WEIGHTED NORM INEQUALITIES, OFF-DIAGONAL ESTIMATES ...

Proceedings of the 8th International Conference on Harmonic Analysisand Partial Differential Equations, El Escorial, June 16-20, 2008Contemporary Mathematics 505 (2010), 61--83**WEIGHTED** **NORM** **INEQUALITIES**, **OFF**-**DIAGONAL** **ESTIMATES**AND ELLIPTIC OPERATORSPASCAL AUSCHER AND JOSÉ MARÍA MARTELLAbstract. We give an overview of the generalized Calderón-Zygmund theory for“non-integral” singular operators, that is, operators without kernels bounds but appropriateoff-diagonal estimates. This theory is powerful enough to obtain weightedestimates for such operators and their commutators with BMO functions. L p − L qoff-diagonal estimates when p ≤ q play an important role and we present them.They are particularly well suited to the semigroups generated by second order ellipticoperators and the range of exponents (p, q) rules the L p theory for many operatorsconstructed from the semigroup and its gradient. Such applications are summarized.1. IntroductionThe Hilbert transform in R and the Riesz transforms in R n are prototypes ofCalderón-Zygmund operators. They are singular integral operators represented bykernels with some decay and smoothness. Since the 50’s, Calderón-Zygmund operatorshave been thoroughly studied. One first shows that the operator in question isbounded on L 2 using spectral theory, Fourier transform or even the powerful T (1), T (b)theorems. Then, the smoothness of the kernel and the Calderón-Zygmund decompositionlead to the weak-type (1,1) estimate, hence strong type (p, p) for 1 < p < 2. Forp > 2, one uses duality or interpolation from the L ∞ to BMO estimate, which involvesalso the regularity of the kernel. Still another way for p > 2 relies on good-λ estimatesvia the Fefferman-Stein sharp maximal function. It is interesting to note that bothCalderón-Zygmund decomposition and good-λ arguments use independent smoothnessconditions on the kernel, allowing different generalizations for each argument.Weighted estimates for these operators can be proved by means of the Fefferman-Stein sharp maximal function, one shows boundedness on L p (w) for every 1 < p < ∞and w ∈ A p , and a weighted weak-type (1, 1) for weights in A 1 . Again, the smoothnessof the kernel plays a crucial role. We refer the reader to [Gra] and [GR] for moredetails on this topic.It is natural to wonder whether the smoothness of the kernel is needed or, evenmore, whether one can develop a generalized Calderón-Zygmund theory in absence ofDate: July 10, 2009.2000 Mathematics Subject Classification. 42B20, 42B25, 47A06, 35J15, 47A60, 58J35.Key words and phrases. Calderón-Zygmund theory, spaces of homogeneous type, Muckenhouptweights, singular non-integral operators, commutators with BMO functions, elliptic operators indivergence form, holomorphic calculi, Riesz transforms, square functions, Riemannian manifolds.This work was partially supported by the European Union (IHP Network “Harmonic Analysisand Related Problems” 2002-2006, Contract HPRN-CT-2001-00273-HARP). The second author wasalso supported by MEC “Programa Ramón y Cajal, 2005”, by MEC Grant MTM2007-60952, andby UAM-CM Grant CCG07-UAM/ESP-1664.1

2 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLkernels. Indeed, one finds Calderón-Zygmund like operators without any (reasonable)information on their kernels which, following the implicit terminology introduced in[BK1], can be called singular “non-integral” operators in the sense that they are stillof order 0 but they do not have an integral representation by a kernel with size and/orsmoothness estimates. The goal is to obtain some range of exponents p for which L pboundedness holds, and because this range may not be (1, ∞), one should abandonany use of kernels. Also, one seeks for weighted estimates trying to determine forwhich class of Muckenhoupt these operators are bounded on L p (w). Again, becausethe range of the unweighted estimates can be a proper subset of (1, ∞) the class A p ,and even the smaller class A 1 , might be too large.The generalized Calderón-Zygmund theory allows us to reach this goal: much ofall the classical results extend. As a direct application, we show in Corollary 3.3 thatassuming that for a bounded (sub)linear operator T on L 2 , the boundedness on L p—and even on L p (w) for A p weights— follows from two basic inequalities involvingthe operator and its action on some functions and not its kernel:∫∫|T f(x)| dx ≤ C |f(x)| dx, (1.1)supx∈BR n \4Bfor any ball B and any bounded function f supported on B with mean 0, and∫|T f(x)| ≤ C− |T f(x)| dx + C inf Mf(x), (1.2)x∈B2Bfor any ball B and any bounded function f supported on R n \ 4 B. The first conditionis used to go below p = 2, that is, to obtain that T is of weak-type (1, 1). On the otherhand, (1.2) yields the estimates for p > 2 and also the weighted norm inequalities inL p (w) for w ∈ A p , 1 < p < ∞. In Proposition 3.6 below, we easily show that classicalCalderón-Zygmund operators with smooth kernels satisfy these two conditions —(1.1) is a simple reformulation of the Hörmander condition [Hör] and (1.2) uses theregularity in the other variable.The previous conditions are susceptible of generalization: in (1.1) one could havean L p 0−L p 0estimate with p 0 ≥ 1, and the L 1 −L ∞ estimate in (1.2) could be replacedby an L p 0− L q 0condition with 1 ≤ p 0 < q 0 ≤ ∞. This would drive us to estimates onL p in the range (p 0 , q 0 ). Still, the corresponding conditions do not involve the kernel.Typical families of operators whose ranges of boundedness are proper subsets of(1, ∞) can be built from a divergence form uniformly elliptic complex operator L =− div(A ∇) in R n . One can consider the operator ϕ(L), with bounded holomorphicfunctions ϕ on sectors; the Riesz transform ∇L −1/2 ; some square functions “à la”Littlewood-Paley-Stein: one, g L , using only functions of L, and the other, G L , combiningfunctions of L and the gradient operator; estimates that control the squareroot L 1/2 by the gradient. These operators can be expressed in terms of the semigroup{e −t L } t>0 , its gradient { √ t ∇e −t L } t>0 , and their analytic extensions to somesector in C. Let us stress that those operators may not be representable with “usable”kernels: they are “non-integral”.The unweighted estimates for these operators are considered in [Aus]. The instrumentaltools are two criteria for L p boundedness, valid in spaces of homogeneous type.One is a sharper and simpler version of a theorem by Blunck and Kunstmann [BK1],based on the Calderón-Zygmund decomposition, where weak-type (p, p) for a given pB

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 3with 1 ≤ p < p 0 is presented, knowing the weak-type (p 0 , p 0 ). We also refer to [BK2]and [HM] where L p estimates are shown for the Riesz transforms of elliptic operatorsfor p < 2 starting from the L 2 boundedness proved in [AHLMT].The second criterion is taken from [ACDH], inspired by the good-λ estimate in thePh.D. thesis of one of us [Ma1, Ma2], where strong type (p, p) for some p > p 0 isproved and applied to Riesz transforms for the Laplace-Beltrami operators on someRiemannian manifolds. A criterion in the same spirit for a limited range of p’s alsoappears implicitly in [CP] towards perturbation theory for linear and non-linear ellipticequations and more explicitly in [Sh1, Sh2].These results are extended in [AM1] to obtain weighted L p bounds for the operatoritself, its commutators with a BMO function and also vector-valued expressions. Usingthe machinery developed in [AM2] concerning off-diagonal estimates in spaces ofhomogeneous type, weighted estimates for the operators above are studied in [AM3].Sharpness of the ranges of boundedness has been also discussed in both the weightedand unweighted case. From [Aus], we learn that the operators that are defined in termsof the semigroup (as ϕ(L) or g L ) are ruled by the range where the semigroup {e −t L } t>0is uniformly bounded and/or satisfies off-diagonal estimates (see the precise definitionbelow). When the gradient appears in the operators (as in the Riesz transform ∇L −1/2or in G L ), the operators are bounded in the same range where { √ t ∇e −t L } t>0 isuniformly bounded and/or satisfies off-diagonal estimates.In the weighted situation, given a weight w ∈ A ∞ , one studies the previous propertiesfor the semigroup and its gradient. Now the underlying measure is no longer dxbut dw(x) = w(x) dx which is a doubling measure. Therefore, we need an appropriatedefinition of off-diagonal estimates in spaces of homogeneous type with the followingproperties: it implies uniform L p (w) boundedness, it is stable under composition, itpasses from unweighted to weighted estimates and it is handy in practice. In [AM2]we propose a definition only involving balls and annuli. Such definition makes clearthat there are two parameters involved, the radius of balls and the parameter of thefamily, linked by a scaling rule independently on the location of the balls. The priceto pay for stability is a somewhat weak definition (in the sense that we can not begreedy in our demands). Nevertheless, it covers examples of the literature on semigroups.Furthermore, in spaces of homogeneous type with polynomial volume growth(that is, the measure of a ball is comparable to a power of its radius, uniformly overcenters and radii) it coincides with some other possible definitions. This is also thecase for more general volume growth conditions, such as the one for some Lie groupswith a local dimension and a dimension at infinity. Eventually, it is operational forproving weighted estimates in [AM3], which was the main motivation for developingthat material.Once it is shown in [AM2] that there exist ranges where the semigroup and itsgradient are uniformly bounded and/or satisfy off-diagonal estimates with respectto the weighted measure dw(x) = w(x) dx, we study the weighted estimates of theoperators associated with L. As in the unweighted situation considered in [Aus],the ranges where the operators are bounded are ruled by either the semigroup or itsgradient. To do that, one needs to apply two criteria in a setting with underlyingmeasure dw. Thus, we need versions of those results valid in R n with the Euclidean

4 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLdistance and the measure dw, or more generally, in spaces of homogeneous type (whenw ∈ A ∞ then dw is doubling).This article is a review on the subject with no proofs except for the section dealingwith Calderón-Zygmund operators. The plan is as follows. In Section 2 we give somepreliminaries regarding doubling measures and Muckenhoupt weights. In Section 3we present the two main results that generalize the Calderón-Zygmund theory. Theeasy application to classical Calderón-Zygmund operators is given with proofs. Wedevote Section 4 to discuss two notions of off-diagonal estimates: one that holds forarbitrary closed sets, and another one, which is more natural in the weighted case,involving only balls and annuli. In Section 5 we introduce the class of elliptic operatorsand present their off-diagonal properties. Unweighted and weighted estimates for thefunctional calculus, Riesz transforms and square functions associated such ellipticoperators are in Section 6. The strategy to prove these results is explained in Section 7.Finally in Section 8 we present some further applications concerning commutators withBMO functions, reverse inequalities for square roots and also vector-valued estimates.We also give some weighted estimates for fractional operators (see [AM5]) and Riesztransforms on manifolds (see [AM4]).2. PreliminariesWe use the symbol A B for A ≤ CB for some constant C whose value is notimportant and independent of the parameters at stake.Given a ball B ⊂ R n with radius r(B) and λ > 0, λ B denotes the concentric ballwith radius r(λ B) = λ r(B).The underlying space is the Euclidean setting R n equipped with the Lebesgue measureor more in general with a doubling measure µ. Let us recall that µ is doublingifµ(2 B) ≤ C µ(B) < ∞for every ball B. By iterating this expression, one sees that there exists D, which iscalled the doubling order of µ, so that µ(λ B) ≤ C µ λ D µ(B) for every λ ≥ 1 and everyball B.Given a ball B, we write C j (B) = 2 j+1 B \ 2 j B when j ≥ 2, and C 1 (B) = 4B. Alsowe set∫− h dµ = 1B µ(B)∫Bh(x) dµ(x),∫− h dµ =C j (B)1µ(2 j+1 B)∫C j (B)h dµ.Let us introduce some classical classes of weights. Let w be a weight (that is, a nonnegative locally integrable function) on R n . We say that w ∈ A p , 1 < p < ∞, if thereexists a constant C such that for every ball B ⊂ R n ,( ∫ ) ( ∫p−1− w dx − w dx) 1−p′ ≤ C.BBFor p = 1, we say that w ∈ A 1 if there is a constant C such that for every ball B ⊂ R n ,∫− w dx ≤ C w(y), for a.e. y ∈ B.B

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 5We write A ∞ = ∪ p≥1 A p . The reverse Hölder classes are defined in the following way:w ∈ RH q , 1 < q < ∞, if there is a constant C such that for any ball B,( ∫ ) 1∫− w q qdx ≤ C − w dx.BThe endpoint q = ∞ is given by the condition w ∈ RH ∞ whenever there is a constantC such that for any ball B,∫w(y) ≤ C − w dx, for a.e. y ∈ B.The following facts are well-known (see for instance [GR, Gra]).Proposition 2.1.(i) A 1 ⊂ A p ⊂ A q for 1 ≤ p ≤ q < ∞.(ii) RH ∞ ⊂ RH q ⊂ RH p for 1 < p ≤ q ≤ ∞.B(iii) If w ∈ A p , 1 < p < ∞, then there exists 1 < q < p such that w ∈ A q .(iv) If w ∈ RH q , 1 < q < ∞, then there exists q < p < ∞ such that w ∈ RH p .(v) A ∞ =⋃A p =⋃RH q .1≤p

6 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLTheorem 3.1. Let µ be a doubling Borel measure on R n , D its doubling order and1 ≤ p 0 < q 0 ≤ ∞. Suppose that T is a sublinear operator bounded on L q 0(µ) and that{A B } is a family indexed by balls of linear operators acting from L ∞ c (µ) (the set ofessentially bounded functions with bounded support) into L q 0(µ). Assume thatand, for j ≥ 1,1µ(4 B)( ∫−∫C j (B)R n \4B( ∫|T (I − A B )f| dµ − |f| p 0dµ) 1 ( ∫|A B f| q q 0dµ0≤ α j −BB) 1p 0, (3.1)) 1|f| p p 0dµ0, (3.2)for all ball B and for all f ∈ L ∞ c (µ) with supp f ⊂ B. If ∑ j α j 2 D j < ∞, then T is ofweak type (p 0 , p 0 ), hence T is of strong type (p, p) for all p 0 < p < q 0 . More precisely,there exists a constant C such that for all f ∈ L ∞ c (µ)‖T f‖ L p (µ) ≤ C ‖f‖ L p (µ).A stronger form of (3.1) is with the notation above,( ∫) 1 ( ∫− |T (I − A B )f| p p 0dµ0≤ α j − |f| p 0dµC j (B)B) 1p 0, j ≥ 2. (3.3)Our second result is based on a good-λ inequality. See [ACDH], [Aus] (in theunweighted case) and [AM1] for more general formulations. In contrast with Theorem3.1, we do not assume any a priori estimate for T . However, in practice, to deal withthe local term (where f is restricted to 4 B) in (3.4), one uses that the operator isbounded on L p 0(µ). Thus, we apply this result to go above p 0 in the unweighted caseand also to show weighted estimates.Theorem 3.2. Let µ be a doubling Borel measure on R n and 1 ≤ p 0 < q 0 ≤ ∞. LetT be a sublinear operator acting on L p 0(µ) and let {A B } be a family indexed by ballsof operators acting from L ∞ c (µ) into L p 0(µ). Assume that( ∫− |T (I − A B )f| p 0dµB) 1p 0≤ ∑ j≥1∫α j(−2 j+1 B) 1|f| p p 0dµ0, (3.4)and( ∫) 1− |T A B f| q q 0dµ0≤ ∑ ∫) 1α j(− |T f| p p 0dµ0, (3.5)Bj≥1 2 j+1 Bfor all f ∈ L ∞ c (µ), all B and for some α j satisfying ∑ j α j < ∞.(a) If p 0 < p < q 0 , there exists a constant C such that for all f ∈ L ∞ c (µ),‖T f‖ L p (µ) ≤ C ‖f‖ L p (µ).(b) Let p ∈ W w (p 0 , q 0 ), that is, p 0 < p < q 0 and w ∈ A pp 0constant C such that for all f ∈ L ∞ c (µ),∩ RH (q 0p′.)There is a‖T f‖ L p (w) ≤ C ‖f‖ L p (w). (3.6)An operator acting from A to B is just a map from A to B. Sublinearity means|T (f + g)| ≤ |T f| + |T g| and |T (λf)| = |λ| |T f| for all f, g and λ ∈ R or C. Next,L p (w) is the space of complex valued functions in L p (dw) with dw = w dµ. However,

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 7all this extends to functions valued in a Banach space and also to functions definedon a space of homogeneous type.Let us notice that in both results, the cases q 0 = ∞ are understood in the sensethat the L q 0-averages are indeed essential suprema. One can weaken (3.5) by addingto the right hand side error terms such as M(|f| p 0)(x) 1/p 0for any x ∈ B (see [AM1,Theorem 3.13]).3.1. Application to Calderón-Zygmund operators. We see that the previous resultsallow us to reprove the unweighted and weighted estimates of classical Calderón-Zygmund operators. We emphasize that the conditions imposed do not involve thekernels of the operators.Corollary 3.3. Let T be a sublinear operator bounded on L 2 (R n ).(i) Assume that, for any ball B and any bounded function f supported on B withmean 0, we have ∫ ∫|T f(x)| dx ≤ C |f(x)| dx. (3.7)R n \4 BBThen, T is of weak-type (1, 1) and consequently bounded on L p (R n ) for every1 < p < 2.(ii) Assume that, for any ball B and any bounded function f supported on R n \4 B,we have∫sup |T f(x)| ≤ C− |T f(x)| dx + C inf Mf(x). (3.8)x∈B2 Bx∈BThen, T is bounded on L p (R n ) for every 2 < p < ∞.(iii) If T satisfies (3.7) and (3.8) then, T is bounded on L p (w), for every 1 < p < ∞and w ∈ A p .Proof of (i). We are going to use Theorem 3.1 with p 0 = 1 and q 0 = 2. By assumptionT is bounded on L 2 (R n ). For every ball B we set A B f(x) = ( ∫− f dx) χB B(x). Then,as a consequence of (3.7) we obtain (3.1) with p 0 = 1:∫∫∫1|T (I − A B )f| dx − |(I − A B )f| dx − |f| dx.|4 B|R n \4 BOn the other hand, we observe that A B ∫f(x) ≡ 0 for x ∈ C j (B) and j ≥ 2, and forx ∈ C 1 (B) = 4 B we have |A B f(x)| ≤ − |f| dx. This shows (3.2) with p B 0 = 1 andq 0 = 2. † Therefore, Theorem 3.1 yields that T is of weak-type (1, 1).By Marcinkiewicz interpolation theorem, it follows that T is bounded on L p (R n )for 1 < p < 2.□Proof of (ii). We use (a) of Theorem 3.2 with p 0 = 2 and q 0 = ∞. For every ball Bwe set A B f(x) = χ R n \4 B (x) f(x). Using that T is bounded on L2 (R n ) we triviallyobtain( ∫( ∫−2 BB) 1|T (I − A B )f| 2 2dx−4 B|f| 2 dx) 12B, (3.9)†In fact, we have (3.2) with p 0 = 1 and q 0 = ∞. We take q 0 = 2 since in Theorem 3.1 we need Tbounded on L q0 (R n ). This shows that one can assume boundedness on L r (R n ) for some 1 < r < ∞(in place of L 2 (R n )), and the argument goes through.

8 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLwhich implies (3.4) with p 0 = 2. On the other hand, (3.8) and (3.9) yield∫‖T A B f‖ L ∞ (B) − |T A B f(x)| dx + inf M(A Bf)(x)2 Bx∈B∫∫≤ − |T (I − A B ) f(x)| dx + − |T f(x)| dx + inf Mf(x)2 B2 Bx∈B( ∫ ) 1∫ − |f| 2 2dx + − |T f(x)| dx + inf Mf(x)4 B2 Bx∈B∫≤ − |T f(x)| dx + infx∈B M(|f|2 )(x) 1 2 .2 BThus, we have obtained (3.5) with q 0 = ∞, p 0 = 2 and α j = 0, j ≥ 2, plus an errorterm inf x∈B M(|f| 2 )(x) 1 2 . Applying part (a) of Theorem 3.2 with the remark thatfollows it, we conclude that T is bounded on L p (R n ) for every 2 < p < ∞. ‡Let us observe that part (b) in Theorem 3.2 yields weighted estimates: T is boundedon L p (w) for every 2 < p < ∞ and w ∈ A p/2 .□Proof of (iii). Note that (ii) already gives weighted estimates. Here we improve thisby assuming (i), that is, by using that T is bounded on L q (R n ) for 1 < q < 2.Fixed 1 < p < ∞ and w ∈ A p , there exists 1 < r < p such that w ∈ A p/r . Then, by(i) (as we can take r very close to 1) we have that T is bounded on L r (R n ). Then, asin (3.9), we have( ∫−2 B) 1|T (I − A B )f| r rdx( ∫ −4 B|f| r dxThis estimate allows us to obtain as before∫‖T A B f‖ L ∞ (B) − |T f(x)| dx + infx∈B M(|f|r )(x) 1 r .2 BThus we can apply part (b) of Theorem 3.2 with p 0 = r and q 0 = ∞ to conclude thatT is bounded on L q (u) for every r < q < ∞ and u ∈ A q/r . In particular, we have thatT is bounded on L p (w).□Remark 3.4. We mention [AM1, Theorem 3.14] and [Sh2, Theorem 3.1] where (3.8)is generalized to( ∫) 1 ( ∫) 1− |T f(x)| q q 0dx0 − |T f(x)| p p 0dx0+ inf M( |f| ) p 0(x) 1p 0 ,x∈BB2Bfor some 1 ≤ p 0 < q 0 ≤ ∞ and f bounded with support away from 4B. In thatcase, if T is bounded on L p 0(R n ), proceeding as in the proofs of (ii) and (iii), oneconcludes that T is bounded on L p (R n ) for every p 0 < p < q 0 and also on L p (w) forevery p 0 < p < q 0 and w ∈ A p/p0 ∩ RH (q0 /p) ′.Remark 3.5. To show that T maps L 1 (w) into L 1,∞ (w) for every w ∈ A 1 one needs tostrengthen (3.7). For instance, we can assume that (3.7) holds with dw(x) = w(x) dxin place of dx and for functions f with mean value zero with respect to dx. In this case,we choose A B as in (i). Taking into account that w ∈ A 1 yields −∫B |f| dx ∫− |f| dw,Bthe proof follows the same scheme replacing everywhere (except for the definition‡As before, if one a priori assumes boundedness on L r (R n ) for some 1 < r < ∞ (in place ofL 2 (R n )) the same computations hold with r replacing 2, see the proof of (iii).) 1r.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 9of A B ) dx by dw. Notice that the boundedness of T on L 2 (w) is needed, this isguaranteed by (iii) as A 1 ⊂ A 2 .Let us observe that we can assume that (3.7) holds with dw(x) = w(x) dx in placeof dx, but for functions f with mean value zero with respect to dw. In that case, theproof goes through by replacing everywhere dx by dw, even in the definition of A B .When applying this to classical operators the first approach is more natural.Proposition 3.6. Let T be a singular integral operator with kernel K, that is,∫T f(x) = K(x, y) f(y) dy, x /∈ supp f, f ∈ L ∞ c ,R nwhere K is a measurable function defined away from the diagonal.(i) If K satisfies the Hörmander condition∫|K(x, y) − K(x, y ′ )| dx ≤ Cthen (3.7) holds.|x−y|>2 |y−y ′ |(ii) If K satisfies the Hölder condition|K(x, y) − K(x ′ , y)| ≤ C |x − x′ | γfor some γ > 0, then (3.8) holds.|x − y| n+γ , |x − y| > 2 |x − x′ |,Remark 3.7. Notice that in (i) the smoothness is assumed with respect to the secondvariable and in (ii) with respect to the first variable. If one assumes the strongerHölder condition in (i), it is easy to see that (3.7) holds with dw(x) = w(x) dx inplace of dx for every w ∈ A 1 . Therefore, the first approach in Remark 3.5 yields thatT maps L 1 (w) into L 1,∞ (w) for w ∈ A 1 .Proof. We start with (i). Let B be a ball with center x B . For every f ∈ L ∞ c (R n ) withsupp f ⊂ B and ∫ f dx = 0 we obtain (3.7):B∫∫∫|T f(x)| dx = ∣ (K(x, y) − K(x, x B )) f(y) dy∣ dxR n \4 BR n \4 B B∫ ∫∫≤ |f(y)||K(x, y) − K(x, x B )| dx dy |f(y)| dy.B|x−y|>2 |y−x B |We see (ii). Let B be a ball and f ∈ L ∞ c be supported on R n \ 4 B. Then, for everyx ∈ B and z ∈ 1 B we have2∫|T f(x) − T f(z)| ≤ |K(x, y) − K(z, y)| |f(y)| dy∞∑∫j=2C j (B)R n \4 B|x − z| γ|x − y| |f(y)| dy ∑ ∞ ∫2 −j γ − |f(y)| dy inf Mf(x).n+γ 2 j+1 Bx∈BThen, for every x ∈ B we have as desired∫∫∫|T f(x)| ≤ − |T f(z)| dz + − |T f(x) − T f(z)| dz −12 B 12 Bj=22 BB|T f(z)| dz + infx∈B Mf(x). □

10 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL4. Off-diagonal estimatesWe extract from [AM2] some definitions and results (sometimes in weaker form)on unweighted and weighted off-diagonal estimates. See there for details and moreprecise statements. Set d(E, F ) = inf{|x − y| : x ∈ E, y ∈ F } where E, F are subsetsof R n .Definition 4.1. Let 1 ≤ p ≤ q ≤ ∞. We say that a family {T t } t>0 of sublinearoperators satisfies L p − L q full off-diagonal estimates, in short T t ∈ F ( L p − L q) , if forsome c > 0, for all closed sets E and F , all f and all t > 0 we have( ∫ |T t (χ Ef)| q dxF) 1q t − 1 2 ( n p − n q ) e − c d2 (E,F )t( ∫ |f| p dxE) 1p. (4.1)Full off-diagonal estimates on a general space of homogenous type, or in the weightedcase, are not expected since L p (µ) − L q (µ) full off-diagonal estimates when p < qimply L p (µ) − L q (µ) boundedness but not L p (µ) boundedness. For example, the heatsemigroup e −t∆ on functions for general Riemannian manifolds with the doublingproperty is not L p −L q bounded when p < q unless the measure of any ball is boundedbelow by a power of its radius (see [AM2]).The following notion of off-diagonal estimates in spaces of homogeneous type involvedonly balls and annuli. Here we restrict the definition of [AM2] to the weightedsituation, that is, for dw = w(x) dx with w ∈ A ∞ . When w = 1, it turns out to beequivalent to full off-diagonal estimates. Also, it passes from unweighted estimates toweighted estimates.We set Υ(s) = max{s, s −1 } for s > 0. Given a ball B, recall that C j (B) =2 j+1 B \ 2 j B for j ≥ 2 and if w ∈ A ∞ we use the notation∫−Bh dw = 1w(B)∫Bh dw,∫−C j (B)h dw =1w(2 j+1 B)∫C j (B)h dw.Definition 4.2. Given 1 ≤ p ≤ q ≤ ∞ and any weight w ∈ A ∞ , we say that a familyof sublinear operators {T t } t>0 satisfies L p (w) − L q (w) off-diagonal estimates on balls,in short T t ∈ O ( L p (w) − L q (w) ) , if there exist θ 1 , θ 2 > 0 and c > 0 such that for everyt > 0 and for any ball B with radius r and all f,and, for all j ≥ 2,( ∫−andB( ∫−( ∫−B) 1|T t (χ Bf)| q qdw) 1|T t (χ Cj (B) f)|q qdwC j (B)) 1|T t (χ Bf)| q qdw( ) θ2r√t( ∫ Υ − |f| p dw( ) 2 j θ 1 2 j θ2rΥ √ e − c 4j r 2tt( ) 2 j θ 1 2 j θ2rΥ √ e − c 4j r 2ttB( ∫−C j (B)) 1p; (4.2)) 1|f| p pdw( ∫− |f| p dwB) 1p(4.3). (4.4)Let us make some relevant comments (see [AM2] for further details and more properties).• In the Gaussian factors the value of c is irrelevant as long as it remains positive.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 11• These definitions can be extended to complex families {T z } z∈Σθ with t replaced by|z| in the estimates.• T t may only be defined on a dense subspace D of L p or L p (w) (1 ≤ p < ∞) thatis stable by truncation by indicator functions of measurable sets (for example,L p ∩ L 2 , L p (w) ∩ L 2 or L ∞ c ).• If q = ∞, one should adapt the definitions in the usual straightforward way.• L 1 (w)−L ∞ (w) off-diagonal estimates on balls are equivalent to pointwise Gaussianupper bounds for the kernels of T t .• Hölder’s inequality implies O ( L p (w) − L q (w) ) ⊂ O ( L p 1(w) − L q 1(w) ) for all p 1 , q 1with p ≤ p 1 ≤ q 1 ≤ q.• If T t ∈ O ( L p (w) − L p (w) ) , then T t is uniformly bounded on L p (w).• This notion is stable by composition: T t ∈ O ( L q (w)−L r (w) ) and S t ∈ O ( L p (w)−L q (w) ) imply T t ◦ S t ∈ O ( L p (w) − L r (w) ) when 1 ≤ p ≤ q ≤ r ≤ ∞.• When w = 1, L p − L q off-diagonal estimates on balls are equivalent to L p − L q fulloff-diagonal estimates.• Given 1 ≤ p 0 < q 0 ≤ ∞, assume that T t ∈ O ( L p − L q) for every p, q withp 0 < p ≤ q < q 0 . Then, for all p 0 < p ≤ q < q 0 and for any w ∈ A pp 0∩ RH (q 0q ) ′ wehave that T t ∈ O ( L p (w) − L q (w) ) , equivalently, T t ∈ O ( L p (w) − L q (w) ) for everyp ≤ q with p, q ∈ W w (p 0 , q 0 ).5. Elliptic operators and their off-diagonal estimatesWe introduce the class of elliptic operators considered. Let A be an n × n matrixof complex and L ∞ -valued coefficients defined on R n . We assume that this matrixsatisfies the following ellipticity (or “accretivity”) condition: there exist 0 < λ ≤ Λ 0 of contractions on L 2 (R n , dx). Define ϑ ∈ [0, π/2)by,ϑ = sup {∣ ∣ }∣ arg〈Lf, f〉 : f ∈ D(L) .Then, the semigroup {e −t L } t>0 has an analytic extension to a complex semigroup{e −zL } z∈Σ π2of contractions on L 2 (R n , dx). Here we have written Σ θ = {z ∈ C ∗ :−ϑ| arg z| < θ}, 0 < θ < π.The families {e −t L } t>0 , { √ t ∇e −t L } t>0 , and their analytic extensions satisfy fulloff-diagonal on L 2 (R n ). These estimates can be extended to some other ranges that,up to endpoints, coincide with those of uniform boundedness.

12 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLWe define ˜J (L), respectively ˜K(L), as the interval of those exponents p ∈ [1, ∞]such that {e −t L } t>0 , respectively { √ t ∇e −t L } t>0 , is a bounded set in L(L p (R n )) (whereL(X) is the space of linear continuous maps on a Banach space X).Proposition 5.1 ([Aus, AM2]). Fix m ∈ N and 0 < µ < π/2 − ϑ.(a) There exists a non empty maximal interval of [1, ∞], denoted by J (L), such that ifp, q ∈ J (L) with p ≤ q, then {e −t L } t>0 and {(zL) m e −z L } z∈Σµ satisfy L p − L q fulloff-diagonal estimates and are bounded sets in L(L p ). Furthermore, J (L) ⊂ ˜J (L)and Int J (L) = Int ˜J (L).(b) There exists a non empty maximal interval of [1, ∞], denoted by K(L), such thatif p, q ∈ K(L) with p ≤ q, then { √ t ∇e −t L } t>0 and { √ z ∇(zL) m e −z L } z∈Σµ satisfyL p − L q full off-diagonal estimates and are bounded sets in L(L p ). Furthermore,K(L) ⊂ ˜K(L) and Int K(L) = Int ˜K(L).(c) K(L) ⊂ J (L) and, for p < 2, we have p ∈ K(L) if and only if p ∈ J (L).(d) Denote by p − (L), p + (L) the lower and upper bounds of the interval J (L) (hence,of Int ˜J (L) also) and by q − (L), q + (L) those of K(L) (hence, of Int ˜K(L) also).We have p − (L) = q − (L) and (q + (L)) ∗ ≤ p + (L).(e) If n = 1, J (L) = K(L) = [1, ∞].(f) If n = 2, J (L) = [1, ∞] and K(L) ⊃ [1, q + (L)) with q + (L) > 2.(g) If n ≥ 3, p − (L) < 2nn+2 , p +(L) > 2nn−2 and q +(L) > 2.We have set q ∗ = q n , the Sobolev exponent of q when q < n and n−q q∗ = ∞ otherwise.Given w ∈ A ∞ , we define ˜J w (L), respectively ˜K w (L), as the interval of thoseexponents p ∈ [1, ∞] such that the semigroup {e −t L } t>0 , respectively its gradient{ √ t ∇e −t L } t>0 , is uniformly bounded on L p (w). As in Proposition 5.1 uniform boundednessand weighted off-diagonal estimates on balls hold essentially in the same ranges.Proposition 5.2 ([AM2]). Fix m ∈ N and 0 < µ < π/2 − ϑ. Let w ∈ A ∞ .(a) Assume W w(p− (L), p + (L) ) ≠ Ø. There is a maximal interval of [1, ∞], denotedby J w (L), containing W w(p− (L), p + (L) ) , such that if p, q ∈ J w (L) with p ≤ q,then {e −t L } t>0 and {(zL) m e −z L } z∈Σµ satisfy L p (w)−L q (w) off-diagonal estimateson balls and are bounded sets in L(L p (w)). Furthermore, J w (L) ⊂ ˜J w (L) andInt J w (L) = Int ˜J w (L).(b) Assume W w(q− (L), q + (L) ) ≠ Ø. There exists a maximal interval of [1, ∞], denotedby K w (L), containing W w(q− (L), q + (L) ) such that if p, q ∈ K w (L) withp ≤ q, then { √ t ∇e −t L } t>0 and { √ z ∇(zL) m e −z L } z∈Σµ satisfy L p (w) − L q (w)off-diagonal estimates on balls and are bounded sets in L(L p (w)). Furthermore,K w (L) ⊂ ˜K w (L) and Int K w (L) = Int ˜K w (L).(c) Let n ≥ 2. Assume W w(q− (L), q + (L) ) ≠ Ø. Then K w (L) ⊂ J w (L). Moreover,inf J w (L) = inf K w (L) and (sup K w (L)) ∗ w ≤ sup J w (L).

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 13(d) If n = 1, the intervals J w (L) and K w (L) are the same and contain (r w , ∞] ifw /∈ A 1 and are equal to [1, ∞] if w ∈ A 1 .We have set qw ∗ = q n rwwhen q < n r n r w−q w and qw ∗ = ∞ otherwise. Recall thatr w = inf{r ≥ 1 :(w ∈ A r } and also that s w = sup{s > 1 : w ∈ RH s }.Note that W w p− (L), p + (L) ) ≠ Ø means p +(L)> r p − (L) w(s w ) ′ . This is a compatibility(condition between L and w. Similarly, W w q− (L), q + (L) ) ≠ Ø means q +(L)> r q − (L) w(s w ) ′ ,which is a more restrictive condition on w since q − (L) = p − (L) and (q + (L)) ∗ ≤ p + (L).In the case of real operators, J (L) = [1, ∞] in all dimensions ( because the kernel e −t Lsatisfies a pointwise Gaussian upper bound. Hence W w p− (L), p + (L) ) = (r w , ∞). Ifw ∈ A 1 , then one has that J w (L) = [1, ∞]. If w /∈ A 1 , since the kernel is also positiveand satisfies a similar ( pointwise lower bound, one has J w (L) ⊂ (r w , ∞]. Hence,Int J w (L) = W w p− (L), p + (L) ) .The situation may change for complex operators. But we lack of examples to saywhether or not J w (L) and W w(p− (L), p + (L) ) have different endpoints.Remark 5.3. Note that by density of L ∞ c in the spaces L p (w) for 1 ≤ p < ∞,the various extensions of e −z L and √ z ∇e −z L are all consistent. We keep the abovenotation to denote any such extension. Also, we showed in [AM2] that as long asp ∈ J w (L) with p ≠ ∞, {e −t L } t>0 is strongly continuous on L p (w), hence it has aninfinitesimal generator in L p (w), which is of type ϑ.6. ApplicationsIn this section we apply the generalized Calderón-Zygmund theory presented aboveto obtain weighted estimates for operators that are associated with L. The off-diagonalestimates on balls introduced above are one of the main tools.Associated with L we have the four numbers p − (L) = q − (L) and p + (L), q + (L).We often drop L in the notation: p − = p − (L), . . . . Recall that the semigroup andits analytic extension are uniformly bounded and satisfy full off-diagonal estimates(equivalently, off-diagonal estimates on balls) in the interval Int J (L) = Int ˜J (L) =(p − , p + ). Up to endpoints, this interval is maximal for these properties. Analogously,the gradient of the semigroup is ruled by the interval Int K(L) = Int ˜K(L) = (q − , q + ).Given w ∈ A ∞ , if W w(p− , p +)≠ Ø, then the open interval Int Jw (L) containsW w(p− , p +)and characterizes (up to endpoints) the uniform L p (w)-boundedness andthe weighted off-diagonal estimates on balls of the semigroup and its analytic extension.For the gradient, we assume that W w(q− , q +)≠ Ø and the correspondingmaximal interval is Int K w (L).

14 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL6.1. Functional calculi. Let µ ∈ (ϑ, π) and ϕ be a holomorphic function in Σ µ withthe following decay|ϕ(z)| ≤ c |z| s (1 + |z|) −2 s , z ∈ Σ µ , (6.1)for some c, s > 0. Assume that ϑ < θ < ν < µ < π/2. Then we have∫∫ϕ(L) = e −z L η + (z) dz + e −z L η − (z) dz, (6.2)Γ + Γ −where Γ ± is the half ray R + e ±i (π/2−θ) ,η ± (z) = 1 ∫e ζ z ϕ(ζ) dζ, z ∈ Γ ± , (6.3)2 π i γ ±with γ ± being the half-ray R + e ±i ν (the orientation of the paths is not needed in whatfollows so we do not pay attention to it). Note that |η ± (z)| min(1, |z| −s−1 ) forz ∈ Γ ± , hence the representation (6.2) converges in norm in L(L 2 ). Usual argumentsshow the functional property ϕ(L) ψ(L) = (ϕ ψ)(L) for two such functions ϕ, ψ.Any L as above is maximal-accretive and so it has a bounded holomorphic functionalcalculus on L 2 . Given any angle µ ∈ (ϑ, π):(a) For any function ϕ, holomorphic and bounded in Σ µ , the operator ϕ(L) can bedefined and is bounded on L 2 withwhere C only depends on ϑ and µ.‖ϕ(L)f‖ 2 ≤ C ‖ϕ‖ ∞ ‖f‖ 2(b) For any sequence ϕ k of bounded and holomorphic functions on Σ µ converginguniformly on compact subsets of Σ µ to ϕ, we have that ϕ k (L) converges stronglyto ϕ(L) in L(L 2 ).(c) The product rule ϕ(L) ψ(L) = (ϕ ψ)(L) holds for any two bounded and holomorphicfunctions ϕ, ψ in Σ µ .Let us point out that for more general holomorphic functions (such as powers), theoperators ϕ(L) can be defined as unbounded operators.Given a functional Banach space X, we say that L has a bounded holomorphicfunctional calculus on X if for any µ ∈ (ϑ, π), and for any ϕ holomorphic and satisfying(6.1) in Σ µ , one has‖ϕ(L)f‖ X ≤ C ‖ϕ‖ ∞ ‖f‖ X , f ∈ X ∩ L 2 , (6.4)where C depends only on X, ϑ and µ (but not on the decay of ϕ).If X = L p (w) as below, then (6.4) implies that ϕ(L) extends to a bounded operatoron X by density. That (a), (b) and (c) hold with L 2 replaced by X for all boundedholomorphic functions in Σ µ , follow from the theory in [McI] using the fact that onthose X, the semigroup {e −t L } t>0 has an infinitesimal generator which is of type ϑ(see Remark 5.3).Theorem 6.1 ([BK1, Aus]). If p ∈ Int J (L) then L has a bounded holomorphicfunctional calculus on L p (R n ). Furthermore, this range is sharp up to endpoints.The weighted version of this result is presented next. We mention [Ma1] wheresimilar weighted estimates are proved under kernel upper bounds assumptions.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 15Theorem 6.2 ([AM3]). Let w ∈ A ∞ be such that W w(p− (L), p + (L) ) ≠ Ø. Letp ∈ Int J w (L) and µ ∈ (ϑ, π). For any ϕ holomorphic on Σ µ satisfying (6.1), we havewith C independent of ϕ and f.calculus on L p (w).‖ϕ(L)f‖ L p (w) ≤ C ‖ϕ‖ ∞ ‖f‖ L p (w), f ∈ L ∞ c , (6.5)Hence, L has a bounded holomorphic functionalRemark 6.3. Fix w ∈ A ∞ with W w(p− (L), p + (L) ) ≠ Ø. If 1 < p < ∞ and L hasa bounded holomorphic functional calculus on L p (w), then p ∈ ˜J w (L). Indeed, takeϕ(z) = e −z . As Int ˜J w (L) = Int J w (L) by Proposition 5.1, this shows that the rangeobtained in the theorem is optimal up to endpoints.6.2. Riesz transforms. The Riesz transforms associated to L are ∂ j L −1/2 , 1 ≤ j ≤ n.Set ∇L −1/2 = (∂ 1 L −1/2 , . . . , ∂ n L −1/2 ). The solution of the Kato conjecture [AHLMT]implies that this operator extends boundedly to L 2 . This allows the representation∇L −1/2 f = √ 1 ∫ ∞ √t ∇e−t L f dtπ t , (6.6)in which the integral converges strongly in L 2 both at 0 and ∞ when f ∈ L 2 . The L pestimates for this operator are characterized in [Aus].Theorem 6.4 ([Aus]). Under the previous assumptions, p ∈ Int K(L) if and only if∇L −1/2 is bounded on L p (R n ).In the weighted case we have the following analog.Theorem 6.5 ([AM3]). Let w ∈ A ∞ be such that W w(q− (L), q + (L) ) ≠ Ø. For allp ∈ Int K w (L) and f ∈ L ∞ c ,Hence, ∇L −1/2 has a bounded extension to L p (w).0‖∇L −1/2 f‖ L p (w) ≤ C ‖f‖ L p (w). (6.7)For a discussion on sharpness issues concerning this result, the reader is referred to[AM3, Remark 5.5].6.3. Square functions. We define the square functions for x∈R n and f ∈L 2 ,( ∫ ∞g L f(x) = |(t L) 1/2 e −t L f(x)| 2 dt ) 12,0t( ∫ ∞G L f(x) = | √ t ∇e −t L f(x)| 2 dt ) 12.tThese square functions satisfy the following unweighted estimates.Theorem 6.6 ([Aus]). (a) If p ∈ Int J (L) then for all f ∈ L p ∩ L 2 ,0‖g L f‖ p ∼ ‖f‖ p .Furthermore, this range is sharp up to endpoints.(b) If p ∈ Int K(L) then for all f ∈ L p ∩ L 2 ,‖G L f‖ p ∼ ‖f‖ p .Furthermore, this range is sharp up to endpoints.

16 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLIn this statement, ∼ can be replaced by : the square function estimates for L(with ) automatically imply the reverse ones for L ∗ . The part concerning g L can beobtained using an abstract result of Le Merdy [LeM] as a consequence of the boundedholomorphic functional calculus on L p . The method in [Aus] is direct. We remind thereader that in [Ste], these inequalities for L = −∆ were proved differently and theboundedness of G −∆ follows from that of g −∆ and of the Riesz transforms ∂ j (−∆) −1/2(or vice-versa) using the commutation between ∂ j and e t ∆ . Here, no such thing ispossible.We have the following weighted estimates for square functions.Theorem 6.7 ([AM3]). Let w ∈ A ∞ .(a) If W w(p− (L), p + (L) ) ≠ Ø and p ∈ Int J w (L) then for all f ∈ L ∞ cwe have‖g L f‖ L p (w) ‖f‖ L p (w).(b) If W w(q− (L), q + (L) ) ≠ Ø and p ∈ Int K w (L) then for all f ∈ L ∞ cwe have‖G L f‖ L p (w) ‖f‖ L p (w).We also get reverse weighted square function estimates as follows.Theorem 6.8 ([AM3]). Let w ∈ A ∞ .(a) If W w(p− (L), p + (L) ) ≠ Ø and p ∈ Int J w (L) then(b) If r w < p < ∞,‖f‖ L p (w) ‖g L f‖ L p (w), f ∈ L p (w) ∩ L 2 .‖f‖ L p (w) ‖G L f‖ L p (w), f ∈ L p (w) ∩ L 2 .Remark 6.9. Let us observe that Int J w (L) is the sharp range, up to endpoints, for‖g L f‖ L p (w) ∼ ‖f‖ L p (w). Indeed, we have g L (e −t L f) ≤ g L f for all t > 0. Hence, theequivalence implies the uniform L p (w) boundedness of e −t L , which implies p ∈ ˜J w (L)(see Proposition 5.2). Actually, Int J w (L) is also the sharp range up to endpoints forthe inequality ‖g L f‖ L p (w) ‖f‖ L p (w). It suffices to adapt the interpolation procedurein [Aus, Theorem 7.1, Step 7]. Similarly, this interpolation procedure also shows thatInt K w (L) is sharp up to endpoints for ‖G L f‖ L p (w) ‖f‖ L p (w).7. About the proofsThey follow a general scheme. First, we choose A B = I − (I − e −r2 L ) m with r theradius of B and m ≥ 1 sufficiently large and whose value changes in each situation.A first application of Theorem 3.1 and Theorem 3.2 yield unweighted estimates,and weighted estimates in a first range. This requires to prove (3.1) (or the stronger(3.3)), (3.2), (3.4) and (3.5) with measure dx, using the full off-diagonal estimates ofProposition 5.1.Then, having fixed w, a second application of Theorems 3.1 and Theorems 3.2 yieldweighted estimates in the largest range. This requires to prove (3.1) (or the stronger(3.3)), (3.2), (3.4) and (3.5) with measure dw, using the off-diagonal estimates onballs of Proposition 5.2.There are technical difficulties depending on whether operators commute or notwith the semigroup. Full details are in [AM3]

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 178. Further resultsWe present some additional results obtained in [AM3], [AM4], [AM5].Let µ be a doubling measure in R n and let b ∈ BMO(µ) (BMO is for bounded meanoscillation), that is,∫‖b‖ BMO(µ) = sup − |b − b B |dµ < ∞,B Bwhere the supremum is taken over balls and b B stands for the µ-average of b on B.When dµ = dx we simply write BMO. If w ∈ A ∞ (so dw is a doubling measure) thenthe reverse Hölder property yields that BMO(w) = BMO with equivalent norms.For T a sublinear operator, bounded in some L p 0(µ), 1 ≤ p 0 ≤ ∞, b ∈ BMO, k ∈ N,we define the k-th order commutatorT k b f(x) = T ( (b(x) − b) k f ) (x), f ∈ L ∞ c (µ), x ∈ R n .Note that T 0 b = T and that T k b f(x) is well-defined almost everywhere when f ∈ L∞ c (µ).If T is linear it can be alternatively defined by recurrence: the first order commutator isT 1 b f(x) = [b, T ]f(x) = b(x) T f(x) − T (b f)(x)and for k ≥ 2, the k-th order commutator is given by T k bTheorem 8.1 ([AM1]). Let k ∈ N and b ∈ BMO(µ).k−1= [b, Tb ].(a) Assume the conditions of Theorem 3.1 with (3.1) replaced by the stronger condition(3.3). Suppose that T and Tbm for m = 1, . . . , k are bounded on L q 0(µ) andthat ∑ j α j 2 D j j k < ∞. Then for all p 0 < p < q 0 ,‖T k b f‖ L p (µ) ≤ C ‖b‖ k BMO(µ) ‖f‖ L p (µ).(b) Assume the conditions of Theorem 3.2. If ∑ j α j j k < ∞, then for all p 0 < p < q 0 ,w ∈ A pp 0∩ RH (q 0p) ′, ‖T k b f‖ L p (w) ≤ C ‖b‖ k BMO(µ) ‖f‖ L p (w).With these results in hand, we obtain weighted estimates for the commutators ofthe previous operators.Theorem 8.2 ([AM3]). Let w ∈ A ∞ , k ∈ N and b ∈ BMO. Assume one of thefollowing conditions:(a) T = ϕ(L) with ϕ bounded holomorphic on Σ µ , W w(p− (L), p + (L) ) ≠ Ø and p ∈Int J w (L).(b) T = ∇ L −1/2 , W w(q− (L), q + (L) ) ≠ Ø and p ∈ Int K w (L).(c) T = g L , W w(p− (L), p + (L) ) ≠ Ø and p ∈ Int J w (L).(d) T = G L , W w(q− (L), q + (L) ) ≠ Ø and p ∈ Int K w (L).Then, for every for f ∈ L ∞ c (R n ), we have‖T k b f‖ L p (w) ≤ C ‖b‖ k BMO ‖f‖ L p (w),where C does not depend on f, b, and is proportional to ‖ϕ‖ ∞ in case (a).Let us mention that, under kernel upper bounds assumptions, unweighted estimatesfor commutators in case (a) are obtained in [DY1].

18 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL8.1. Reverse inequalities for square roots. The method described above can beused to consider estimates opposite to (6.7). In the unweighted case, [Aus] shows thatif f ∈ S and p is such that max { 1, n p −(L)n+p − (L)}< p < p+ (L), then‖L 1/2 f‖ p ‖∇f‖ p .The weighted counterpart of this estimate is considered in [AM3]. Let w ∈ A ∞ andassume that W w(p− (L), p + (L) ) ≠ Ø, then‖L 1/2 f‖ L p (w) ‖∇f‖ L p (w), f ∈ S, (8.1)for all p such that max { n rr w , w ̂p − (L)n r w+̂p − (L)}< p < ̂p+ (L), where r w = inf{r ≥ 1 :w ∈ A r }, and ̂p − (L), ̂p + (L) are the endpoints of J w (L), that is, (̂p − (L), ̂p + (L) ) =Int J w (L).Let us define Ẇ 1,p (w) as the completion of S under the semi-norm ‖∇f‖ L p (w).Arguing as in [AT] (see [Aus]) combining Theorem 6.5 and (8.1), it follows that L 1/2extends to an( isomorphism from Ẇ 1,p (w) into L p (w) for all p ∈ Int K w (L) with p > r w ,provided W w q− (L), q + (L) ) ≠ Ø.8.2. Vector-valued estimates. In [AM1], by using an extrapolation result “à laRubio de Francia” for the classes of weights A p ∩ RHp 0 (q 0p′,)it follows automaticallyfrom Theorem 3.2, part (b), that for every p 0 < p, r < q 0 and w ∈ A p ∩ RHp 0 (q 0p′,)onehas( ∥∑ ) 1|T f k | r r( ∥ C ∥∑ ) 1|f k | r r∥ (8.2)L p (w) L (w). pkAs a consequence, one can show weighted vector-valued estimates for the previousoperators (see [AM3] for more details). Given w ∈ A ∞ , we have• If W w(p− (L), p + (L) ) ≠ Ø, and T = ϕ(L) (ϕ bounded holomorphic in an appropriatesector) or T = g L then (8.2) holds for all p, r ∈ Int J w (L)• If W w(q− (L), q + (L) ) ≠ Ø, and T = ∇L −1/2 or T = G L then (8.2) holds for allp, r ∈ Int J w (L) ∩ (r w , ∞).8.3. Maximal regularity. Other vector-valued inequalities of interest are( ∥∑ ) 1|e −ζkL f k | 2 2( ∥ ≤ C ∥∑ ) 1|f k | 2 2 ∥L q (w)1≤k≤Nk1≤k≤N∥L q (w)(8.3)for ζ k ∈ Σ α with 0 < α < π/2−ϑ and f k ∈ L p (w) with a constant C independent of N,the choice of the ζ k ’s and the f k ’s. We restrict to 1 < q < ∞ and w ∈ A ∞ . By [Wei,Theorem 4.2], we know that the existence of such a constant is equivalent to themaximal L p -regularity of the generator −A of e −tL on L q (w) with one/all 1 < p < ∞,that is the existence of a constant C ′ such that for all f ∈ L p ((0, ∞), L q (w)) thesolution u of the parabolic problem on R n × (0, ∞),satisfiesu ′ (t) + Au(t) = f(t), u(0) = 0,‖u ′ ‖ L p ((0,∞),L q (w)) + ‖Au‖ L p ((0,∞),L q (w)) ≤ C ′ ‖f‖ L p ((0,∞),L q (w)).

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 19Proposition 8.3 ([AM3]). Let w ∈ A ∞ be such that W w(p− (L), p + (L) ) ≠ Ø. Thenfor any q ∈ Int J w (L), (8.3) holds with C = C q,w,L independent of N, ζ k , f k .This result follows from an abstract result of Kalton-Weis [KW, Theorem 5.3] togetherwith the bounded holomorphic functional calculus of L on those L q (w) thatwe established in Theorem 6.2. However, the proof in [AM3] uses extrapolation andl 2 -valued versions of Theorems 3.1 and 3.2. Note that q = 2 may not be contained inInt J w (L) and the interpolation method of [BK2] may not work here.8.4. Fractional operators. The fractional operators associated with L are formallygiven by, for α > 0,∫L −α/2 1 ∞=t α/2 −t L dteΓ(α/2)t .Theorem 8.4 ([Aus]). Let p − < p < q < p + and α/n = 1/p − 1/q. Then L −α/2 isbounded from L p (R n ) to L q (R n ).Remark 8.5. A special case of this result with p − = 1 and p + = ∞ is when L = −∆as one has that L −α/2 = I α , the classical Riesz potential whose kernel is c |x| −(n−α) . Ifone has a Gaussian kernel bounds, then |L −α/2 f| I α (|f|) and the result follows atonce from the well known estimates for I α . For a more general result see [Var].Theorem 8.6 ([AM5]). Let p − < p < q < p + and α/n = 1/p − 1/q. Then L −α/2 isbounded from L p (w p ) to L q (w q ) for every w ∈ A 1+1− 1 ∩ RH pp − p q ( + . Furthermore, forq)′every k ∈ N and b ∈ BMO, we have that (L −α/2 ) k b —the k-th order commutator ofL −α/2 — satisfies the same estimates.0The proof of this result is based on a version of Theorem 3.2 adapted to the caseof fractional operators and involving fractional maximal functions.Remark 8.7. In the classical case of the commutator with the Riesz potential, unweightedestimates were considered in [Cha]. Weighted estimates were established in[ST] by means of extrapolation. Another proof based on a good-λ estimate was givenin [CF]. For k = 1 and elliptic operators L with Gaussian kernel bounds, unweightedestimates were studied in [DY2] using the sharp maximal function introduced in [Ma1],[Ma2]. In that case, a simpler proof, that also yields the weighted estimates, was obtainedin [CMP] using the pointwise estimate ∣ ∣[b, L −α/2 ]f(x) ∣ ∣ I α (|b(x) − b| |f|)(x).A discretization method inspired by [Per] is used to show that the latter operator iscontrolled in L 1 (w) by M L log L,α f for every w ∈ A ∞ . From here, by the extrapolationtechniques developed in [CMP], this control can be extended to L p (w) for 0 < p < ∞,w ∈ A ∞ and consequently the weighted estimates of [b, L −α/2 ] reduce to those ofM L log L,α which are studied in [CF].8.5. Riesz transform on manifolds. Let M be a complete non-compact Riemannianmanifold with d its geodesic distance and µ the volume form. Let ∆ be the positiveLaplace-Beltrami operator on M given by∫〈∆f, g〉 = ∇f · ∇g dµMwhere ∇ is the Riemannian gradient on M and · is an inner product on T M. TheRiesz transform is the tangent space valued operator ∇∆ −1/2 and it is bounded fromL 2 (M, µ) into L 2 (M; T M, µ) by construction.

20 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLThe manifold M verifies the doubling volume property if µ is doubling:(D) µ(B(x, 2 r)) ≤ C µ(B(x, r)) < ∞,for all x ∈ M and r > 0 where B(x, r) = {y ∈ M : d(x, y) < r}. A Riemannianmanifold M equipped with the geodesic distance and a doubling volume form is aspace of homogeneous type. Non-compactness of M implies infinite diameter, whichtogether with the doubling volume property yields µ(M) = ∞ (see for instance [Ma2]).One says that the heat kernel p t (x, y) of the semigroup e −t∆ has Gaussian upperbounds if for some constants c, C > 0 and all t > 0, x, y ∈ M,C(GUB)p t (x, y) ≤µ(B(x, √ d2 (x,y)t)) e−c t .It is known that under doubling it is a consequence of the same inequality only aty = x [Gri, Theorem 1.1].Theorem 8.8 ([CD]). Under (D) and (GUB), then(R p )∥ |∇∆ −1/2 f| ∥ p≤ C p ‖f‖ pholds for 1 < p < 2 and all f ∈ L ∞ c (M).Here, | · | is the norm on T M associated with the inner product.We shall setq + = sup { p ∈ (1, ∞) : (R p ) holds } .which satisfies q + ≥ 2 under the assumptions of Theorem 8.8. It can be equalto 2 ([CD]). It is bigger than 2 assuming further the stronger L 2 -Poincaré inequalities([AC]). It can be equal to +∞ (see below).Let us turn to weighted estimates.Theorem 8.9 ([AM4]). Assume (D) and (GUB). Let w ∈ A ∞ (µ).(i) For p ∈ W w (1, q + ), the Riesz transform is of strong-type (p, p) with respect tow dµ, that is, ∥ ∥ |∇∆ −1/2 f| ∥ ∥L p (M,w) ≤ C p,w ‖f‖ L p (M,w) (8.4)for all f ∈ L ∞ c (M).(ii) If w ∈ A 1 (µ) ∩ RH (q+ )′(µ), then the Riesz transform is of weak-type (1, 1) withrespect to w dµ, that is,∥ |∇∆ −1/2 f| ∥ ≤ C L 1,∞ (M,w) 1,w ‖f‖ L 1 (M,w) (8.5)for all f ∈ L ∞ c (M).Here, the strategy of proof is a little bit different. Following ideas of [BZ], part(i) uses the tools to prove Theorem 3.2, namely a good-λ inequality, together witha duality argument. For part (ii), it uses a weighted variant of Theorem 3.1. Theoperator A B is given by I − (I − e −r2 ∆ ) m with m large enough and r the radiusof B. Note that here, the heat semigroup satisfies unweighted L 1 − L ∞ off-diagonalestimates on balls from (GUB), so the kernel of A B has a pointwise upper bound.Remark 8.10. Given k ∈ N and b ∈ BMO(M, µ) one can consider the k-th ordercommutator of the Riesz transform (∇∆ −1/2 ) k b . This operator satisfies (8.4), that is,(∇∆ −1/2 ) k b is bounded on Lp (M, w) under the same conditions on M, w, p.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 21If q + = ∞ then the Riesz transform is bounded on L p (M, w) for r w < p < ∞, thatis, for w ∈ A p (µ), and we obtain the same weighted theory as for the Riesz transformon R n :Corollary 8.11 ([AM4]). Let M be a complete non-compact Riemannian manifoldsatisfying the doubling volume property and Gaussian upper bounds. Assume that theRiesz transform has strong type (p, p) with respect to dµ for all 1 < p < ∞. Thenthe Riesz transform has strong type (p, p) with respect to w dµ for all w ∈ A p (µ) and1 < p < ∞ and it is of weak-type (1, 1) with respect to w dµ for all w ∈ A 1 (µ).Unweighted L p bounds for Riesz transforms in different specific situations werereobtained in a unified manner in [ACDH] assuming conditions on the heat kerneland its gradient. The methods used there are precisely those which allowed us tostart the weighted theory in [AM1].Let us recall three situations in which this corollary applies (see [ACDH], wheremore is done, and the references therein): manifolds with non-negative Ricci curvature,co-compact covering manifolds with polynomial growth deck transformationgroup, Lie groups with polynomial volume growth endowed with a sublaplacian. A situationwhere q + < ∞ is conical manifolds with compact basis without boundary. Theconnected sum of two copies of R n is another (simpler) example of such a situation.[Aus][AC][ACDH]ReferencesP. Auscher, On necessary and sufficient conditions for L p estimates of Riesz transformassociated to elliptic operators on R n and related estimates, Mem. Amer. Math. Soc. 186(2007), no. 871.P. Auscher and T. Coulhon, Riesz transforms on manifolds and Poincaré inequalities,Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 531–555.P. Auscher, T. Coulhon, X.T. Duong and S. Hofmann, Riesz transforms on manifolds andheat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911–957.[AHLMT] P. Auscher, S. Hofmann, M. Lacey, A. M c Intosh and Ph. Tchamitchian, The solution ofthe Kato square root problem for second order elliptic operators on R n , Ann. of Math. (2)156 (2002), 633–654.[AM1][AM2][AM3][AM4][AM5][AT]P. Auscher and J.M. Martell, Weighted norm inequalities, off-diagonal estimates andelliptic operators. I. General operator theory and weights, Adv. Math. 212 (2007), no. 1,225–276.P. Auscher and J.M. Martell, Weighted norm inequalities, off-diagonal estimates andelliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ.7 (2007), no. 2, 265–316.P. Auscher and J.M. Martell, Weighted norm inequalities, off-diagonal estimates andelliptic operators. III. Harmonic analysis of elliptic operators, J. Funct. Anal. 241 (2006),no. 2, 703–746.P. Auscher and J.M. Martell, Weighted norm inequalities, off-diagonal estimates andelliptic operators. IV. Riesz transforms on manifolds and weights, Math. Z. 260 (2008),no. 3, 527–539.P. Auscher and J.M. Martell, Weighted norm inequalities for fractional operators, IndianaUniv. Math. J. 57 (2008), no. 4, 1845–1870.P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and relatedtopics, Astérisque Vol. 249, Soc. Math. France, 1998.[BZ] F. Bernicot and J. Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008), no. 7,1761–1796.

22 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL[BK1]S. Blunck and P. Kunstmann, Calderón-Zygmund theory for non-integral operators andthe H ∞ -functional calculus, Rev. Mat. Iberoamericana 19 (2003), no. 3, 919–942.[BK2] S. Blunck and P. Kunstmann, Weak-type (p, p) estimates for Riesz transforms, Math. Z.247 (2004), no. 1, 137–148.[CP]L.A. Caffarelli and I. Peral, On W 1,p estimates for elliptic equations in divergence form,Comm. Pure App. Math. 51 (1998), 1–21.[Cha] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7–16.[CD][CF][CMP][DY1][DY2][GR][Gra][Gri][HM][Hör]T. Coulhon and X.T. Duong, Riesz transforms for 1 ≤ p ≤ 2, Trans. Amer. Math. Soc.351 (1999), 1151–1169.D. Cruz-Uribe and A. Fiorenza, Endpoint estimates and weighted norm inequalities forcommutators of fractional integrals, Publ. Mat. 47 (2003), 103–131.D. Cruz-Uribe, J.M. Martell and C. Pérez, Extrapolation from A ∞ weights and applications,J. Funct. Anal. 213 (2004), 412–439.X.T. Duong and L. Yan, Commutators of BMO functions and singular integral operatorswith non-smooth kernels, Bull. Austral. Math. Soc. 67 (2003), no. 2, 187–200.X.T. Duong and L. Yan, On commutators of fractional integrals, Proc. Amer. Math. Soc.132 (2004), no. 12, 3549–3557.J. García-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics,North Holland Mathematics Studies 116, North-Holland Publishing Co., Amsterdam,1985.L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper SaddleRiver, NJ, 2004.A. Grigor’yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. DifferentialGeom. 45 (1997), no. 1, 33–52.S. Hofmann and J.M. Martell, L p bounds for Riesz transforms and square roots associatedto second order elliptic operators, Publ. Mat. 47 (2003), 497–515.L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math.104 (1960), 93–140.[KW] N. Kalton and L. Weis, The H ∞ -calculus and sums of closed operators, Math. Ann. 321(2001), 319–345.[LeM][Ma1][Ma2][McI]C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math.France 132 (2004), no. 1, 137–156.J.M. Martell, Sharp maximal functions associated with approximations of the identity inspaces of homogeneous type and applications, Studia Math. 161 (2004), 113–145.J.M. Martell, Desigualdades con pesos en el Análisis de Fourier: de los espacios de tipohomogéneo a las medidas no doblantes, Ph.D. Thesis, Universidad Autónoma de Madrid,2001.A. M c Intosh, Operators which have an H ∞ functional calculus, Miniconference on operatortheory and partial differential equations (North Ryde, 1986), Proc. Centre Math.Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, 1986, 210–231.[Per] C. Pérez, Sharp L p -weighted Sobolev inequalities, Ann. Inst. Fourier (Grenoble) 45 (1995),no. 3, 809–824.[Sh1][Sh2][Ste]Z. Shen, The L p Dirichlet problem for elliptic systems on Lipschitz domains, Math Res.Letters 13 (2006), 143–159.Z. Shen, Bounds of Riesz transforms on L p spaces for second order elliptic operators, Ann.Inst. Fourier 55 (2005), no. 1, 173–197.E.M. Stein, Singular integrals and differentiability properties of functions, Princeton MathematicalSeries, No. 30, Princeton University Press, Princeton, N.J. 1970.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 23[ST][Var][Wei]J.O. Strömberg and A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics1381, Springer-Verlag, Berlin, 1989.N.Th. Varopoulos, Une généralisation du théorème de Hardy-Littlewood-Sobolev pour lesespaces de Dirichlet, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 14, 651–654.L. Weis, Operator-valued Fourier multiplier theorems and maximal L p -regularity, Math.Ann. 319 (2001), 735–758.Pascal Auscher, Université de Paris-Sud et CNRS UMR 8628, 91405 Orsay Cedex,FranceE-mail address: pascal.auscher@math.u-psud.frJosé María Martell, Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, ConsejoSuperior de Investigaciones Científicas, C/ Serrano 121, E-28006 Madrid, SpainE-mail address: chema.martell@uam.es