(B)α−n|B|1 q+1 p0kwkΨ,Bkv−1kΦ,B ≤K, (4) where kfkΨ,B (similarly kfkΦ,B) denotes the localized Luxemburg norm kfkΨ,B = inf λ > 0 : 1 |B|ZB Ψ|f(y)| λdy ≤1. See section 4 for more information. There is a similar situation for maximal functions. Indeed, it was also shown in [P1] that a sufficient condition for the analogue of (1) with Iαfreplaced by the fractional maximal function Mα(f)(x) = sup r(B)α−nZB |f(y)|dy, 0≤α < n, where the supremum is taken over all balls Bcontaining x, is just the condition (4) with no “bump” on the weight w, namely with Ψ(t) = tq. In this case, the condition on the weights 4 becomes simply r(B)α−n|B|1/p0ZB wqdx1/q kv−1kΦ,B ≤K for some Φ as in (3) and all balls B. An antecedent of the results in [P1], [P3] was given by Neugebauer in [N] for the case α= 0. Another goal of this paper is to extend these results for Mαto more general maximal operators and to spaces of homogeneous type. In case wqand v−p0are A∞weights (in the sense of C. Fefferman and B. Muckenhoupt), as is well-known, condition (2) holds for some r > 1 if and only if it holds for r= 1. Thus, in this case, no bump is needed in the condition imposed on the weights in order to obtain (1). In a sequel [PW] to this paper, we will show that no bump is needed for classes which are larger than A∞, and in spaces which are more general than Rn. This extends earlier results of the same kind in [SW] in the usual Euclidean case as well as in spaces of homogeneous type. See also [BSa] for results about A∞weights in spaces of homogeneous type. For these more general spaces, and for larger classes of weight functions than A∞, we also study in [PW] extensions of results in [MW] relating norms of integral operators of potential type to norms of maximal functions. 2 Statements of the main results Following [SW], we consider potential operators T=TKof the form Tf(x) = T(fdµ)(x) = ZS f(y)K(x, y)dµ(y),(5) where Sis a space of homogeneous type with underlying doubling measure µ. See §3 for the exact definition of a space of homogeneous type; by a doubling measure, we mean a Borel measure µwith the property that there is a constant Csuch that for every “ball” B⊂ S, µ(2B)≤Cµ(B), 5 where 2Bdenotes the ball with the same center as Bbut twice the radius. If d(x, y) denotes the corresponding quasimetric in S, we will always assume that the kernel K(x, y) is nonnegative and satisfies the following growth conditions: there exist constants C1, C2>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 fractional integrals Iαf mentioned 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] 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 (5). 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 the kernel Kis a functional ϕ=ϕKwhich acts on balls Band is defined by ϕ(B) = sup x,y∈B d(x,y)≥cr(B) K(x, y) (8) for a sufficiently small positive geometric constant c(see [SW]), where r(B) denotes the radius of B. 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.In the subelliptic case (7), note that ϕ(B)≈r(B)/µ(B). 6 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 (4.3)] that if θ > 1, there is a constant Cdepending only on θ, C1, C2, the constant cin (8), and geometric properties of Sso that ϕ(B)≤Cϕ(θB) for all balls B⊂ S.(9) Also, for such a constant C(but now Cis independent of θ), ϕ(B)≤Cϕ(B0) for all pairs of balls B0⊂B. (10) We shall also assume in some of our results that ϕsatisfies the following condition for some  > 0: ϕ(B1)µ(B1)≤Cr(B1) r(B2) ϕ(B2)µ(B2) if B1⊂B2.(11) Observe that in the case of the fractional integrals Iα, we can pick =αin (11); for the operator in (7), we can choose = 1. Next we define a class of Young functions that plays a key role in our results. Some further facts about Young functions and Orlicz spaces are listed in §4. Definition 2.1 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<∞.(12) Simple examples of functions which satisfy Bpare tp−βand tp(log(1 + t))−1−β, both when β > 0. The relevance of condition Bpstems from its relationship to the boundedness of a maximal function that is defined in terms of Φ. In fact, given a Young function Φ, let kfkΦ,B = inf{λ > 0 : 1 µ(B)ZB Φ(|f| λ)dµ ≤1}, 7 and define the corresponding maximal function MΦf(x) = sup B:x∈B kfkΦ,B.(13) We will prove the following characterization of Bp, 1 <p<∞, in Theorem 5.1 below: Φ∈Bpif and only if MΦ:Lp(S, µ)→Lp(S, µ). For example, in the standard case when Φ(t) = trwith r≥1, so that kfkΦ,B =µ(B)−1RB|f|rdµ1/r, this statement 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 Bpmentioned 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 will play a main role in the proof of the boundedness of Tas stated in Theorem 2.2 below. For other applications to different operators ¿from harmonic analysis, see [P1], [P5], [P6], [CP1] and [CP2]. A Young function Φ has a conjugate function ¯ Φ satisfying t≤¯ Φ(t)¯ Φ−1(t)≤2t for all t > 0 (cf. §4). For example, if 1 <p<∞and 1 p+1 p0= 1, the conjugate of tpis tp0, and the conjugate of tp(log(1 + t))−1−β,β > 0, is tp0(log(1 + t))(p0−1)(1+β)(cf. [O], p. 275). We can now state our main result about the boundedness of the potential operators (5). Theorem 2.2 Let 1< p ≤q < ∞and Tbe an integral operator of type (5) with a kernel K such that (6) holds and ϕsatisfies (11). Let (w, v)be a pair of weights for which ϕ(B)µ(B)1 q+1 p0kwkΨ,Bkv−1kΦ,B ≤C(14) 8 for all balls Bin S, where Ψand Φare Young functions whose corresponding conjugate functions ¯ Ψand ¯ Φsatisfy ¯ Ψ∈Bq0and ¯ Φ∈Bp. Then ZS (|Tf|w)qdµ1/q ≤CZS (|f|v)pdµ1/p (15) with Cindependent of f. In the Euclidean setting, condition (14) is a generalization of the sort of condition first considered in [CWW] (see also [P1], [ChW2]). For example, given p, q with 1 < p ≤q < ∞, if we choose Ψ(t) = trq and Φ(t) = trp0for any r > 1, then condition (14) becomes φ(B)µ(B)1 q+1 p01 µ(B)ZB wrq dµ1 rq 1 µ(B)ZB v−rp0dµ1 rp0 ≤C. Assuming this r–bump condition, the conclusion (15) was proved in [SW] in the usual Euclidean situation (with dµ =dx); it was also proved there for spaces of homogeneous type but with one of the additional assumptions that both wrqdµ, v−rp0dµ are doubling measures or that there is an appropriate group stucture for S. In case v= 1 and if Tis an integral operator of type (7) with dµ =dx, (15) was proved in [D] without either of these additional assumptions. It is easy to see that when p > 1 and Φ satisfies the doubling property Φ(2t)≤cΦ(t), then Z∞ c Φ(t) tp dt t≈Z∞ ctp0 ¯ Φ(t)p−1dt t for c > 0. Hence, if both ¯ Φ and ¯ Ψ satisfy this doubling condition, then the assumption in Theorem 2.2 that ¯ Ψ∈Bq0and ¯ Φ∈Bpis equivalent to assuming both Z∞ ctq Ψ(t)q0−1dt t<∞and Z∞ ctp0 Φ(t)p−1dt t<∞ for some c > 0. 9 with cβ=κβ +κ2β+κ2.To verify (29), note that if z∈B∩Pand y∈P, then d(y, xB)≤κ[d(y, xP) + d(xP, xB)] ≤κ[r(P) + κ(d(xP, z) + d(z, xB))] ≤κ[r(P) + κ(r(P) + r(B))] ≤κ[βr(B) + κ(βr(B) + r(B))] = cβr(B), which implies (29). We will use a grid of dyadic sets in Swhich are “almost balls”, as constructed in [SW]. In fact, the following has been proved there: If ρ= 8K5, 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 D as 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, there may be no nestedness properties of the cubes in Dm1relative to the cubes in Dm2if m1, m2are different. As usual, we say that wis a weight if w(x) is a nonnegative locally integrable function with respect to µ, and for a measurable set E, we write w(E) = REw(x)dµ(x). Thus, w(E) = |E|wdµ. 16 4 Orlicz spaces We next recall some basic definitions and facts about Orlicz spaces, 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, and in particular, that Φ(γt)≥γΦ(t) if γ≥1 and t≥0. For Orlicz spaces, we are usually only concerned about the behavior of Young functions for t large. If A, B are two Young functions, we write A(t)≈B(t) if there are constants c, c1, c2>0 with c1A(t)≤B(t)≤c2A(t) for t>c. 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(30) 17 for all t > 0. 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). Another example that will be used frequently is Φ(t)≈tp(log t)−1−for large t, 1 <p<∞, > 0, with complementary function ¯ Φ(t)≈tp0(log t)(p0−1)(1+)(cf. [O]). A very important property of Orlicz spaces is the generalized H¨older inequality ZS |fg|dµ ≤ kfkΦkgk¯ Φ.(31) We will sometimes assume that Φ satisfies the doubling condition Φ(2t)≤CΦ(t). If Φ is doubling then Φ0(t)≈Φ(t)/t almost everywhere. Recall that if Xis a rearrangement–invariant function space with respect to the measure µ, then the fundamental function of X,ϕX(t), is defined so that if t > 0 and Eis any measurable set with µ(E) = t, then ϕX(t) = χEX. See [BS] for more information. In particular, it is shown there that for any Young function Φ, LΦis a rearrangement–invariant space with fundamental function given by ϕLΦ(t) = 1 Φ−1(1 t).(32) In particular, if Eis a measurable subset of X, then χEΦ,µ =1 Φ−1(1 µ(E)).(33) 18 5 Auxiliary exotic maximal functions In order to define another maximal function which will play a key role, we need to introduce local versions of Orlicz norms. If Φ is a Young function, let kfkΦ,B =kfkΦ,B,µ = inf{λ > 0 : 1 µ(B)ZB Φ(|f| λ)dµ ≤1}. For this norm, we will use the fact that if λ > 0, then kfkΦ,B > λ if and only if 1 µ(B)RBΦ(|f|/λ)dµ > 1. Furthermore, the local version of the generalized H¨older inequality (31) is 1 µ(B)ZB fg dµ ≤ kfkΦ,Bkgk¯ Φ,B.(34) As in (13), there is a corresponding maximal function defined by MΦf(x) = sup B:x∈B kfkΦ,B.(35) This maximal function has been used in the usual Euclidean context in [P3] as a tool to derive sharp weighted estimates for the Hardy-Littlewood maximal function. Also, it was considered in the work of T. Iwaniec and Greco [GI] and in [WW] in case Φ(t)≈tlog t. Note that from 32 we have with the corresponding normalization that χKΦ,B =1 Φ−1µ(B) µ(B∩K), and therefore MΦ(χK)(x) = sup B:x∈B 1 Φ−1µ(B) µ(B∩K).(36) The main results that we will prove about MΦare summarized in the next theorem. Theorem 5.1 Let 1<p<∞and Φbe a doubling Young function. Then the following statements are equivalent. 19 i) Φ∈Bp, i.e., there is a constant c > 0such that Z∞ c Φ(t) tp dt t<∞.(37) ii) There is a constant C > 0such that ZS MΦf(x)pdµ(x)≤CZS f(x)pdµ(x) (38) 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) (39) for all nonnegative fand w, where Mw is the Hardy–Littlewood maximal function defined in (18). iv) There is a constant C > 0such that ZS Mf(x)pw(x) [M¯ Φ(u1/p)(x)]pdµ(x)≤CZS f(x)pMw(x) u(x)dµ(x) (40) for all nonnegative f,wand u, where Magain denotes the operator defined in (18). The proof of Theorem 5.1 is based on the following lemma. Lemma 5.2 Let Φbe a Young function and fbe a bounded nonnegative function with bounded support. For λ > 0, let Ωλ={x∈ S :MΦf(x)> λ}. If Ωλis not empty, then given σ > 1, there exists a countable family {Bi}of pairwise disjoint balls such that i) ∪iBi⊂Ωλ⊂ ∪iB∗ i,where B∗=κ(4κ+ 1)B(κis the quasimetric constant), ii) kfkΦ,Bi> λ for all i, iii) kfkΦ,B ≤λif Bis any ball with Bi⊂Band r(B)≥σr(Bi)for some i. 20 Consequently, µ(Ωλ)≤CZS Φ(f λ)dµ. (41) Proof: The proof uses a sort of Calder´on–Zygmund decomposition combined with Vitali’s lemma. A similar method occurs in [MP]. Fix fand λ. If x∈Ωλ, there is a ball Bsuch that x∈Band kfkΦ,B > λ. Define R=R(f, λ) by R= sup B:kfkΦ,B>λ r(B). We claim that Ris finite. Indeed, suppose that the support of fis contained in a ball B0, and let Bsatisfy kfkΦ,B > λ. Then by definition of the Luxemburg norm, Bsatisfies 1<1 µ(B)ZB Φf λdµ =1 µ(B)ZB∩B0 Φf λdµ ≤ΦkfkL∞ λµ(B∩B0) µ(B). Therefore, Band B0must intersect. Assuming as we may that r(B)≥r(B0), we easily obtain from (29) that B0⊂κ(2κ+ 1)B, and then 1<ΦkfkL∞ λµ(B∩B0) µ(B)≤ΦkfkL∞ λµ(B0) µ(B) ≤ΦkfkL∞ λcκ,µ r(B0) r(B)δ for some fixed δ > 0 by the reverse doubling property of µ(see (28)). In particular, r(B)≤cΦkfkL∞ λ1/δ r(B0).(42) This shows that Ris finite. It also shows that each ball Bfor which kfkΦ,B > λ is contained in cB0with cdepending only on λand f, since then Bintersects B0and (42) holds. Now, for each x, let Rx=Rx(f, λ) = sup B:x∈B,kfkΦ,B>λ r(B), and note that Rxis finite since Rx≤R. Fix σ > 1. If x∈Ωλ, there is a ball Bxwhich contains x, whose radius r(Bx) satisfies Rx/σ < r(Bx)≤Rx, and for which kfkΦ,Bx> λ. If B 21 is any ball with Bx⊂Band r(B)≥σr(Bx), then r(B)> Rx, and consequently kfkΦ,B ≤λ since x∈B. Thus the ball Bxsatisfies ii) and iii). Also note that Ωλ=Sx∈ΩλBx. Picking a Vitali type subcover of {Bx}x∈Ωλas in [SW], Lemma 3.3, then provides a family of pairwise disjoint balls {Bi}⊂{Bx}x∈Ωλsatisfying i). Therefore {Bi}satisfies i), ii) and iii).Finally, (41) follows in a standard way from i) and ii) by the doubling property of µand the disjointness of the Bi: µ(Ωλ)≤Xµ(B∗ i)≤XCµ(Bi) ≤CXZBi Φf λdµ ≤CZS Φf λdµ. 2 Proof of Theorem 5.1 We may assume that fis bounded with bounded support and that f≥0. We start by proving that i) implies ii). Let Ωλ={x∈ S :MΦf(x)> λ}. For each λ > 0, we split fas usual: f=f1+f2where f1(x) = f(x) if f(x)> λ/2 and f1(x) = 0 otherwise. We may assume without loss of generality that Φ is normalized so that Φ(1) = 1. Since f2≤λ/2, it then follows that MΦ(f2)≤λ/2, and consequently that MΦ(f)≤ MΦ(f1) + MΦ(f2)≤ MΦ(f1) + λ/2. Using this combined with (41), we get µ(Ωλ)≤CZx∈S:f(x)>λ/2 Φ2f(x) λdµ(x). Then ZS MΦ(f)pdµ =pZ∞ 0 λpµ(Ωλ)dλ λ ≤CZ∞ 0 λpZx∈S:f(x)>λ/2 Φ2f(x) λdµ(x)dλ λ=CZSZ2f(x) 0 λpΦ2f(x) λdλ λdµ(x) =CZS f(x)pZ∞ 1 Φ(t) tp dt tdµ(x) = CZS f(x)pdµ(x) since Φ ∈Bp. This proves that i) implies ii). 22 We now show that ii) implies iii). For k∈Zand γ > 1, γto be chosen, let Ωk={x∈ S :MΦ(f)(x)> γk}. For σ > 1 to be chosen, by applying Lemma 5.2 to each Ωk, we obtain balls {Bk j}jwith SjBk j⊂Ωk⊂SjcBk j, and for each kthe balls {Bk j}are disjoint in j. Furthermore, for all k and j,kfkΦ,Bk j> γkand kfkΦ,B ≤γkif Bis any ball with Bk j⊂Band r(B)≥σr(Bk j). Then ZS MΦ(f)pw dµ ≤X kZΩk\Ωk+1 MΦ(f)pw dµ ≤X k γ(k+1)pw(Ωk) ≤cX k,j kfkp Φ,Bk jw(cBk j) = cX k,j kfkp Φ,Bk j w(cBk j) µ(cBk j)µ(cBk j) ≤cX k,j w(cBk j) µ(cBk j)1/p f p Φ,Bk j µ(Bk j).(43) Now consider the family of sets {Ek j}k,j defined by Ek j=Bk j\Ωk+1. Observe that the Ek jare disjoint in both k, j. We will show that for sufficiently large γthere exists a constant csuch that for all k, j, µ(Bk j)≤c µ(Ek j) Assuming this for the moment, we obtain from (43) that ZS MΦ(f)pw dµ ≤cX k,j (infBk jMw)1/pfp Φ,Bk j µ(Ek j)≤cX k,j ZEk j MΦ(f(Mw)1/p)pdµ ≤cZS MΦ(f(Mw)1/p)pdµ ≤CZS fpMw dµ by hypothesis, and thus iii) would be proved. To prove that µ(Bk j)≤cµ(Ek j) if γis large, it is enough to show that µ(Bk j∩Ωk+1)<1 2µ(Bk j)if γis large.(44) 23 Recall that by Lemma 5.2, the sets {Bk+1 m}are disjoint in mfor each k, and that Ωk+1 ⊂SmcBk+1 mwith c=κ(4κ+ 1). Moreover, kfkΦ,Bk+1 m> γk+1, and kfkΦ,B ≤γkif B satisfies Bk j⊂Band r(B)≥σr(Bk j) for some j, k (with σ > 1 still to be chosen). Thus µ(Bk j∩Ωk+1)≤X m µ(Bk j∩cBk+1 m).(45) We claim that if Bk j∩cBk+1 m6=∅then r(Bk j)> r(Bk+1 m). To see this, first note that if Bk jand cBk+1 jintersect and r(Bk j)≤r(Bk+1 m), then Bk j⊂c1Bk+1 mfor a geometric constant c1>1. Since Φ is a Young function, Φ(t)/t is increasing, so that Φf γk= Φ γ f γk+1 ≥γΦf γk+1 , γ > 1, and therefore by the doubling property of µ, there is a geometric constant C > 1 such that 1 µ(c1Bk+1 m)Zc1Bk+1 m Φf γkdµ ≥γ C µ(Bk+1 m)ZBk+1 m Φf γk+1 dµ >γ C>1, if we choose γ > C. (Recall that for any λ > 1, the inequality kfkΦ,B > λ is the same as 1 µ(B)RBΦ(f/λ)dµ > 1.) This implies that kfkΦ,c1Bk+1 m> γk. However, if we now choose σwith 1< σ ≤c1, we obtain from the inequality r(Bk j)≤r(Bk+1 m) that the ball c1Bk+1 mhas radius at least σr(Bk j), and therefore since Bk j⊂c1Bk+1 m, we must have kfkΦ,c1Bk+1 m≤γkby construction of Bk j. This contradiction shows our claim. Thus if Bk j∩cBk+1 m6=∅, then r(Bk j)> r(Bk+1 m), and consequently Bk+1 m⊂c1Bk j. Hence by (45) and the doubling property of µ, µ(Bk j∩Ωk+1)≤X m:Bk+1 m⊂c1Bk j µ(Bk j∩cBk+1 m)≤cX m:Bk+1 m⊂c1Bk j µ(Bk+1 m) ≤cX m:Bk+1 m⊂c1Bk jZBk+1 m Φf γk+1 dµ ≤cZc1Bk j Φf γk+1 dµ 24 since the sets {Bk+1 m}mare disjoint. Again using the fact that Φ(t)/t is increasing together with property iii) of Lemma 5.2 for Bk j, we can continue the last chain of inequalities with ≤c γZc1Bk j Φf γkdµ ≤c γµ(c1Bk j)≤C γµ(Bk j)<1 2µ(Bk j) if γis large. This completes the proof of (44) and also shows that part ii) of the theorem implies part iii). We prove now that iii) implies iv). Assume then that iii) holds. Since (40) is equivalent to ZS M(fg)(x)pw(x) [M¯ Φ(g)(x)]pdµ(x)≤cZS f(x)pMw(x)dµ(x), for all nonnegative functions f,g, and w, iv) follows immediately from (39) after an application of the inequality M(fg)(x)≤ MΦf(x)M¯ Φg(x), x ∈ S, which is a consequence of the local version (34) of the generalized H¨older’s inequality. To prove that iv) implies i), we let w= 1 in (40), obtaining ZS Mf(x)p1 [M¯ Φ(u1/p)(x)]pdµ(x)≤CZS f(x)p1 u(x)dµ(x) (46) for all nonnegative functions fand u. Fix any z∈ S and r > 0, and let K=B(z, r). WE USE HERE THAT THE SPACE HAS MORE THAN ONE POINT BY THE ANNULI CONDITION. Choosing f=u=χKin (46) gives ZS M(χK)(x)p1 [M¯ Φ(χK)(x)]pdµ(x)≤C (where Cdepends on µ(K)). On the other hand, by (36), M¯ Φ(f)(x) = sup B:x∈B 1 ¯ Φ−1µ(B) µ(B∩K). 25 containing ball of Q, and let xQdenote its center xB(Q). Thus, 1 ρB(Q)⊂Q⊂B(Q) and r(B(Q)) = ρ`+1. We then have d(y, xQ)≤κ(d(y, x) + d(x, xQ)) ≤κ(ρ`+1 +ρ`+1)=2κr(B(Q)), so that y∈2κB(Q). Since d(x, y)> ρ`=r(2κB(Q))/2κρ, then by definition and property (10) of ϕ, K(x, y)≤ϕ(2κB(Q)) ≤Cϕ(B(Q)). Hence, K(x, y)≤cϕ(B(Q))χQ(x)χ2κB(Q)(y)≤cX Q∈Dm ϕ(B(Q))χQ(x)χ2κB(Q)(y), where the last estimate holds for all x, y with d(x, y)> ρm. Therefore, Tmf(x)≤cX Q∈Dm ϕ(B(Q))χQ(x)Z2κB(Q) f(y)dµ(y),(53) and then if g≥0, we obtain ZS (Tmf)g w dµ ≤cX Q∈Dm ϕ(B(Q)) Z2κB(Q) f dµ ZQ g w dµ. (54) For k∈Zand γ > 1 to be chosen, let Ck={Q∈Dm:γk<1 µ(Q)ZQ g w dµ ≤γk+1}. Assuming as we may that gis bounded and has bounded support, we can choose maximal cubes {Qk j}jin Dmwith γk<1 µ(Qk j)ZQk j g w dµ. If Ik jis the next largest dyadic cube containing Qk j, then γk<1 µ(Qk j)ZQk j g w dµ ≤cµ,ρ 1 µ(Ik j)ZIk j g w dµ ≤cµ,ργk≤γk+1 (55) 32 by choosing γ≥cµ,ρ. Thus Qk j∈ Ck. Since each cube Q∈Dmmust lie in some Ck, it must be contained in some Qk j. Of course, the sets {Qk j}jare pairwise disjoint for fixed k. Then ZS (Tmf)g w dµ ≤cX kX Q∈Ck ϕ(B(Q))µ(Q)Z2κB(Q) f dµ 1 µ(Q)ZQ g w dµ ≤cX k γk+1 X jX Q∈∆m(Qk j) ϕ(B(Q))µ(Q)Z2κB(Q) f dµ, (56) where we have used the notation ∆m(Q0) = {Q∈Dm:Q⊂Q0}. Lemma 7.1 Let f≥0and ϕsatisfy (52), and let ∆m(Q0) = {Q∈Dm:Q⊂Q0}if Q0∈Dm. There exists a geometric constant Csuch that for each Q0∈Dm, X Q∈∆m(Q0) ϕ(B(Q))µ(Q)Z2κB(Q) f dµ ≤C ϕ(B(Q0))µ(Q0)Zκ(2κ+1)B(Q0) f dµ. (57) Proof: The left side of (57) equals ∞ X `=0 X Q∈∆m(Q0) `(Q)=ρ−``(Q0) ϕ(B(Q)) µ(Q)Z2κB(Q) f dµ, which by (52) is at most c ∞ X `=0 X Q∈∆m(Q0) `(Q)=ρ−``(Q0) ρ−`  ϕ(B(Q0)) µ(Q0)Z2κB(Q) f dµ =c ϕ(B(Q0)) µ(Q0) ∞ X `=0 ρ−` X Q∈∆m(Q0) `(Q)=ρ−``(Q0)Z2κB(Q) f dµ. (58) To estimate the last expression, first observe that if Q⊂Q0and `(Q)≤`(Q0), then 2κB(Q)⊂κ(2κ+ 1)B(Q0), since if y∈2κB(Q) then d(y, xQ0)≤κ[d(y, xQ) + d(xQ, xQ0)] ≤κ[2κr(B(Q)) + r(B(Q0))] =κ[2κρ`(Q) + r(B(Q0))] ≤κ[2κρ`(Q0) + r(B(Q0))] = κ(2κ+ 1) r(B(Q0)). 33 Thus (58) is at most c ϕ(B(Q0)) µ(Q0) ∞ X `=0 ρ−` Zκ(2κ+1)B(Q0)X Q∈Dm `(Q)=ρ−``(Q0) χ2κB(Q)(x)f(x)dµ(x), and therefore (57) will follow if we show that X Q∈Dm `(Q)=ρ−``(Q0) χ2κB(Q)(x)≤C(59) uniformly in x, j, k, l, m. To prove this, fix x, j, k, l, m and write r=ρ−``(Q0). If Q∈Dm, `(Q) = rand x∈2κB(Q), then for any y∈Qwe have d(x, y)≤κ[d(x, xQ) + d(xQ, y)] ≤κ[2κr(B(Q)) + r(B(Q))] ≤κ(2κ+ 1))ρ`(Q) = c1r, so that Q⊂B(x, c1r). But those Q∈Dmwith `(Q) = rare disjoint, and consequently by doubling, since each Qhas sidelength comparable to the radius of B(x, c1r), the number of such Q⊂B(x, c1r) is bounded uniformly in xand r. This proves (59) and so also (57). We will apply Lemma 7.1 to each Qk j. In fact, by combining (56), (57) and (55), we obtain ZS (Tmf)g w dµ ≤cX k,j γk+1ϕ(B(Qk j))µ(Qk j)Zκ(2κ+1)B(Qk j) f dµ ≤cγ X k,j ϕ(B(Qk j)) Zκ(2κ+1)B(Qk j) f dµ ZQk j g w dµ. (60) This completes the process of discretizing Tm. 7.2 Applying the condition on the weights For simplicity, let ˜ Qk j=κ(2κ+ 1)B(Qk j). We estimate (60) by using the generalized H¨older inequality (34) and the growth condition (9): ZS (Tmf)g w dµ 34 ≤cX k,j ϕ(B(Qk j))µ(˜ Qk j) 1 µ(˜ Qk j)Z˜ Qk j f dµ! 1 µ(Qk j)ZQk j g w dµ!µ(Qk j) ≤cX k,j ϕ(˜ Qk j)µ(˜ Qk j)kfvk¯ Φ,˜ Qk jkv−1kΦ,˜ Qk jkgk¯ Ψ,Qk jkwkΨ,Qk jµ(Qk j), where we have used (10) to majorize ϕ(B(Qk j)) by a multiple of ϕ(˜ Qk j). By the doubling of µ, we can also majorize kwkΨ,Qk jby a fixed multiple of kwkΨ,˜ Qk j. Then using condition (14) on the weights and H¨older inquality, we can continue the estimates above with ≤cX k,j ϕ(˜ Qk j)µ(˜ Qk j)1 q+1 p0kwkΨ,˜ Qk jkv−1kΦ,˜ Qk jkfvk¯ Φ,˜ Qk jµ(˜ Qk j)1 pkgk¯ Ψ,Qk jµ(Qk j)1 q0 ≤cX k,j kfvk¯ Φ,˜ Qk jµ(˜ Qk j)1 pkgk¯ Ψ,Qk jµ(Qk j)1 q0 ≤c X k,j kfvkq ¯ Φ,˜ Qk j µ(˜ Qk j)q p!1/q X k,j kgkq0 ¯ Ψ,Qk jµ(Qk j)!1/q0 ≤c X k,j kfvkp ¯ Φ,˜ Qk j µ(Qk j)!1/p X k,j kgkq0 ¯ Ψ,˜ Qk j µ(Qk j)!1/q0 (61) since q≥pand µis doubling. 7.3 Patching the pieces together Recall that the family {Qk j}jconsists of maximal dyadic cubes satisfying γk<1 µ(Qk j)ZQk j g w dµ, and that we also have 1 µ(Qk j)ZQk j g w dµ ≤cµ,ργk.(62) Let Ωk={x:Md mg(x)> γk}where Md mgis the dyadic maximal function defined by Md mg(x) = sup Q:x∈Q Q∈Dm 1 µ(Q)ZQ |g|dµ, 35 and note that Ωk=∪jQk j. As before, consider the sets Ek j=Qk j\Ωk+1. These are pairwise disjoint in both jand k, and we will show that there is a universal constant csuch that for each j, k, µ(Qk j)≤cµ(Ek j). The proof is somewhat easier than before due to the dyadic structure. In fact, since µ(Qk j) = µ(Qk j∩Ωk+1) + µ(Ek j), it is enough to show that µ(Qk j∩Ωk+1)≤cµ,ρ γµ(Qk j) and then pick γ > cµ,ρ. We have µ(Qk j∩Ωk+1) = X ` µ(Qk j∩Qk+1 `). If Qk j∩Qk+1 `is nonempty, then by the dyadic structure, either Qk j⊂Qk+1 `or Qk+1 `⊂Qk j. If Qk jwere strictly contained in Qk+1 `, then by the maximality of Qk jwe would have 1 µ(Qk+1 `)ZQk+1 ` g w dµ ≤γk, which contradicts the fact that this average exceeds γk+1. Consequently, Qk+1 `⊂Qk jif these sets intersect, and therefore µ(Qk j∩Ωk+1) = X `:Qk+1 `⊂Qk j µ(Qk+1 `)≤X `:Qk+1 `⊂Qk j 1 γk+1 ZQk+1 ` g w dµ ≤1 γk+1 ZQk j g w dµ ≤cµ,ρ γµ(Qk j), which proves the assertion above. Consequently, by (61), ZS (Tmf)g w dµ ≤c X k,j kfvkp ¯ Φ,˜ Qk j µ(Qk j)!1/p X k,j kgkq0 ¯ Ψ,˜ Qk j µ(Qk j)!1/q0 36 ≤c X k,j kfvkp ¯ Φ,˜ Qk j µ(Ek j)!1/p X k,j kgkq0 ¯ Ψ,˜ Qk j µ(Ek j)!1/q0 ≤c X k,j ZEk j M¯ Φ(fv)pdµ!1/p X k,j ZEk j M¯ Ψ(g)q0dµ!1/q0 ≤cZS M¯ Φ(fv)pdµ1/p ZS M¯ Ψ(g)q0dµ1/q0 ≤cZS (fv)pdµ1/p ZS gq0dµ1/q0 by using Theorem 5.1, since ¯ Φ∈Bpand ¯ Ψ∈Bq0by hypothesis. Since the constant cis independent of m, Theorem 2.2 now follows from duality by letting m→ ∞. 2 8 Proof of Theorem 2.5 8.1 A local version of a classical lemma of Wiener We will derive a local version of a result of N. Wiener which leads to a way to control the L logL norm of a function by the L1norm of its Hardy–Littlewood maximal function. We first need a local version of the classical Calder´on–Zygmund decomposition as shown in [MP]. We adapt the arguments there to our context. Fix δ > 0 and a ball B0, and consider the following family of balls adapted to B0: B=BB0,δ ={B:xB∈B0and r(B)≤δr(B0)}.(63) This family has the properties listed in the next lemma. We use the notation c B0= (1 + δ)κB0, where κis the quasimetric constant of d, and we also denote fB=1 µ(B)RBf dµ. In order to obtain our local version of the Calder´on–Zygmund lemma, we begin with the following observations. 37 Lemma 8.1 Let B0be a ball, Bbe defined as in (63), and Dbe the doubling order of µ relative to c B0, i.e., µ(c B0) µ(B)≤cµ r(c B0) r(B)!D if B⊂c B0. Let fbe a nonnegative function which is integrable on c B0. a) If B∈ B then B⊂c B0. b) If B∈ B and fB> λ, then r(B)≤cµ fc B0 λ1/D r(c B0). If we also assume that λ≥γfc B0, where γ=cµM δand M > 0, then r(B)≤δ M1/D r(c B0). Proof: The first observation follows from the quasimetric inequality, since if B∈ B and x∈B, then d(x, xB0)≤κ[d(x, xB) + d(xB, xB0)] < κ[r(B) + r(B0)] ≤κ[δr(B0) + r(B0)] = κ(1 + δ)r(B0). To show b), let B∈ B and fB> λ. Then by using a) and the doubling property of µ, we have λ < fB≤µ(c B0) µ(B)fc B0≤cµ r(c B0) r(B)!D fc B0, and the first part of b) follows. The second part of b) is a simple corollary of the first part. 2 Given an integrable function fon c B0, the maximal function of fassociated to Bis defined by MBf(x) = sup B:x∈B∈B 1 µ(B)ZB |f|dµ 38 if xbelongs to an element of the basis B, and MBf(x) = 0 otherwise. Therefore, if λ > 0 and we define Ωλ={x∈ S :MBf(x)> λ}, then Ωλ⊂c B0. The version of the Calder´on–Zygmund lemma that we will use is given in the next lemma. Lemma 8.2 Let fbe a nonnegative and integrable function on c B0,γbe as in part c) of Lemma 8.1, and λ≥γfc B0. If Ωλis not empty, then given σ > 1, there exists a countable family {Bi}of pairwise disjoint balls such that i) ∪iBi⊂Ωλ⊂ ∪iB∗ i,where B∗=κ(4κ+ 1)Band κis the quasimetric constant ii) r(Bi)≤δ M1/D r(c B0)for all i, where Dis the doubling order of µ, iii) 1 µ(Bi)RBif dµ > λ for all i, iv) 1 µ(σBi)RσBif dµ ≤λif σBi∈ B. Proof: The proof is similar to that of Lemma 5.2. Fix fand λ. If x∈Ωλ, there exists a ball B0∈ B with x∈B0and 1 µ(B0)RB0f dµ > λ. Define R=R(x, f, λ) by R= sup{r(B) : B∈ B, x ∈Band fB> λ}. Lemma 8.1 implies that R≤δ M1/D r(c B0). Then there is a ball Bxwith x∈B∈ B whose radius satisfies R σ< r(Bx)≤Rand for which fBx> λ. For this ball, ii), iii) and iv) hold with Bxin place of Bi. Part iii) implies that Ωλ=Sx∈ΩλBx. Picking a Vitali type subcover of {Bx}x∈Ωλas in [SW], Lemma 3.3, we obtain 39 a family of pairwise disjoint balls {Bi} ⊂ {Bx}x∈Ωλsatisfying i) as well as the rest of the properties. 2 In our context, we have the following version of a classical estimate due to Wiener. Lemma 8.3 Let fbe a nonnegative locally integrable function, δ > 0,B0be any ball, and γ be as in Lemma 8.1 with Mchosen to satisfy M≥[κ2(4κ+ 1)(1 + δ)]Dδ1−D, where κis the quasimetric constant and Dis the doubling order of µ. Then there exists a constant A=Aκ,µ such that for each λ≥γfc B0, 1 λZ{x∈ c B0:f(x)>λ} f dµ ≤A µ({x∈c B0:MB(f)(x)> λ}).(64) Remark 8.4 In Rn, if we consider a cube Qinstead of a ball, and if Md Qdenotes the usual dyadic maximal operator with respect to Q, then it is not difficult to see that a corresponding inequality holds with no “blow-up” in the constant. To be more precise, we then have 1 λZ{x∈Q:f(x)>λ} f dµ ≤2nµ({x∈Q:Md Qf(x)> λ}) for λ>fQ. Proof of Lemma 8.3: Fix λ≥γfc B0with γ=cµM/δ and Mto be chosen. Note that Ωλ={x∈c B0:MB(x)> λ}. We may assume without loss of generality that fis bounded. We may also assume that Ωλis not empty since {x∈c B0:f(x)> λ} ⊂ Ωλ(except possibly for a set of µ-measure zero) by the Lebesgue differentiation theorem. Applying Lemma 8.2 to fand λwith σ=κ(4κ+ 1), we obtain a family of disjoint balls {Bi}satisfying SiBi⊂Ωλ⊂SiσBi. Furthermore, for all i,1 µ(Bi)RBifdµ > λ and 1 µ(σBi)RσBif dµ ≤λsince 40 σBi∈ B if we choose Mwith M1/D ≥κ2(4κ+ 1)(1 + δ)δ1/D−1, because σBiis centered in B0 and r(σBi) = σr(Bi)≤σδ M1/D r(c B0) = σδ M1/D (1 + δ)κr(B0)≤δr(B0) for such M. Then, since {x∈c B0:f(x)> λ} ⊂ Ωλa.e., these properties together with the doubling of µ imply that Z{x∈ c B0:f(x)>λ} f dµ ≤ZΩλ f dµ ≤X iZσBi f dµ ≤λX i µ(σBi)≤A λ X i µ(Bi)≤A λµ(Ωλ), which proves the lemma. 2 8.2 Proof of Theorem 2.5 We will show that Theorem 2.5 follows from Corollary 2.3. Thus, we must show that there are positive constants Cand βso that the pair of weights (w1/p,{M˜ϕ(M[p]w)}1/p) satisfies the condition LB:= ϕ(B)µ(B)w1/pLp(logL)p−1+β(B){M˜ϕ(M[p]w)}−1/pLp0(logL)p0−1+β(B)≤C for all balls Bin S. Recalling that ˜ϕ(B)=(ϕ(B)µ(B))pµ(B)−1and that b B= (1 + δ)κB, we have that M˜ϕ(M[p]w)(x)−1/p ≤[ϕ( b B)µ( b B)]p µ( b B)Rb BM[p]w dµ−1/p for all x∈B. Hence, Lp B≤[ϕ(B)µ(B)]pw1/pp Lp(logL)p−1+β(B) [ϕ(b B)µ(b B)]p µ(b B)Zb B M[p]w dµ!−1 ≤ckwkL(logL)p−1+β(B) 1 µ(b B)Zb B M[p]w dµ!−1 , by the reverse doubling properties of ϕ(B) and µ(B) (see (9) for ϕ). 41 [PW] C. P´erez, R. L. Wheeden, Norm estimates relating fractional integrals and maximal functions, preprint. [R] Y. Rakotondratsimba, Improved Muckenhoupt–Wheeden inequality and weighted inequalities for potential operators, Publicacions Mat. 39 (1995), 23–41. [RR] M. Rao, Z. Ren, Theory of Orlicz spaces , Marcel Dekker, Inc. New York, 1991. [RS] M. Reed and B. 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Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 251–272. 48 [WW] R. L. Wheeden, J. M. Wilson, Weighted norm estimates for gradients of half-space extensions, Indiana U. Math. J. 44 (1995), 917–969. [WZ] R. L. Wheeden, S. Zhao, Weak type estimates for operators of potential type, Studia Math. 119 (1996), 149–160. C. P´erez: Departmento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid, Spain carlos.p[email protected] Research partially supported by DGICYT grant PB940192, Spain R. L. Wheeden: Department of Mathematics Rutgers University New Brunswick, New Jersey 08903, USA [email protected] Research partially supported by NSF Grant DMS95-00799 49