WEIGHTED NORM INEQUALITIES, OFF-DIAGONAL ESTIMATES ...

2 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL(c) Typical square functions “à la” Littlewood-Paley-Stein: one, g L , using only functionsof L, and the other, G L , combining functions of L and the gradient operator(Section 7).(d) Vector-valued inequalities for the operators above and the so-called R-boundednessof the analytic semigroup {e −z L } which is linked to maximal regularity (Section8).Let us stress that those operators may not be representable with “usable” kernels:they are “non-integral”. But they still are singular in the sense that they are of order 0.Hence, usual methods for singular integrals have to be strengthened. The unweightedL p estimates are described in [Aus] for the operators in (a) − (c), with emphasison the sharpness of the ranges of p. The instrumental tools are two criteria for L pboundedness, valid in spaces of homogeneous type: one was a sharper and simplerversion of a theorem by Blunck and Kunstmann [BK1] in the spirit of Hörmander’scriterion via the Calderón-Zygmund decomposition, and the other one a criterion ofthe first author, Coulhon, Duong and Hofmann [ACDH] in the spirit of Feffermanand Stein’s sharp maximal function via a good-λ inequality. The main interest ofthose results were that they yield L p boundedness of (sub)linear operators on spacesof homogeneous type for p in an arbitrary interval. Such theorems are extended inPart I of our series [AM1] to obtain weighted L p bounds for the operator itself, itscommutators with a BMO function and also vector-valued expressions.In Part II [AM2], we studied one-parameter families of operators satisfying localL p − L q estimates called off-diagonal estimates on balls (the setting is that of aspace of homogeneous type). Among other things, such estimates imply uniform L p -boundedness and are stable under composition. In case of one-parameter semigroups,we showed that as soon as there exists one pair (p, q) of indices with p < q for whichthese local L p − L q estimates hold, then they hold for all pairs of indices taken inthe interior of the range of L p boundedness. This fact is of utmost importance forapplications as we often need to play with exponents. We showed that such estimatespass from the unweighted case to the weighted case. Eventually, we made a thoroughstudy of weighted off-diagonal estimates on balls for the semigroup arising from theoperator L above.Our strategy here has the same two steps in each of the four situations describedabove. The first step consists in obtaining a first range of exponents p (depending onthe weight) by applying the abstract machinery from Part I. This range turns out tobe the best possible for both classes of operators and weights.However, given one operator and one weight, the range of p obtained above maynot be sharp, and this leads us to the second step. The sharp range is in fact relatedto the one for weighted off-diagonal estimates established in Part II. At this point,we use the main results of Part I in the Euclidean space but now equipped with thedoubling measure w(x) dx.We wish to point out that some of our results can be obtained by different methods(essentially from geometric theory of Banach spaces) once the bounded holomorphicfunctional calculus is established in (a). We give the references in the text.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 3We wish to say that our proofs are technically simpler than the ones in [Aus] evenfor the unweighted case, because the notion of off-diagonal estimates used here is moreappropriate.Finally, thanks to the general results in Part I, the same technology allows us toprove in passing weighted L p estimates for commutators of the operators in (a) − (c)with BMO functions in the same ranges of exponents (see Section 9).2. General criteria for boundedness and the set W w (p 0 , q 0 )The underlying space is the Euclidean setting R n equipped with Lebesgue measureor a doubling measure obtained from an A ∞ weight. We state two results used in thiswork, referring to [AM1] for statements in stronger form and for references to earlierworks.Given a ball B, we write∫− h dx = 1 ∫h(x) dx.B |B|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.BThe reverse Hölder classes are defined in the following way: w ∈ RH q , 1 < q < ∞, ifthere 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 q1≤p

4 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL(vii) If w ∈ A ∞ , then the measure dw = w dx is a Borel doubling measure.If the Lebesgue measure is replaced by a Borel doubling measure µ, then all theabove properties remain valid with the notation change [ST].Given 1 ≤ p 0 < q 0 ≤ ∞ and w ∈ A ∞ (with respect to a Borel doubling measure µ)we define the setW w (p 0 , q 0 ) = { p : p 0 < p < q 0 , w ∈ A pp 0∩ RH (q 0p) ′ }.If w = 1, then W 1 (p 0 , q 0 ) = (p 0 , q 0 ). As it is shown in [AM1], if not empty, we have(q)0W w (p 0 , q 0 ) = p 0 r w ,(s w ) ′wherer w = inf{r ≥ 1 : w ∈ A r }, s w = sup{s > 1 : w ∈ RH s }.We use the following notation: if B is a ball with radius r(B) and λ > 0, λ Bdenotes the concentric ball with radius r(λ B) = λ r(B), C j (B) = 2 j+1 B \ 2 j B whenj ≥ 2, C 1 (B) = 4B, and∫− h dµ =C j (B)1µ(2 j+1 B)∫C j (B)h dµ.Theorem 2.2. Let µ be a doubling Borel measure on R n and 1 ≤ p 0 < q 0 ≤ ∞.Let T be a sublinear operator acting on L p 0(µ), {A r } r>0 a family of operators actingfrom a subspace D of L p 0(µ) into L p 0(µ) and S an operator from D into the space ofmeasurable functions on R n . Assume that( ∫− |T (I − A r(B) )f| p 0dµB) 1p 0≤ ∑ j≥1( ∫g(j) −2 j+1 B) 1|Sf| p p 0dµ0, (2.1)and( ∫−B) 1|T A r(B) f| q q 0dµ0≤ ∑ j≥1( ∫g(j) −2 j+1 B) 1|T f| p p 0dµ0, (2.2)for all f ∈ D, all ball B where r(B) denotes its radius for some g(j) with ∑ g(j) < ∞(with usual changes if q 0 = ∞). Let p ∈ W w (p 0 , q 0 ), that is, p 0 < p < q 0 andw ∈ A p ∩ RHp 0 (q 0p′.)There is a constant C such that for all f ∈ D‖T f‖ L p (w) ≤ C ‖Sf‖ L p (w). (2.3)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 (althoughthe second property is not needed in this section). Next, L p (w) is the space of complexvalued functions in L p (dw) with dw = w dµ. However, all this extends to functionsvalued in a Banach space.Remark 2.3. In the applications below, we have, either Sf = f with f ∈ L ∞ c thespace of compactly supported bounded functions on R n , or Sf = ∇f with f ∈ S theSchwartz class on R n (see Section 6).

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 5Let us recall that the doubling order D of a doubling measure µ is the smallestnumber κ ≥ 0 such that there exists C ≥ 0 for which µ(λ B) ≤ C µ λ κ µ(B) for everyball B and for any λ > 1.The other criterion we are going to use is the following.Theorem 2.4. 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 r } r>0 is family of linear operators acting from L ∞ c into L q 0(µ). Assume that forj ≥ 2,and for j ≥ 1,( ∫−C j (B)( ∫−) 1|T (I − A r(B) )f| p p 0dµ0C j (B)) 1|A r(B) f| q q 0dµ0( ∫≤ g(j) −( ∫≤ g(j) −BB) 1|f| p p 0dµ0(2.4)) 1|f| p p 0dµ0(2.5)for all ball B with r(B) its radius and for all f ∈ L ∞ c supported in B. If ∑ j g(j) 2D j 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 has an analytic extension to a complex semigroup {e −z L } z∈Σπ/2−ϑof contractions on L 2 (R n , dx). Here we have written for 0 < θ < π,Σ θ = {z ∈ C ∗ : | arg z| < θ}.

6 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLLet w ∈ A ∞ . Here and thereafter, we write L p (w) for L p (R n , wdx) and if w = 1, wedrop w in the notation. We define ˜J w (L) and ˜K w (L) as the (possibly empty) intervalsof those exponents p ∈ [1, ∞] such that {e −t L } t>0 is a bounded set in L(L p (w)) and{ √ t ∇e −t L } t>0 is a bounded set in L(L p (w)) respectively (where L(X) is the space oflinear continuous maps on a Banach space X).We extract from [Aus, AM2] some definitions and results (sometimes in weakerform) on unweighted and weighted off-diagonal estimates. See there for details andmore precise statements. Set d(E, F ) = inf{|x − y| : x ∈ E, y ∈ F } where E, F aresubsets of R n .Definition 3.1. 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 dxFB) 1q t − 1 2 ( n p − n q ) e − c d2 (E,F )t( ∫ |f| p dxE) 1p. (3.1)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∫− h dw = 1 ∫ ∫∫1h dw, − h dw =h dw.w(B)w(2 j+1 B)BC j (B)C j (B)Definition 3.2. 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,( ∫− |T t (χ Bf)| q dwand, for all j ≥ 2,( ∫− |T t (χ Cj (B) f)|q dwandB( ∫−C j (B)B) 1q) 1|T t (χ Bf)| q qdw) 1q( ) 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)B) 1p; (3.2)) 1|f| p pdw( ∫− |f| p dw) 1p(3.3). (3.4)Let us make some relevant comments for this work (see [AM2] for further details).• In the Gaussian factors the value of c is irrelevant as long as it remains non negative.We will freely use the same letter from line to line even if its value changes.• These definitions extend to complex families {T z } z∈Σθ with t replaced by |z| in theestimates.• In both definitions, T t may only be defined on a dense subspace D of L p or L p (w)(1 ≤ p < ∞) that is stable by truncation by indicator functions of measurable sets(for example, L p ∩ L 2 , L p (w) ∩ L 2 or L ∞ c ).†Here and thereafter, for two positive quantities A, B, by A B we mean that there exists aconstant C > 0 (independent of the various parameters) such that A ≤ CB.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 7• 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 .• Both notions are stable by composition: T t ∈ O ( L q (w) − L r (w) ) and S t ∈O ( L p (w) − L q (w) ) then T t ◦ S t ∈ O ( L p (w) − L r (w) ) when 1 ≤ p ≤ q ≤ r ≤ ∞and similarly for full off-diagonal estimates.• When w = 1, L p − L q off-diagonal estimates on balls are equivalent to L p − L q fulloff-diagonal estimates.• Notice the symmetry between (3.3) and (3.4).If I is a subinterval of [1, ∞], Int I denotes the interior in R of I ∩ R.Proposition 3.3. Fix m ∈ N and 0 < µ < π/2 − ϑ.(a) There exists a non empty maximal interval in [1, ∞], denoted by J (L), suchthat if p, q ∈ J (L) with p ≤ q, then {(zL) m e −z L } z∈Σµ satisfies L p − L q full offdiagonalestimates and is a bounded set 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 { √ z ∇(zL) m e −z L } z∈Σµ satisfies L p − L q full offdiagonalestimates and is a bounded set 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 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 havep − (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 nn−q , the Sobolev exponent of q when q < n and q∗ = ∞ otherwise.Proposition 3.4. 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 {(zL) m e −z L } z∈Σµ satisfies L p (w)−L q (w) off-diagonal estimates on balls andis a bounded set in L(L p (w)). Furthermore, J w (L) ⊂ ˜J w (L) and Int 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) with

8 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLp ≤ q, then { √ z ∇(zL) m e −z L } z∈Σµ satisfies L p (w) − L q (w) off-diagonal estimateson balls and is a bounded set in L(L p (w)). Furthermore, K w (L) ⊂ ˜K w (L) andInt 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).(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 3.5. Note that by density of L ∞ c in the spaces L p (w) for 1 ≤ p < ∞, thevarious extensions of e −z L and ∇e −z L are all consistent. We keep the above notationto denote any such extension. Also, we showed in [AM2] that as long as p ∈ J w (L)with p ≠ ∞, {e −t L } t>0 is strongly continuous on L p (w), hence it has an infinitesimalgenerator in L p (w), which is of type ϑ.From now on, L denotes an operator as defined in this section with the four numbersp − (L) = q − (L) and p + (L), q + (L). We often drop L in the notation: p − = p − (L),. . . . For a given weight w ∈ A ∞ , we set W w(p− , p +)= (˜p− , ˜p + ) (when it is notempty) and Int J w (L) = (̂p − , ̂p + ). We have ̂p − ≤ ˜p − < ˜p + ≤ ̂p + . Similarly, we setW w(q− , q +)= (˜q− , ˜q + ) (when it is not empty) and Int K w (L) = (̂q − , ̂q + ). We havêq − ≤ ˜q − < ˜q + ≤ ̂q + .4. Functional CalculiLet µ ∈ (ϑ, π) (do not confuse with the measure µ used in Section 2) and ϕ be aholomorphic function in Σ µ with the following decay|ϕ(z)| ≤ c |z| s (1 + |z|) −2 s , z ∈ Σ µ , (4.1)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, (4.2)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 ± , (4.3)2 π i γ ±

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 9with 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 ), z ∈ Γ ± , (4.4)hence the representation (4.2) converges in norm in L(L 2 ). Usual arguments show thefunctional 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), for any ϕ holomorphic and satisfying(4.1) in Σ µ one has‖ϕ(L)f‖ X ≤ C ‖ϕ‖ ∞ ‖f‖ X , f ∈ X ∩ L 2 , (4.5)where C depends only on X, ϑ and µ (but not on the decay of ϕ).If X = L p (w) as below, then (4.5) 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 (seethe last remark of previous section). We skip such classical arguments of functionalcalculi.Theorem 4.1. [BK1, Aus] The interior of the set of exponents p ∈ (1, ∞) such thatL has a bounded holomorphic functional calculus on L p is equal to Int J (L) definedin Proposition 3.3.Our first result is a weighted version of this theorem. We mention [Mar] wheresimilar weighted estimates are proved under kernel upper bounds assumptions.Theorem 4.2. Let w ∈ A ∞ be such that W w(p− (L), p + (L) ) ≠ Ø. Let p ∈ Int J w (L)and µ ∈ (ϑ, π). For any ϕ holomorphic on Σ µ satisfying (4.1), 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 , (4.6)Hence, L has a bounded holomorphic functional

10 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLRemark 4.3. 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 3.3, this shows that the rangeobtained in the theorem is optimal up to endpoints.Proof. It is enough to assume µ < π/2. Note that the operators e −z L are uniformlybounded on L p (w) when z ∈ Σ µ , hence, by (4.4), the representation (4.2) converges innorm in L(L p (w)). Of course, this simple argument does not yield the right estimate,(4.6), which is our goal. It is no loss of generality to assume that ‖ϕ‖ ∞ = 1. We splitthe argument into three cases: p ∈ (˜p − , ˜p + ), p ∈ (˜p − , ̂p + ), p ∈ (̂p − , ˜p + ).Case p ∈ (˜p − , ˜p + ). By (iii) and (iv) in Proposition 2.1, there exist p 0 , q 0 such thatp − < p 0 < p < q 0 < p + and w ∈ A pp 0∩ RH (q 0p) ′.The desired bound (4.6) follows on applying Theorem 2.2 for the underlying measuredx and weight w to T = ϕ(L) with p 0 , q 0 , A r = I − (I − e −r2 L ) m where m ≥ 1 isan integer to be chosen and S = I . As ϕ(L) and (I − e −r2 L ) m are bounded on L p 0(uniformly with respect to r for the latter) by Proposition 3.3 and Theorem 4.1, itremains to checking both (2.1) and (2.2) on D = L ∞ c .We start by showing (2.2). We fix f ∈ L ∞ c and a ball B. We will use several timesthe following decomposition of any given function h:h = ∑ j≥1h j , h j = h χ Cj (B) . (4.7)Fix 1 ≤ k ≤ m. Since p 0 ≤ q 0 and p 0 , q 0 ∈ J (L), we have e −t L ∈ O ( L p 0− L 0) q (weare using the equivalence between the two notions of off-diagonal estimates for theLebesgue measure), hence( ∫−B) 1|e −k r2 L h j | q q 0dx0and by Minkowski’s inequality( ∫− |e −k r2 L h| q 0dxB( ∫) 1 2 j (θ 1+θ 2 ) e −α 4j − |h| p p 0dx02 j+1 B) 1q 0 ∑ j≥1( ∫g(j) −2 j+1 B) 1|h| p 0dxp 0(4.8)with g(j) = 2 j (θ 1+θ 2 ) e −α 4j for any h ∈ L p 0. This estimate with h = ϕ(L)f ∈ L p 0yields (2.2) since, by the commutation rule, ϕ(L)e −k r2 L f = e −k r2 L h.We next show (2.1). Let f ∈ L ∞ c and B be a ball. Write f = ∑ j≥1 f j as before.For j = 1, we use the L p 0boundedness of ϕ(L) and (I − e −r2 L ) m , hence( ∫) 1 ( ∫ ) 1− |ϕ(L)(I − e −r2 L ) m f 1 | p p 0dx0 − |f| p p 0dx0. (4.9)BFor j ≥ 2, the functions η ± associated with ψ(z) = ϕ(z) (1 − e −r2 z ) m by (4.3) satisfy4 B|η ± (z)| r2 m|z| m+1 , z ∈ Γ ±.

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 11Since p 0 ∈ J (L), {e −z L } z∈Γ± ∈ O ( L p 0− L 0) p and so( ∫∫− ∣ η + (z) e −z L f j dz∣ p ) 1 ( ∫0 pdx0≤ − |eB Γ +∫Γ −z L f j | p 0dx+ B∫ ( ) 2 j θ 2 j θ2r1Υ √ e − α 4j r 2r 2 m ( ∫|z|Γ + |z| |z| |dz| −m+1( ∫) 1 2 j (θ 1−2 m)− |f| p p 0dx0C j (B)) 1C j (B)p 0|η + (z)| |dz|) 1|f| p p 0dx0provided 2 m > θ 2 . We have used, after a change of variable, that∫ ∞Υ(s) θ 2e −c s2 s 2 m ds0s < ∞.The same is obtained when one deals with the term corresponding to Γ − . Pluggingboth estimates into the representation of ψ(L) given by (4.2) one obtains( ∫) 1( ∫) 1− |ϕ(L)(I − e −r2 L ) m f j | p p 0dx0 2 j (θ 1−2 m)− |f| p p 0dx0, (4.10)BC j (B)therefore, (2.1) holds when 2m > max{θ 1 , θ 2 } since C j (B) ⊂ 2 j+1 B.Case p ∈ (˜p − , ̂p + ): Take p 0 , q 0 such that ˜p − < p 0 < ˜p + and p 0 < p < q 0 < ̂p + . LetA r = I − (I − e −r2 L ) m for some large enough m ≥ 1. Remark that by the previouscase, ϕ(L) has the right norm in L(L p 0(w)) and so does A r by Proposition 3.4. Weapply Theorem 2.2 with the Borel doubling measure dw and no weight. Thus, it isenough to see that ϕ(L) satisfies (2.1) and (2.2) for dw on D = L ∞ c ⊂ L p 0(w). Butthis follows by adapting the preceding argument replacing everywhere dx by dw andobserving that e −z L ∈ O ( L p 0(w) − L q 0(w) ) since p 0 , q 0 ∈ Int J w (L) and p 0 ≤ q 0 . Weskip details.Case p ∈ (̂p − , ˜p + ): Take p 0 , q 0 such that ˜p − < q 0 < ˜p + and ̂p − < p 0 < p < q 0 . SetA r = I − (I − e −r2 L ) m for some integer m ≥ 1 to be chosen later. Since q 0 ∈ (˜p − , ˜p + ),by the first case, ϕ(L) has the right norm in L(L q 0(w)) and so does A r by Proposition3.4. We apply Theorem 2.4 with underlying measure dw. It is enough to show (2.4)and (2.5). Fix a ball B and f ∈ L ∞ c supported in B.We begin with (2.5) for A r . It is enough to show it for e −k r2 L with 1 ≤ k ≤ m.Since p 0 , q 0 ∈ J w (L) and p 0 ≤ q 0 we have e −t L ∈ O ( L p 0(w) − L q 0(w) ) , hence( ∫−C j (B)|e −k r2 L f| q 0dw) 1q 0 2 j (θ 1+θ 2 ) e −c 4j( ∫− |f| p 0dwB) 1p 0. (4.11)This implies (2.5) with g(j) = C 2 j (θ 1+θ 2 ) −c 4je and ∑ j≥1 g(j) 2D j < ∞ holds whereD is the doubling order of dw.We turn to (2.4). Let j ≥ 2. The argument is the same as the one for (4.10) byreversing the roles of C j (B) and B, and using dw and e −z L ∈ O ( L p 0(w) − L p 0(w) )(since p 0 ∈ J w (L)) instead of dx and e −z L ∈ O ( L p 0− L 0) p . We obtain( ∫− |ϕ(L)(I − e −r2 L ) m f| p 0dw) 1p 0 2 j (θ 1−2 m)( ∫− |f| p 0dwC j (B)B) 1p 0

12 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLprovided 2 m > θ 2 and it remains to impose further 2 m > θ 1 + D to conclude.Remark 4.4. If W w (p − , p + ) ≠ Ø, the last part of the proof yields weighted weak-type(̂p − , ̂p − ) of ϕ(L) provided ̂p − ∈ J w (L). To do so, one only has to take p 0 = ̂p − .5. Riesz transformsThe 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 [AHLMcT] implies thatthis operator extends boundedly to L 2 (we ignore the C n -valued aspect of things).This allows the representation∇L −1/2 f = √ 1 ∫ ∞∇e −t L f √ dt , (5.1)π tin which the integral converges strongly in L 2 both at 0 and ∞ when f ∈ L 2 . Notethat for an arbitrary f ∈ L 2 , h = L −1/2 f makes sense in the homogeneous Sobolevspace Ḣ1 which is the completion of C0 ∞ (R n ) for the semi-norm ‖∇h‖ 2 and ∇ becomesthe extension of the gradient to that space. This construction can be forgotten if n ≥ 3as Ḣ1 ⊂ L 2∗ but not if n ≤ 2. To circumvent this technical difficulty, we introduce∫S ε = √ 1 1/ε −t L dtπe √tfor 0 < ε < 1 and, in fact, ∇S ε are uniformly bounded on L 2εand converge strongly in L 2 . This defines ∇L −1/2 .Theorem 5.1 ([Aus]). The maximal interval of exponents p ∈ (1, ∞) for which∇L −1/2 has a bounded extension to L p is equal to Int K(L) defined in Proposition3.3 and for p ∈ Int K(L), ‖∇f‖ p ∼ ‖L 1/2 f‖ p for all f ∈ D(L 1/2 ) = H 1 (the Sobolevspace).Again, the operators ∇S ε are uniformly bounded on L p and converge strongly inL p as ε → 0. Indeed, for f ∈ L ∞ c , S ε f ∈ D(L) ⊂ D(L 1/2 ) and ‖∇S ε f‖ p ‖L 1/2 S ε f‖ p .Observe that L 1/2 S ε = ϕ ε (L), where ϕ ε is a bounded holomorphic function in Σ µfor any 0 < µ < π/2 with sup ε ‖ϕ ε ‖ ∞ < ∞ and {ϕ ε } ε converges uniformly to 1 oncompact subsets of Σ µ as ε → 0. The claim follows by Theorem 4.1 and density.We turn to weighted norm inequalities. Remark that by Proposition 3.4, for all p ∈J w (L), S ε is bounded on L p (w) (the norm must depend on ε) and for all p ∈ K w (L),∇S ε is bounded on L p (w) with no control yet on the norm with respect to ε.Theorem 5.2. Let w ∈ A ∞(be such that W w q− (L), q + (L) ) ≠ Ø. For all p ∈Int K w (L) and f ∈ L ∞ c ,‖∇L −1/2 f‖ L p (w) ‖f‖ L p (w). (5.2)Hence, ∇L −1/2 has a bounded extension to L p (w).We note that for a given p ∈ Int K w (L), once (5.2) is established, similar argumentsusing Theorem 4.2 imply convergence in L p (w) of ∇S ε f to ∇L −1/2 f for f ∈ L ∞ c .Proof. We split the argument in three cases: p ∈ (˜q − , ˜q + ), p ∈ (˜q − , ̂q + ), p ∈ (̂q − , ˜q + ).Case p ∈ (˜q − , ˜q + ): By (iii) and (iv) in Proposition 2.1, there exist p 0 , q 0 such thatq − < p 0 < p < q 0 < q + and w ∈ A pp 0∩ RH (q 0p) ′.The desired estimate (5.2) is obtained by applying Theorem 2.2 with underlying measuredx and weight w. Hence, it suffices to verify (2.1) and (2.2) on D = L ∞ c for0□

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 13T = ∇L −1/2 , S = I and A r = I − (I − e −r2 L ) m , with m a large enough integer. Theseconditions will be proved as in [Aus], but here we use the whole range of exponents forwhich the Riesz transforms are bounded on unweighted L p spaces, that is, (q − , q + ).Lemma 5.3. Fix a ball B. For f ∈ L ∞ c and m large enough,( ∫) 1− |∇L −1/2 (I − e −r2 L ) m f| p p 0dx0≤ ∑ ( ∫g 1 (j) −Bj≥1and for f ∈ L p 0such that ∇f ∈ L p 0and 1 ≤ k ≤ m,( ∫) 1− |∇e −k r2 L f| q q 0dx0≤ ∑ ( ∫g 2 (j) −Bj≥12 j+1 BC j (B)) 1|f| p 0dxp 0(5.3)) 1|∇f| p 0dxp 0, (5.4)where g 1 (j) = C m 2 j θ 4 −m j and g 2 (j) = C m 2 j ∑ l≥j 2l θ e −α 4l for some θ > 0.Assume this is proved. Note that if 2m > θ then ∑ j≥1 g 1(j) < ∞ and the firstestimate is (2.1).Next, expanding A r = I − (I − e −r2 L ) m , the latter estimate applied to S ε f in placeof f (since S ε f ∈ L p 0and ∇S ε f ∈ L p 0) and the commuting rule A r S ε = S ε A r give us( ∫− |∇S ε A r f| q 0dxB) 1q 0≤ ∑ j≥1( ∫g 2 (j) −2 j+1 B|∇S ε f| p 0dx) 1p 0.By letting ε go to 0 (the justification uses the observations made at the beginningof this section and is left to the reader), we obtain (2.2) using ∑ j≥1 g 2(j) < ∞.Therefore, by Theorem 2.2, (5.2) holds for f ∈ L ∞ c .Proof of Lemma 5.3. We begin with the first estimate. Decomposing f as in (4.7),( ∫) 1− |∇L −1/2 (I − e −r2 L ) m f| p p 0dx0≤ ∑ ( ∫) 1− |∇L −1/2 (I − e −r2 L ) m f j | p p 0dx0.Bj≥1 BFor j = 1, since q − < p 0 < q + , ∇L −1/2 and e −r2 L are bounded on L p 0by Theorem 5.1and Proposition 3.3. Hence,( ∫) 1 ( ∫ ) 1− |∇L −1/2 (I − e −r2 L ) m f 1 | p p 0dx0 − |f| p p 0dx0.BFor j ≥ 2, we use a different approach. If h ∈ L 2 , by (5.1)∇L −1/2 (I − e −r2 L ) m h = √ 1 ∫ ∞ √ dtt ∇ϕ(L, t)h π t ,where ϕ(z, t) = e −t z (1 − e −r2 z ) m . The functions η ± (·, t) associated with ϕ(·, t) by(4.3) satisfyr 2 m|η ± (z, t)| (|z| + t) , z ∈ Γ ±, t > 0.m+1Since √ z ∇e −z L ∈ O ( L p 0−L 0) p (note that p0 ∈ K(L) and we are using the equivalencebetween the two notions of off-diagonal estimates for the Lebesgue measure),( ∫∫− ∣ η + (z) √ t ∇e −z L f j dz∣ p ) 10 pdx0B Γ +04 B

14 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL≤∫Γ +( 2 j θ 1 2 j θ 1∫∫−BΥΓ +∫ ∞0| √ ) 1z ∇e −z L f j | p p 0dx0()2 j θ2r√ e − α 4j r 2|z||z|( ) 2 j θ2rΥ √ e − α 4j r 2ssObserving that when 2 m > θ 2∫ ∞ ∫ ∞( ) 2 j θ2rΥ √ e − α 4j r 2ss00√t√|z||η + (z)| |dz|√t√|z||η + (z)| |dz|√t√ s√t√ sr 2 m ((s + t) ds m+1( ∫−∫−C j (B)C j (B)r 2 mdsdt(s + t)m+1t = C 4−j m ,) 1|f| p p 0dx0) 1|f| p p 0dx0.and plugging this, plus the corresponding term for Γ − , into the representation (4.2),we obtain( ∫) 1∫ ∞ ( ∫− |∇e −t L (I − e −r2 L ) m f j | p p 0dx0 − | √ ) 1t ∇ϕ(L, t)f j | p p 0dx0dtB0 Bt( ∫) 1 2 j (θ 1−2 m)− |f| p p 0dx0. (5.5)C j (B)This readily yields the first estimate in the lemma.Let us get the second one. Fix 1 ≤ k ≤ m. Let f ∈ L p 0such that ∇f ∈ L p 0. Wewrite h = f − f 4 B where f λ B is the dx-average of f on λ B. Then by the conservationproperty (see [Aus]) e −t L 1 = 1 for all t > 0, we have∇e −k r2 L f = ∇e −k r2 L (f − f 4 B ) = ∇e −k r2 L h = ∑ j≥1∇e −k r2 L h j ,with h j = h χ Cj (B) . Hence,( ∫) 1− |∇e −k r2 L f| q q 0dx0≤ ∑ ( ∫−Bj≥1B) 1|∇e −k r2 L h j | q q 0dx0.Since p 0 ≤ q 0 and p 0 , q 0 ∈ K(L), √ t ∇e −t L ∈ O ( L p 0−L 0) q . This and the Lp 0-Poincaréinequality for dx yield( ∫) 1− |∇e −k r2 L h j | q q 0dx0 2j (θ 1+θ 2 ) −α 4je( ∫) 1− |h j | p p 0dx0BrC j (B)≤2j (θ 1+θ 2 ) −α 4je( ∫) 1− |f − f 4 B | p p 0dx0r≤2j (θ 1+θ 2 ) −α 4jer2j (θ 1+θ 2 ) −α 4jer2 j+1 B( ( ∫−2 j+1 Bj∑ ( ∫−l=1) 1|f − f 2 j+1 B| p p 0dx0+2 l+1 Bj∑ ( ∫ 2 j (θ 1+θ 2 ) e −α 4j 2 l −l=12 l+1 B) 1|f − f 2 B| p p 0dx0l+1j∑l=2)|f 2 l B − f 2 B| l+1) 1|∇f| p 0dxp 0, (5.6)

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 15which is the desired estimate with θ = θ 1 + θ 2 .□Case p ∈ (˜q − , ̂q + ): Take p 0 , q 0 such that ˜q − < p 0 < ˜q + and p 0 < p < q 0 < ̂q + .Let A r = I − (I − e −r2 L ) m for some m ≥ 1 to be chosen later. As p 0 ∈ (˜q − , ˜q + ),both ∇L −1/2 and A r are bounded on L p 0(w) (we have just shown it for the Riesztransforms and A r are bounded uniformly in r by Proposition 3.4). By Theorem 2.2with underlying doubling measure dw and no weight, it is enough to verify (2.1) and(2.2) on D = L ∞ c for T = ∇L −1/2 , S = I and A r . To do so, it suffices to copythe proof of Lemma 5.3 in the weighted case by changing systematically dx to dw,off-diagonal estimates with respect to dx by those with respect to dw given the choiceof p 0 , q 0 . Also in the argument with dx we used a Poincaré inequality. Here, sincep 0 ∈ W w (q − , q + ), w ∈ A p0 /q −and in particular w ∈ A p0 (since q − ≥ 1). Therefore wecan use the L p 0(w)-Poincaré inequality (see [FPW]):( ∫− |f − f B,w | p 0dwB) 1p 0( ∫ r(B) −B) 1|∇f| p p 0dw0for all f ∈ L 1 loc (w) such that ∇f ∈ Lp 0loc (w), where f B,w is the dw-average of f over B.We leave further details to the reader.Case p ∈ (̂q − , ˜q + ): Take p 0 , q 0 such that ˜q − < q 0 < ˜q + and ̂q − < p 0 < p < q 0 . SetA r = I − (I − e −r2 L ) m for some integer m ≥ 1 to be chosen later. Since q 0 ∈ (˜q − , ˜q + ),it follows that ∇L −1/2 is already bounded on L q 0(w) and so is A r . That ∇L −1/2 isbounded on L p (w) will follow on applying Theorem 2.4 with underlying measure w.Hence it is enough to check both (2.4) and (2.5).We begin with (2.5). By Proposition 3.4, inf J w (L) = inf K w (L) = ̂q − . Sincep 0 > ̂q − and p 0 ≤ q 0 ∈ W w (q − , q + ) ⊂ W w (p − , p + ) ⊂ J w (L), we have p 0 , q 0 ∈ J w (L)and so e −t L ∈ O ( L p 0(w) − L q 0(w) ) . This yields (4.11), hence (2.5) with g(j) =−c 4jwhichC 2 j (θ 1+θ 2 ) e clearly satisfies ∑ j g(j) 2D j < ∞, with D the doubling orderof dw.We next show (2.4). Let f ∈ L ∞ c be supported on a ball B and j ≥ 2. Theargument is the same as the one for (5.5) by reversing the roles of C j (B) and B, andusing dw and √ z ∇e −z L ∈ O ( L p 0(w) − L p 0(w) ) (since p 0 ∈ K w (L)) instead of dx and√ z ∇e−z L ∈ O ( L p 0− L p 0). Hence, we obtain( ∫−C j (B)) 1|∇L −1/2 (I − e −r2 L ) m f| p 0dwp 0 2 j (θ 1−2 m)( ∫−B) 1|f| p p 0dw0provided 2 m > θ 2 and it remains to impose further 2 m > θ 1 + D to conclude.□Remark 5.4. If W w (q − , q + ) ≠ Ø, the last part of the proof yields weighted weak-type(̂q − , ̂q − ) of ∇L −1/2 provided ̂q − ∈ K w (L): one only needs to take p 0 = ̂q − .Remark 5.5. Theorem 5.1 asserts that Int K(L) is the exact range of L p boundednessfor the Riesz transforms when w = 1. When w ≠ 1, we cannot repeat thesame argument as it used Sobolev embedding which has no simple counterpart in theweighted situation. However, if we insert in the integral of (5.1) a function m(t) withm ∈ L ∞ (0, ∞), then (5.2) holds with a constant proportional to ‖m‖ ∞ . Indeed, Letϕ m (z) = ∫ ∞z 1/2 e −t z m(t) √ dt0 tfor z ∈ Σ µ , ϑ < µ < π/2. Then, ϕ m is holomorphic in

16 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLΣ µ and bounded with ‖ϕ m ‖ ∞ ≤ c µ ‖m‖ ∞ . Now for f ∈ L ∞ c ,∫ ∞0∇e −t L f m(t) dt √t= ∇L −1/2 ϕ m (L)fhence, combining Theorems 4.2 and 5.2, we obtain for p ∈ Int K w (L),∫ ∞√∥ t ∇e−t L f m(t) dt∥ ‖m‖ ∞ ‖f‖ Ltp (w).L p (w)0Conversely, given an exponent p ∈ (1, ∞), assume that this L p (w) estimate holds forall m ∈ L ∞ . Using randomization techniques which we skip (see Section 8 for someaccount on such techniques), this implies( ∫ ∞∥0| √ t ∇e −t L f| 2 dtt) 12∥ ‖f‖ L p (w).L p (w)This square function estimate is proved directly in Section 7 and we indicate at theend of that section why this inequality implies p ∈ ˜K w (L). Thus, the range in p issharp up to endpoints (see Proposition 3.4).6. Reverse inequalities for square rootsWe continue on square roots by studying when the inequality opposite to (5.2) hold.First we recall the unweighted case.Theorem 6.1. [Aus] If max { 1, n p −(L)n+p − (L)}< p < p+ (L) then for f ∈ S,‖L 1/2 f‖ p ‖∇f‖ p . (6.1)To state our result, we need a new exponent. For p > 0, definep w,∗ =n r w pn r w + p ,where r w = inf{r ≥ 1 : w ∈ A r }. Set also p ∗ w = n rw p for p < n r n r w−p w and p ∗ w = ∞otherwise. Note that (p w,∗ ) ∗ w = p.Theorem 6.2. Let w ∈ A ∞ with W w(p− (L), p + (L) ) ≠ Ø. If max { r w , (̂p − ) w,∗} r w , thenL 1/2 extends to an isomorphism from Ẇ 1,p (w) into L p (w).Proof of Theorem 6.2. We split the argument in three cases:(˜p − , ̂p + ), p ∈ (max { }r w , (̂p − ) w,∗ , ˜p+ ).Case p ∈ (˜p − , ˜p + ): It relies on the following lemma.p ∈ (˜p − , ˜p + ), p ∈

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 19Proof. Since w ∈ A p , we have an L p (w) Poincaré inequality (see [FPW]). On theother hand, as w ∈ A p and 1 ≤ q < p ∗ w (if p = 1, i.e. w ∈ A 1 , it holds also atq = 1 ∗ w =n when n ≥ 2) we can apply [FPW, Corollary 3.2] (when checking then−1“balance condition” in that reference we have used that w ∈ A r implies (|E|/|B|) r w(E)/w(B) for any ball B and any E ⊂ B). Thus there is an L p (w)−L q (w) Poincaréinequality:( ∫−B) 1|f − f B,w | q qdw( ∫ r(B) −for all locally Lipschitz functions f and all balls B.needed to invoke [AM1, Proposition 9.1].We use the following resolution of L 1/2 :L 1/2 f = 1It suffices to work with ∫ Rε√ π∫ ∞0BLe −t L f dt √t.) 1|∇f| p pdw(6.12)These are all the ingredients□. . ., to obtain bounds independent of ε, R, and then tolet ε ↓ 0 and R ↑ ∞: indeed, the truncated integrals converge to L 1/2 f in L 2 whenf ∈ S and a use of Fatou’s lemma concludes the proof. For the truncated integrals,all the calculations are justified. We write L 1/2 where it is understood that it shouldbe replaced by its approximation at all places.Take q 0 so that ˜p − < q 0 < ˜p + . By the first case of the proof,‖L 1/2 f‖ L q 0 (w) ‖∇f‖ L q 0 (w) . (6.13)We may assume that max{ r w , (̂p − ) w,∗ } < p < ˜p − , otherwise there is nothing toprove. We claim that it is enough to show that‖L 1/2 f‖ L p,∞ (w) ‖∇f‖ L p (w). (6.14)Assuming this estimate we want to interpolate. To this end, we use the followinglemma.Lemma 6.7. Assume r > r w . Then D = { (−∆) 1/2 f : f ∈ S, supp ̂f ⊂ R n \ {0} } isdense in L r (w), where ̂f denotes the Fourier transform of f.Proof. It is easy to see that D ⊂ S hence D ⊂ L r (w). As in [Gra, p. 353], using thatthe classical Littlewood-Paley series converges in L r (w) since w ∈ A r , it follows thatthe set˜D = { g ∈ S : supp ĝ ⊂ R n \ {0}, supp ĝ is compact }is dense in L r (w). We see that ˜D ⊂ D and so D is dense in L r (w). For g ∈ ˜D,f = (−∆) −1/2 g is well-defined in S as ̂f(ξ) = c |ξ| −1/2 ĝ(ξ) and supp ̂f ⊂ R n \ {0}.Hence, g = (−∆) 1/2 f ∈ D.□If r > r w , the usual Riesz transforms, ∇(−∆) −1/2 , are bounded on L r (w) (this canbe reobtained from the results in Section 5). Also, for g ∈ L r (w), one has‖g‖ L r (w) ∼ ‖∇(−∆) −1/2 g‖ L r (w)using the identity −I = R 2 1 + · · · + R 2 n where R j = ∂ j (−∆) −1/2 . Thus, for g ∈ D,L 1/2 (−∆) −1/2 g = L 1/2 f if f = (−∆) −1/2 g and ‖∇f‖ L r (w) ∼ ‖g‖ L r (w) for r > r w . Asr w < p < q 0 , (6.13) and (6.14) reformulate into weighted strong type (q 0 , q 0 ) and weaktype (p, p) of T = L 1/2 (−∆) −1/2 a priori defined on D. Since D is dense in all L r (w)

20 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLwhen r > r w by the above lemma, we can extend T by density in both cases and theirrestrictions to the space of simple functions agree. Hence, we can apply Marcinkiewiczinterpolation and conclude again by density that (6.13) holds for all q with p < q < q 0which leads to the desired estimate.Our goal is thus to establish (6.14), more precisely: for f ∈ S and α > 0,w{|L 1/2 f| > α} = w{x ∈ R n : |L 1/2 f(x)| > α} ≤ C α p ∫R n |∇f| p dw. (6.15)Since p > r w , we have w ∈ A p . From the condition (̂p − ) w,∗ < p, we have ̂p − < p ∗ w.Therefore, there exists q ∈ (̂p − , ̂p + ) = Int J w (L) such that ̂p − < q < p ∗ w. Thus, wecan apply the Calderón-Zygmund decomposition of Lemma 6.6 to f at height α forthe measure dw and write f = g + ∑ i b i. Using (6.13), (6.7) and q 0 > p, we have{w |L 1/2 g| > α } 1 ∫|L 1/2 g| q 0dw 13 α q 0R α n q 0 1 ∫|∇f| p dw + 1 ∫α p R α n p∫R n |∇g| q 0dw 1 α p ∫|∇g| p dwR∣ n ∣∣∑ ∣ ∫∣∣p 1∇b i dw |∇f| p dw,R α n p R nwhere the last estimate follows by applying (6.10), (6.8), (6.9).To compute L 1/2 ( ∑ i b i), let r i = 2 k if 2 k ≤ r(B i ) < 2 k+1 , hence r i ∼ r(B i ) for all i.Writeand then{∣ ∣∣ ∑wiL 1/2 = 1 √ π∫ r 2i0−t L dtLe √ + 1t√ π∫ ∞∣L 1/2 ∣∣ 2 α} ( ⋃ )b i > ≤ w 4 B i + w3ir 2 i((+ w R n \ ⋃ ii{∣ ∣∣ ∑ 1 α p ∫R n |∇f| p dw + I + II,−t L dtLe √ = T i + U i ,t∣ ∣∣ α}U i b i >3i) ⋂ {∣ ∣∣ ∑ ∣ ∣∣ α})4 B i T i b i >3where we have used (6.9). We estimate II. Since q ∈ J w (L), it follows that t L e −t L ∈O ( L q (w) − L q (w) ) by Proposition 3.4, henceII 1 α 1 α ∑ i∑ ∑∫ij≥2C j (B i )∑ ∑2 j D w(B i )ij≥2|T i b i | dw 1 α∫ r 2i∑2 j D e −c 4j w(B i ) ∑ ij≥20∑ ∑w(2 j B i )ij≥2( )2 j θ 1 2 j θ2r iΥ √te − c 4j ri2t∫ r 2i0∫−C j (B i )idt( ∫−t 3/2w(B i ) 1 α p ∫R n |∇f| p dw,|t L e −t L b i | dw dtt 3/2B i|b i | q dwwhere we have used (6.11) and (6.9), and D is the doubling order of dw.) 1q

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 21It remains to handling the term I. Using functional calculus for L one can computeU i as r −1i ψ(ri 2 L) with ψ the holomorphic function on the sector Σ π/2 given byψ(z) = c∫ ∞1−tz dtz e √ . (6.16)tIt is easy to show that |ψ(z)| ≤ C|z| 1/2 e −c|z| , uniformly on subsectors Σ µ , 0 ≤ µ < π.2We claim that, since q ∈ Int J w (L),∑ ∥ ∥ ψ(4 k ∥∥L ( ∑ ) 1L) β k ∥ |β k | 2 2∥ (6.17)q (w) L (w). qk∈Z∑The proof of this inequality is postponed until the end of Section 7. We set β k =b ii : r i =2 k r i. Then,∑U i b i = ∑ ( ∑)ψ(4 k b iL)= ∑ ψ(4 k L)β k .riik∈Z i : r i =2 k k∈ZUsing (6.17), the bounded overlap property (6.10), (6.11), r i ∼ r(B i ) and (6.9), onehasI 1 ∥ ∥∥ ∑ ∥ ∥∥qUα q i b i 1 ∥ ∥∥∥ ( ∑ ) 1 q|βL q (w) α q k | 2 2∥ 1 ∫∑ |b i | qα q R r q dwn iik∈Zk∈ZL q (w)i ∑ iw(B i ) 1 α p ∫R n |∇f| p dw.Collecting the obtained estimates, we conclude (6.14) as desired.Remark 6.8. If w ∈ A 1 , W w(p− (L), p + (L) ) ≠ Ø and (̂p − ) w,∗ < 1 then for all f ∈ S‖L 1/2 f‖ L 1,∞ (w) ‖∇f‖ L 1 (w).This (that is (6.15) with p = 1) uses a similar argument (left to the reader) oncewe have chosen an appropriate q for which L 1 (w) − L q (w) Poincaré inequality holds:since w ∈ A 1 , one needs q ≤n . As r n−1 w = 1, the assumption (̂p − ) w,∗ < 1 means that̂p −

22 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLandInt { 1 < p < ∞ : ‖G L f‖ p ∼ ‖f‖ p , ∀ f ∈ L p ∩ L 2} = ( q − (L), q + (L) ) .In 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 7.2. Let w ∈ A ∞ .(a) If W w(p− (L), p + (L) ) ≠ Ø and p ∈ Int J w (L) then for all f ∈ L ∞ c‖g L f‖ L p (w) ‖f‖ L p (w).we have(b) If W w(q− (L), q + (L) ) ≠ Ø and p ∈ Int K w (L) then for all f ∈ L ∞ c‖G L f‖ L p (w) ‖f‖ L p (w).we haveNote that the operators (t L) 1/2 e −t L and ∇e −t L extend to L p (w) when p ∈ Int J w (L)and p ∈ Int K w (L) respectively. By seeing g L and G L as linear operators from scalarfunctions to H-valued functions (see below for definitions), the above inequalities extendto all f ∈ L p (w) by density (see the proof).We also get reverse weighted square function estimates as follows.Theorem 7.3. 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 .The restriction that f ∈ L 2 can be removed provided g L and G L are appropriatelyinterpreted: see the proofs. We add a comment about sharpness of the ranges of p atthe end of the section.As a corollary, g L (resp. G L ) defines a new norm on L p (w) when p ∈ Int J w (L)(resp. p ∈ Int K w (L) and p > r w ). Again, Le Merdy’s result cited above [LeM] alsogives such a result for g L , but not for G L . The restriction p > r w in part (b) comesfrom the argument. We do not know whether it is necessary for a given weight nonidentically 1.Before we begin the arguments, we recall some basic facts about Hilbert-valuedextensions of scalar inequalities. To do so we introduce some notation: by H we meanL 2 ((0, ∞), dt)and |· | denotes the norm in H. Hence, for a function h: R n ×(0, ∞) →tC, we have for x ∈ R n ( ∫ ∞|h(x, ·) | =0|h(x, t)| 2 dt ) 1/2.t

In particular,**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 23with ϕ(z, t) = (t z) 1/2 e −t z andg L f(x) = |ϕ(L, ·)f(x) |G L f(x) = |∇ϕ(L, ·)f(x) |with ϕ(z, t) = √ t e −t z . Let L p H (w) be the space of H-valued Lp (w)-functions equippedwith the norm(∫) 1‖h‖ LpH (w) = |h(x, ·) | p pdw(x) .R nLemma 7.4. Let µ be a Borel measure on R n (for instance, given by an A ∞ weight).Let 1 ≤ p ≤ q < ∞. Let D be a subspace of M, the space of measurable functions inR n . Let S, T be linear operators from D into M. Assume there exists C 0 > 0 suchthat for all f ∈ D, we have∑‖T f‖ L q (µ) ≤ C 0 α j ‖Sf‖ L p (F j ,µ),j≥1where F j are subsets of R n and α j ≥ 0. Then, there is an H-valued extension with thesame constant: for all f : R n ×(0, ∞) → C such that for (almost) all t > 0, f(·, t) ∈ D,∑‖T f‖ LqH (µ) ≤ C 0 α j ‖Sf‖ LpH (F j,µ).j≥1The extension of a linear operator T on C-valued functions to H-valued functions isdefined for x ∈ R n and t > 0 by (T h)(x, t) = T ( h(·, t) ) (x), that is, t can be consideredas a parameter and T acts only on the variable in R n . This result is essentially thesame as the Marcinkiewicz-Zygmund theorem and the fact that H is isometric to l 2 .That the norm decreases uses p ≤ q. We refer to, for instance, [Gra, Theorem 4.5.1]for an argument that extends straightforwardly to our setting.Proof of Theorem 7.2. Part (a). We split the argument in three cases: p ∈ (˜p − , ˜p + ),p ∈ (˜p − , ̂p + ), p ∈ (̂p − , ˜p + ).Case p ∈ (˜p − , ˜p + ): By Proposition 2.1, there exist p 0 , q 0 such thatp − < p 0 < p < q 0 < p + and w ∈ A pp 0∩ RH (q 0p) ′.We are going to apply Theorem 2.2 with T = g L , S = I, A r = I − (I − e −r2 L ) m ,m large enough, underlying measure dx and weight w. We first see that (2.2) holdsfor all f ∈ L ∞ c . Here, we could have used the approach in [Aus], but the one belowadapts to the other two cases with minor changes.As p 0 , q 0 ∈ J (L) and p 0 ≤ q 0 , we know that e −t L ∈ O ( L p 0− L 0) q . If B is a ball,j ≥ 1 and g ∈ L p 0with supp g ⊂ C j (B) we have( ∫− |e −k r2 L g| q 0dxB) 1q 0( ∫≤ C 0 2 j (θ 1+θ 2 ) e −α 4j −C j (B)) 1|g| p 0dxp 0. (7.1)Lemma 7.4 applied to S = I, T : L p 0= L p 0(R n , dx) −→ L q 0= L q 0(R n , dx) given byT g =(C 0 2 j (θ 1+θ 2 ) e −α 4j) −1 |2 j+1 B| 1p 0|B| 1q 0χ Be −k r2 L (χ Cj (B) g)

24 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLyields( ∫−B) 1|e −k r2 L g(x, ·) | q q 0dx0( ∫≤ C 0 2 j (θ 1+θ 2 ) e −α 4j −C j (B)for all g ∈ L p 0H with supp g(·, t) ⊂ C j(B) for each t > 0.As in (4.7), for h ∈ L p 0Hwriteh(x, t) = ∑ j≥1h j (x, t), x ∈ R n , t > 0,) 1|g(x, ·) | p 0dxp 0(7.2)where h j (x, t) = h(x, t) χ Cj (B)(x). Using (7.2), we have for 1 ≤ k ≤ m,( ∫) 1− |e −k r2 L h(x, ·) | q q 0dx0≤ ∑ ( ∫) 1− |e −k r2 L h j (x, ·) | q q 0dx0Bj B( ∫) 1 2 j (θ 1+θ 2 ) e −α 4j − |h(x, ·) | p p 0dx0. (7.3)∑j≥12 j+1 BTake h(x, t) = (t L) 1/2 e −t L f(x). Since g L f(x) = |h(x, ·) | and f ∈ L ∞ c , h ∈ L p 0HTheorem 7.1 and(∫ ∞g L (e −k r2 L f)(x) = |(t L) 1/2 e −t L e −k r2 L f(x)| 2 dt ) 12= |e −k r2 L h(x, ·) |.tThus (7.3) implies( ∫− |g L (e −k r2 L f)| q 0dxB0) 1q 0 ∑ j≥1( ∫) 12 j (θ 1+θ 2 ) e −α 4j − |g L f| p p 0dx02 j+1 Bbyand it follows that g L satisfies (2.2).It remains to show that (2.1) with Sf = f holds for all f ∈ L ∞ c . Write f = ∑ j≥1 f jas before. If j = 1 we use that both g L and (I − e −r2 L ) m are bounded on L p 0(seeTheorem 7.1 and Proposition 3.3):For j ≥ 2, we observe that( ∫ ∞g L (I − e −r2 L ) m f j (x) =( ∫− |g L (I − e −r2 L ) m f 1 | p 0dxB0) 1p 0( ∫ −4 B|(t L) 1/2 e −t L (I − e −r2 L ) m f j (x)| 2 dt ) 12t) 1|f| p 0dxp 0. (7.4)= |ϕ(L, ·)f j (x) |where ϕ(z, t) = (t z) 1/2 e −t z (1 − e −r2 z ) m . As in [Aus], the functions η ± (·, t) associatedwith ϕ(·, t) by (4.3) verifyThus,|η ± (z, t)| t 1/2(|z| + t) 3/2( ∫ ∞|η ± (z, ·) | ≤0t(|z| + t) 3r 2 m(|z| + t) m , z ∈ Γ ±, t > 0.r 4 m dt) 12(|z| + t) 2 m tr2 m. (7.5)|z|m+1

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 25Next, applying Minkowski’s inequality and e −z L ∈ O ( L p 0− L 0) p , since p0 ∈ J (L), wehave( ∫∫−∣ e −z L f j (x) η + (z, ·)dz∣ p ) 10 pdx0B Γ +( ∫ ( ∫≤ − |e −z L p0) 1pf j (x)| ||η + (z, ·) | |dz|)dx0Γ +≤B( ∫Γ + 2 j θ 1∫−B∫ ∞( ∫ 2 j (θ 1−2 m)−0) 1|e −z L f j | p p 0dx0( ) 2 j θ2rΥ √ e − α 4j r 2ssC j (B)) 1|f| p p 0dx0r 2 m|dz||z|m+1r 2 ms mdss( ∫−C j (B)) 1|f| p p 0dx0provided 2 m > θ 2 . This plus the corresponding term for Γ − yield( ∫) 1( ∫− |g L (I − e −r2 L ) m f j | p p 0dx0 2 j (θ 1−2 m)− |f| p 0dxBC j (B)) 1p 0. (7.6)Collecting the latter estimate and (7.4), we obtain that (2.1) holds whenever 2 m >max{θ 1 , θ 2 }.Case p ∈ (˜p − , ̂p + ): Take p 0 , q 0 such that ˜p − < p 0 < ˜p + and p 0 < p < q 0 < ̂p + . LetA r = I − (I − e −r2 L ) m for some m ≥ 1 to be chosen later. Remark that by theprevious case, g L is bounded in L p 0(w) and so does A r by Proposition 3.4. We applyTheorem 2.2 to T = g L and S = I with underlying measure dw and no weight: it isenough to see that g L satisfies (2.1) and (2.2) on L ∞ c . But this follows by adaptingthe preceding argument replacing everywhere dx by dw and observing that e −z L ∈O ( L p 0(w) − L q 0(w) ) . We skip details.Case p ∈ (̂p − , ˜p + ): Take p 0 , q 0 such that ˜p − < q 0 < ˜p + and ̂p − < p 0 < p < q 0 . SetA r = I − (I − e −r2 L ) m for some integer m ≥ 1 to be chosen later. Since q 0 ∈ (˜p − , ˜p + ),by the first case, g L is bounded on L q 0(w) and so does A r by Proposition 3.4. ByTheorem 2.4 with underlying Borel doubling measure dw, it is enough to show (2.4)and (2.5). Fix a ball B, f ∈ L ∞ c supported on B.Observe that (2.5) follows directly from (4.11) since p 0 , q 0 ∈ J w (L) and p 0 ≤ q 0 .We turn to (2.4). Assume j ≥ 2. The argument is the same as the one for (7.6) byreversing the roles of C j (B) and B, and using dw and e −z L ∈ O ( L p 0(w) − L p 0(w) )(since p 0 ∈ J w (L)) instead of dx and e −z L ∈ O ( L p 0− L 0) p . We obtain( ∫−C j (B)|g L (I − e −r2 L ) m f| p 0dw( ∫p 0 2 j (θ 1−2 m)− |f| p 0dw) 1provided 2 m > θ 2 and it remains to impose further 2 m > θ 1 + D to conclude, whereD is the doubling order of w.□Proof of Theorem 7.2. Part (b). We split the argument in three cases: p ∈ (˜q − , ˜q + ),p ∈ (˜q − , ̂q + ), p ∈ (̂q − , ˜q + ).B) 1p 0

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 27 2 j θ 1∫ ∞( ∫ 2 j (θ 1−2 m)−0( ) 2 j θ2rΥ √ e − α 4j r 2ssC j (B)) 1|f| p p 0dx0r 2 ms mdss( ∫−C j (B)) 1|f| p p 0dx0provided 2 m > θ 2 . This, plus the corresponding term for Γ − , yields( ∫− |G L (I − e −r2 L ) m f j | p 0dx) 1p 0 2 j (θ 1−2 m)( ∫− |f| p 0dxBC j (B)Collecting the latter estimate and (7.7), we obtain by Minkowski’s inequality( ∫) 1− |G L (I − e −r2 L ) m f| p p 0dx0∑ ( ∫) 1 2 j (θ 1−2 m)− |f| p p 0dx0.Bj≥1Therefore, (2.1) holds on taking 2 m > sup(θ 1 , θ 2 ).C j (B)) 1p 0. (7.9)Case p ∈ (˜q − , ̂q + ): Take p 0 , q 0 such that ˜q − < p 0 < ˜q + and p 0 < p < q 0 < ̂q + . LetA r = I − (I − e −r2 L ) m for some m ≥ 1 to be chosen later. As p 0 ∈ (˜q − , ˜q + ), bothG L and A r are bounded on L p 0(w) (we have just shown it for G L and Proposition 3.4yields it for A r with a uniform norm in r). By Theorem 2.2 with underlying doublingmeasure dw and no weight, it is enough to verify (2.1) and (2.2) on D = L ∞ c forT = G L , S = I. It suffices to copy the preceding argument replacing everywhere dxby dw, observing that p 0 , q 0 ∈ K w (L) implies weighted off-diagonal estimates and anL p 0(w) Poincaré inequality, and applying Lemma 7.4 to obtain an H-valued extension.We leave the details to the reader.Case p ∈ (̂q − , ˜q + ): Take p 0 , q 0 such that ˜q − < q 0 < ˜q + and ̂q − < p 0 < p < q 0 . SetA r = I − (I − e −r2 L ) m for some m ≥ 1 to be chosen later. Since q 0 ∈ (˜q − , ˜q + ), itfollows that G L is bounded on L q 0(w) and so is A r by Proposition 3.4. By Theorem2.4 with underlying measure dw, it is enough to show (2.4) and (2.5).Observe that (2.5) is nothing but (4.11) since p 0 , q 0 ∈ K w (L) ⊂ J w (L). The proofof (2.4) is again analogous to (7.9) in the weighted setting exchanging the roles ofC j (B) and B. We skip details.□To prove Theorem 7.3, part (a), we introduce the following operator. Define forf ∈ L 2 H and x ∈ Rn ,T L f(x) =∫ ∞0(t L) 1/2 e −t L f(x, t) dtt .Recall that (t L) 1/2 e −t L f(x, t) = (t L) 1/2 e −t L (f(·, t))(x). Hence, T L maps H-valuedfunctions to C-valued functions. We note that, for f ∈ L 2 H and h ∈ L2 , we have∫∫ ∞T L f h dx = f(x, t) (t LR∫R ∗ ) 1/2 e −t L∗ h(x) dtt dx,n nwhere L ∗ is the adjoint (on L 2 ) of L, hence,∫∫∣ T L f h dx∣ ≤ |f(x, ·) | g L ∗(h)(x) dx.R n R n0

28 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLLet p − (L) < p < p + (L). Since p − (L ∗ ) = ( p + (L) ) ′< p ′ < ( p − (L) ) ′= p + (L ∗ ), g L ∗ isbounded on L p′ . This and a density argument imply that T L has a bounded extensionfrom L p H to Lp . The weighted version is as follows.Theorem 7.5. Let w ∈ A ∞ . If W w(p− (L), p + (L) ) ≠ Ø and p ∈ Int J w (L) then forall f ∈ L ∞ c (R n × (0, ∞)) we have‖T L f‖ L p (w) ‖f‖ LpH (w) .Hence, T L has a bounded extension from L p H (w) to Lp (w).The duality argument above works for exponents in W w(p− (L), p + (L) ) , but we donot know how to extend it to all of Int J w (L). Hence, we proceed via a direct proofwhere duality is used only when w = 1.Proof. We split the argument in three cases: p ∈ (˜p − , ˜p + ), p ∈ (˜p − , ̂p + ), p ∈ (̂p − , ˜p + ).Case p ∈ (˜p − , ˜p + ): By Proposition 2.1, there exist p 0 , q 0 such thatp − < p 0 < p < q 0 < p + and w ∈ A pp 0∩ RH (q 0p) ′.We are going to apply Theorem 2.2 (in fact, its vector-valued extension) with underlyingmeasure dx and weight w to the linear operator T = T L with S = I andA r = I − (I − e −r2 L ) m , m large enough. Here, A r denotes both the scalar operatorand its H-valued extension. We first see that T L satisfies (2.2) with p 0 , q 0 forf ∈ L ∞ c (R n × (0, ∞)). Let B be a ball. Note that T L A r f = A r T L f with our confusionof notation. Hence (2.2) is a simple consequence of (7.1) applied to g = T L f.Next, it remains to check (2.1). Let f ∈ L ∞ c (R n × (0, ∞)) and let B be a ball. Asin (4.7), we writef(x, t) = ∑ j≥1f j (x, t),where f j (x, t) = f(x, t) χ Cj (B) (x). For T L(I − A r )f 1 , we use the boundedness of T Lfrom L p 0H to Lp 0noted above and the L p 0Hboundedness of A r to obtain( ∫) 1 ( ∫) 1− |T L (I − A r )f 1 | p p 0dx0 − |f(x, ·) | p p 0dx0.BFor j ≥ 2, the functions η ± (z, t) associated with ϕ(z, t) = (t z) 1/2 e −t z (1 − e −r2 z ) m by(4.3) satisfy (7.5). Hence,( ∫∫ ∞ ∫−∣ e −z L f j (x, t)η + (z, t) dz dt0 ) 1p∣ dx0∣pB 0 Γ +t( ∫ ( ∫ − |e −z L p0) 1pf j (x, ·) | |η + (z, ·) | |dz|)dx0Γ +B( ∫Γ + 2 j θ 1∫−B∫ ∞04 B) 1|e −z L f j (x, ·) | p p 0dx0|η + (z, ·) | |dz|( ) 2 j θ2rΥ √ e − c 4j r 2ssr 2ms mdss( ∫−C j (B)) 1|f(x, ·) | p p 0dx0

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 29( ∫ 2 j (θ 1−2 m)−C j (B)) 1|f(x, ·) | p p 0dx0where we used the H-valued extension of e −z L ∈ O ( L p 0− L 0) p and assumed 2 m > θ2 .This, plus the corresponding term for Γ − , yields( ∫) 1( ∫) 1− |T L (I − A r )f j | p p 0dx0 2 j (θ 1−2 m)− |f(x, ·) | p p 0dx0(7.10)Band therefore (2.1) follows on taking also 2 m > θ 1 .C j (B)Case p ∈ (˜p − , ̂p + ): Take p 0 , q 0 with ˜p − < p 0 < ˜p + and p 0 < p < q 0 < ̂p + . It suffices toapply the H-valued extension of Theorem 2.2 with underlying doubling measure dwand no weight to the linear operator T = T L , S = I and A r = I − (I − e −r2 L ) m , mlarge enough. This is done exactly as in the previous case. At some step we have touse that T L is bounded from L p 0H (w) to Lp 0(w) which follows by the previous case. Weleave the details to the reader.Case p ∈ (̂p − , ˜p + ): Take p 0 , q 0 with ˜p − < q 0 < ˜p + and ̂p − < p 0 < p < q 0 . Sinceq 0 ∈ (˜p − , ˜p + ), by the first case, T L is bounded from L q 0H (w) to Lq 0(w) and so does A r =I − (I − e −r2 L ) m , m ≥ 1, on L q 0H(w) by Proposition 3.4 and Lemma 7.4. By Theorem2.4 (in fact, its H-valued extension) with underlying Borel doubling measure dw, it isenough to show (2.4) and (2.5) on D = L ∞ c (R n ×(0, ∞)) for T = T L and A r with largeenough m. As usual, the latter is a mere consequence of e −t L ∈ O ( L q 0(w) − L q 0(w) )and its H-valued analog. The first condition is again a repetition of the argument for(7.10) in the weighted setting switching C j (B) and B. We skip details. □Proof of ( Theorem 7.3. We begin with part (a). Fix p ∈ Int J w (L) where w ∈ A ∞ sothat W w p− (L), p + (L) ) ≠ Ø. Let f ∈ L 2 and define F by F (x, t) = (t L) 1/2 e −t L f(x).Note that F ∈ L 2 H since ‖F ‖ L 2 = ‖g Lf‖H 2 . By functional calculus on L 2 , we havef = 2∫ ∞0(t L) 1/2 e −t L F (·, t) dtt = 2 T LF (7.11)with convergence in L 2 . Note that for p ∈ Int J w (L), e −t L has an infinitesimal generatoron L p (w) as recalled in Remark 3.5. Let us call L p,w this generator. In particulare −t L −t Lp,wand e agree on L p (w) ∩ L 2 . Our results assert that L p,w has a boundedholomorphic functional calculus on L p (w), hence replacing L by L p,w and f ∈ L 2 byf ∈ L p (w), we see that F ∈ L p H (w) with ‖F ‖ L p (w) = ‖g Lp,w f‖ L p (w) and (7.11) is validwith convergence in L p (w) (this is standard fact from functional calculus and we skipdetails). Thus, by Theorem 7.5,‖f‖ L p (w) = 2‖T Lp,w F ‖ L p (w) ‖F ‖ LpH (w) = ‖g Lp,w f‖ L p (w).Noting that g L f = g Lp,w f when f ∈ L 2 ∩ L p (w) and T L F = T Lp,w F when F ∈L 2 H ∩ Lp H(w), part (a) is proved.Let us show part (b), that is the corresponding inequality for G L . Fix w ∈ A ∞ . Weuse the following estimate from [Aus]: for f, h ∈ L 2∫∫∣ f h dx∣ ≤ (1 + ‖A‖ ∞) G L f G −∆ h dx,R n R n

30 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLwhere G −∆ is the square function associated with the operator −∆. It is well knownthat G −∆ is bounded on L q (u) for all 1 < q < ∞ and all u ∈ A q . Let us emphasizethat, indeed, the results that we have proved can be applied ( to the operator −∆ and soG −∆ is bounded on L q (u) for u ∈ A ∞ and all q ∈ W u q− (−∆), q + (−∆) ) = W u (1, ∞),that is, for all 1 < q < ∞ and u ∈ A q .Coming back to the argument, let p > r w , hence w ∈ A p . Let f ∈ L 2 ∩L p (w). Then∫∫|f| p dw = lim f h dw NR n N,k,R→∞R nwith w N = min{w, N} and h = f|f| p−2 χ B(0,R)χ {0 0. Hence, the equivalence impliesthe uniform L p (w) boundedness of e −t L , which implies p ∈ ˜J w (L) (see Proposition3.4). Actually, Int J w (L) is also the sharp range up to endpoints for the inequality‖g L f‖ L p (w) ‖f‖ L p (w). It suffices to adapt the interpolation procedure in [Aus,Theorem 7.1, Step 7]. We skip details.Similarly, this interpolation procedure also shows that Int K w (L) is also sharp upto endpoints for ‖G L f‖ L p (w) ‖f‖ L p (w).8. Some vector-valued estimatesIn [AM1], we also obtained vector-valued inequalities.Proposition 8.1. Let µ, p 0 , q 0 , T, A r , D be as in Theorem 2.2 and assume (2.1) and(2.2) with S = I. Let p 0 < p, r < q 0 . Then, there is a constant C such that for allf k ∈ D( ∥∑ ) 1|T f k | r r( ∥ ≤ C ∥∑ ) 1|f k | r r∥ (8.1)L p (µ) L (µ). pkk□

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 31Let us see how it applies here.First, let T = ϕ(L) (ϕ bounded holomorphic in an appropriate sector). Theorem4.2 says that T is bounded on L p (w) for all p ∈ Int J w (L). Also, for p 0 , q 0 ∈ Int J w (L)with p 0 < q 0 , we have L p 0(w) − L q 0(w) off-diagonal estimates on balls for A r =I − (I − e −r2L ) m . Hence, we can prove (2.1) and (2.2) with S = I where dx is nowreplaced by w(x)dx by mimicking the first case of the proof of Theorem 4.2 in theweighted context. Hence, one can apply the proposition above with dµ = w dx toabove weighted vector-valued estimates for ϕ(L) with all p, r ∈ Int J w (L).The same weighted vector-valued estimates hold with all p, r ∈ Int J w (L) withT = g L starting from Theorem 7.2 and mimicking the proof of its first case with dxreplaced with w(x)dx.If T = ∇L −1/2 or T = G L , then the same reasoning applies modulo the Poincaréinequality used towards obtaining (2.2). Hence, we conclude that for both ∇L −1/2and G L , one has (8.1) with dµ = wdx and p, r ∈ Int K w (L) ∩ (r w , ∞).Other vector-valued inequalities of interest are( ∑ ) 1∥ |e −ζkL f k | 2 2( ∥ ≤ C∑∥L q (w)1≤k≤N1≤k≤N) 1|f k | 2 2 ∥∥L q (w)(8.2)for ζ k ∈ Σ α with 0 < α < π/2 − ϑ and f k ∈ L p (w) with a constant C independent ofN, the choice of the ζ k ’s and the f k ’s. We restrict to 1 < q < ∞ and w ∈ A ∞ (wekeep working on R n ). By a theorem of L. Weis [Wei, Theorem 4.2], we know thatthe existence of such a constant is equivalent to the maximal L p -regularity of L onL q (w) with one/all 1 < p < ∞, that is the existence of a constant C ′ such that for allf ∈ L p ((0, ∞), L q (w)) there is a solution u of the parabolic problem on R n × (0, ∞),withu ′ (t) + Lu(t) = f(t), t > 0, u(0) = 0,‖u ′ ‖ L p ((0,∞),L q (w)) + ‖Lu‖ L p ((0,∞),L q (w)) ≤ C ′ ‖f‖ L p ((0,∞),L q (w)).Proposition 8.2. Let w ∈ A ∞ be such that W w(p− (L), p + (L) ) ≠ Ø. Then for anyq ∈ Int J w (L), (8.2) 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) that weestablished in Theorem 4.2. However, we wish to give a different proof using extrapolationand preceding ideas. Note that q = 2 may not be contained in Int J w (L) andthe interpolation method of [BK2] may not work here.Proof. There are three steps.First step: Extrapolation. Letting N, ζ k ’s and f k ’s vary at will, we denote F thefamily of all ordered pairs (F, G) of the form( ∑ ) 1( ∑ ) 1F = |e −ζkL f k | 2 2and G = |f k | 2 2.1≤k≤NThen we have for all (F, G) ∈ F,1≤k≤N‖F ‖ L 2 (u) ≤ C u ‖G‖ L 2 (u), for all u ∈ A 2/p− ∩ RH (p+ /2)′. (8.3)

32 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLRecall that 2 ∈ (p − , p + ) = Int J (L) and u ∈ A 2/p− ∩RH (p+ /2) ′ means 2 ∈ W )u(p− , p + .In particular, {e −ζL : ζ ∈ Σ α } is bounded in L(L 2 (u)). This inequality is triviallychecked with C u equal to the upper bound of this family. Applying our extrapolationresult [AM1, Theorem 4.7], we deduce that, for all p − < q < p + and (F, G) ∈ F wehave‖F ‖ L q (u) ≤ C q,u ‖G‖ L q (u), for all u ∈ A q/p− ∩ RH (p+ /q)′. (8.4)( ) ( )In other words, for all u ∈ A ∞ with W u p− , p + ≠ Ø, (8.2) holds for q ∈ Wu p− , p +with C depending on q and w. This applies to our fixed weight w of the statementwith q ∈ W w (p − , p + ). It remains to push the range of q’s to all of Int J w (L).Step 2: Pushing to the right. Take p 0 ∈ W w(p− , p +), q0 ∈ Int J w (L) with p 0 < q < q 0 .Fix N and the ζ k ’s. To prove (8.2) for that q, it suffices to apply the l 2 -valued versionof Theorem 2.2 with underlying measure dw and no weight to T given byTf = (e −ζ 1 L f 1 , . . . , e −ζ N L f N ), f = (f 1 , . . . , f N ),with S = I. To check (2.1) and (2.2) we use A r = I − (I − e −r2 L ) m with m largeenough (here, the l 2 -valued extension). Pick a ball B and f ∈ (L ∞ c ) l 2. Using thate −t L ∈ O ( L p 0(w) − L q 0(w) ) we can obtain (7.2), replacing H by l 2 and with dwin place of dx. This and the fact that TA r = A r T yield (2.2). We are left withchecking (2.1). As usual we split f as ∑ j≥1χ Cj (B)f (componentwise). The termwith χ C1 (B) f = χ 4 B f is treated using the Lp 0l(w) boundedness of T (first step) and2of I − A r (l 2 -valued extension of Proposition 3.4). The terms χ Cj (B)f, j ≥ 2, aretreated using off-diagonal estimates injecting the Khintchine inequality in the process:Let F (x) = ‖T(I − A r )(χ Cj (B) f)(x)‖ l 2. Let r 1, . . . , r N be the N first Rademacherfunctions on [0, 1]. Then, by Khintchine’s inequalities (see [Ste] for instance)( ∫ 1∑F (x) = ∣ r k (t)ϕ ζk (L)(χ Cj (B) f k)(x) ∣ 2 ) 12dt0( ∫ 1∼ ∣01≤k≤N∑1≤k≤Nr k (t)ϕ ζk (L)(χ Cj (B) f k)(x) ∣ p ) 10 pdt0where z ↦−→ ϕ ζ (z) = e −ζ z (1 − e −r2 z ) m for ζ ∈ Σ α is bounded on Σ µ when ϑ < µ

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 33plus the similar term on Γ − . Using e −z L ∈ O ( L p 0(w) − L p 0(w) ) for z ∈ Γ + , the righthand side in the above inequality is bounded by∫ ( )2 j θ 2 j θ2r1√ e − α 4j r 2 ( ∫ 1∫) 1|z|− |h(x, t, z)| p p 0dw(x) dt0|dz|.|z|Γ +Υ0C j (B)Using again Khintchine’s inequality, this is comparable to∫ ( )2 j θ 2 j θ2r1√ e − α 4j r 2 ( ∫ ( ∑) p 0|z|−|η +,ζk (z) f k (x)| 2 2) 1pdw(x)0|dz|.|z|Γ +ΥC j (B)1≤k≤NAt this point, we use the upper bound on η ζk and integrate in z if 2 m > θ 2 to obtainthat the latter is controlled by( ∫ ( ∑ )2 j(θ 1−2 m)−|f k (x)| 2 p0 /2 ) 1/p0.dw(x)C j (B)1≤k≤NThe condition (2.1) follows readily if 2 m > θ 1 as well.Step 3: Pushing to the left. This time, it suffices to use the l 2 -valued version ofTheorem 2.4 with underlying measure dw and exponents p 0 , q 0 such that ˜p − < q 0 < ˜p +and ̂p − < p 0 < q < q 0 . Then (2.5) follows from the l 2 -valued extension of e −t L ∈O ( L p 0(w) − L q 0(w) ) , and (2.4) is obtained with a similar argument for the one justabove to prove (2.1), by switching the role of B and C j (B) (j ≥ 2), and using e −z L ∈O ( L p 0(w) − L p 0(w) ) .□Remark 8.3. When w = 1, (8.2) holds for p ∈ Int J (L) and recall that this intervalcontains 2. Our proof contains two ways of seeing this. First, apply the extrapolationstep and specialize to u = 1. Second, apply steps 2 and 3 with w = 1 and transitionexponent 2 pushing to its right or to its left. Note that one could even reduce thingsto one of those two steps by using duality as, if we denote T by T L then T ∗ = T L ∗.In [BK2], Step 3 and duality is used. However, duality does not seem to work whenw ≠ 1 on all of Int J w (L).9. Commutators with bounded mean oscillation functionsLet µ be a doubling measure in R n . Let b ∈ BMO(µ) (BMO is for bounded meanoscillation), that is,∫∫1‖b‖ BMO(µ) = sup − |b − b B |dµ = sup |b(y) − b B | dµ < ∞B BB µ(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 Tb0 = T . If T is linear they can be alternatively defined by recurrence: thefirst order commutator isT 1 b f(x) = [b, T ]f(x) = b(x) T f(x) − T (b f)(x)

34 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLand for k ≥ 2, the k-th order commutator is given by Tbk k−1= [b, Tb]. As it is observedin [AM1], Tb kf(x) is well-defined almost everywhere when f ∈ L∞ c (µ) and it suffices toobtain boundedness with b ∈ L ∞ with norm depending only on ‖b‖ BMO(µ) . We statethe results for commutators obtained in [AM1].Theorem 9.1. Let µ be a doubling measure on R n , 1 ≤ p 0 < q 0 ≤ ∞ and k ∈ N.Suppose that T is a sublinear operator bounded on L p 0(µ), and let {A r } r>0 be a familyof operators acting from L ∞ c (µ) into L p 0(µ). Assume that (2.1) and (2.2) hold withS = I. Let p 0 < p < q 0 and w ∈ A p ∩ RHp 0 (q 0p) ′. If ∑ j g(j) jk < ∞ then there is aconstant C independent of f and b ∈ BMO(µ) such thatfor all f ∈ L ∞ c (µ).‖T k b f‖ L p (w) ≤ C ‖b‖ k BMO(µ) ‖f‖ L p (w), (9.1)Theorem 9.2. Let k ∈ N, µ be a doubling Borel measure on R n with doubling orderD and 1 ≤ p 0 < q 0 ≤ ∞. Suppose that T is a sublinear operator and that T and T m bfor m = 1, . . . , k are bounded on L q 0(µ). Let {A r } r>0 be a family of operators actingfrom L ∞ c (µ) into L q 0(µ). Assume that (2.4) and (2.5) hold. If ∑ j g(j) 2D j j k < ∞,then for all p 0 < p < q 0 , there exists a constant C (independent of b) such that for allf ∈ L ∞ c (µ) and b ∈ BMO(µ),‖T k b f‖ L p (µ) ≤ C ‖b‖ k BMO(µ) ‖f‖ L p (µ).With these results in hand, we have the following theorem.Theorem 9.3. Let w ∈ A ∞ , k ∈ N and b ∈ BMO. Assume one of the followingconditions:(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 f ∈ L ∞ c (R n ),‖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 [DY].Proof of Theorem 9.3. Part (a). We fix p ∈ Int J w (L) and take p 0 , q 0 ∈ Int J w (L) sothat p 0 < p < q 0 . We are going to apply Theorem 9.1 with dµ = dw and no weight toT = ϕ(L) where ϕ satisfies (4.1).First, as p 0 ∈ Int J w (L), Theorem 4.2 yields that ϕ(L) is bounded on L p 0(w). Then,choosing A r = I − (I − e −r2 L ) m with m ≥ 1 large enough, we proceed exactly as inthe second case of the proof of Theorem 4.2. That is, we repeat the computations ofthe first case with dw replacing dx and using the corresponding weighted off-diagonal

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 35estimates on balls. Applying (4.8) with dw in place of dx to h = ϕ(L) we conclude(2.2). Besides, (4.9) and (4.10) with dw replacing dx lead us to (2.1) (with S =I). Therefore, Theorem 9.1 shows the boundedness of the commutators with BMOfunctions since ‖b‖ BMO (w) ≈ ‖b‖ BMO as noticed earlier.It remains to remove the assumption on ϕ. This is done easily if one assumes thatb ∈ L ∞ . Then the general case with b ∈ BMO follows as mentioned above. □Remark 9.4. The argument is the same as in the second case in Theorem 4.2 butfor the whole range Int J w (L) (in place of working with p ∈ (˜p − , ̂p + )) since we alreadyproved that ϕ(L) is bounded in Int J w (L) by Theorem 4.2. That is, T = ϕ(L) aposteriori satisfies (2.1) and (2.2) for dµ = dw and for all p 0 , q 0 ∈ Int J w (L) withp 0 < q 0 .Proof of Theorem 9.3. Part (b). We write T = ∇L −1/2 and we already know that Tis bounded on L p (w) for p ∈ Int K w (L) by Theorem 5.2.First consider the case p ∈ (˜q − , ̂q + ). We take p 0 , q 0 so that ˜q − < p 0 < ˜q + andp 0 < p < q 0 < ̂q + . Let A r = I − (I − e −r2 L ) m where m ≥ 1 is an integer to be chosen.As mentioned in the second case of the proof of Theorem 5.2, Lemma 5.3 holds withdw replacing dx. Thus, the hypotheses of Theorem 9.1 are fulfilled with dµ = dw andwe can apply it with no weight.Next we consider the case p ∈ (̂q − , ˜q + ). We take p 0 , q 0 so that ˜q − < q 0 < ˜q + and̂q − < p 0 < p < q 0 . Set A r = I − (I − e −r2 L ) m where m ≥ 1 is an integer to be chosen.Notice that we have just proved that the operators Tb l for l = 0, . . . , k are boundedon L q 0(w) as q 0 ∈ (˜q − , ̂q + ). We have already seen in the third case of the proof ofTheorem 5.2that T satisfies (2.4) and (2.5) with dµ = dw. Choosing m large enough yields theneeded condition for g(j) to apply Theorem 9.2 with dµ = dw.□Remark 9.5. In contrast with part (a), we do not know if Lemma 5.3 holds in thewhole range Int K w (L) with dw replacing dx. Indeed, its proof relies on an L p 0(w)-Poincaré inequality which is known only if p 0 > r w . We get around this obstacle withTheorem 9.2 .Proof of Theorem 9.3. Part (c). We proceed exactly as in part (a) using the argumentsin Theorem 7.2, Part (a), in place of those in Theorem 4.2. Details are left tothe reader.□Proof of Theorem 9.3. Part (d). We follow the same scheme as in part (b) using thearguments in Theorem 7.2, Part (b), in place of those in Theorem 5.2. Details are leftto the reader.□Similar results can be proved for the multilinear commutators considered in [PT](see also [AM1]) which are defined by replacing (b(x) − b) k in Tbk by ∏ kj=1 (b j(x) − b j )with b j ∈ BMO for 1 ≤ j ≤ k. Details are left to the reader.10. Real operators and power weightsLet us illustrate our results on Riesz transforms in a specific case and in particulardiscuss sharpness issues. Assume in this section that L has real coefficients. Then oneknows that q − (L) = p − (L) = 1, p + (L) = ∞.

36 PASCAL AUSCHER AND JOSÉ MARÍA MARTELLIf n = 1, one has also q + (L) = ∞, so that we have obtained for all 1 < p < ∞ andw ∈ A p ,‖L 1/2 f‖ L p (w) ∼ ‖f ′ ‖ L p (w).For p = 1, there are two weak-type (1,1) estimates for A 1 weights. In fact, all this canbe seen from [AT2] where it is shown that L 1/2 = R d and d = M ˜RL 1/2 with R anddx dx˜R being classical Calderón-Zygmund operators and M being the operator of pointwisemultiplication by 1/a(x). Thus the usual weighted norm theory for Calderón-Zygmundoperators applies.Let us assume next that n ≥ 2. In this case q + (L) > 2. The next result will helpus to study sharpness.Proposition 10.1. For each q > 2, there exists a real symmetric operator L on R 2for which q + (L) = q.Proof. This is the example of Meyers-Kenig [AT1, p. 120]. Let q > 2 and set β =−2/q ∈ (−1, 0). Consider the operator L = − div A∇ obtained from −∆ by pullingback the associated quadratic form ∫ ∇u · ∇v by the quasi-conformal applicationϕ(x) = |x| β x, x ∈ R 2 . That is, A is obtained by writing out the change of variable inthe relation ∫ ∫A(x)∇u(x) · ∇v(x) dx = ∇(u ◦ ϕ −1 )(y) · ∇(v ◦ ϕ −1 )(y) dy,with u, v ∈ C0 ∞ (R n ). It is easy to see that A is bounded and uniformly elliptic. Hence,u is a weak solution (in W 1,2loc ) of L if and only if u ◦ ϕ−1 is a weak solution of −∆.In other words, weak solutions of L are harmonic functions composed with ϕ. Thus,the local L p integrability of the gradient of such a solution is exactly that of ∇ϕ. Thelatter is in L p near 0 if and only if p < −2/β and is bounded locally away from 0.Thus, for any weak solution u of L defined on a ball 2B, ∇u ∈ L p (B) for p < −2/βand this is optimal if B is the unit ball. With this in hand, we can apply a result byShen [She] which asserts that q + (L) is the supremum of those p for which all weaksolutions of L defined on an arbitrary ball have ∇u in L p locally inside that ball. Inour case, q + (L) = −2/β = q.□Remark 10.2. Let us also stress that if η is a smooth compactly supported functionwhich is equal to 1 in a neighborhood of 0, then v = ϕη satisfies |∇v(x)| ∼ |x| β near0, whereas |L 1/2 v(x)| ≤ c(1 + |x|) −1 . See [AT1, p. 120], for this last fact.Let us come back to a( general situation and consider the power weights w α (x) = |x| α .Then, one has p ∈ W wα 1, q+ (L) ) if and only if( ) p1 < p < q + (L) and nq + (L) − 1 < α < n(p − 1). (10.1)For (p, α) tight with these relations Theorem 5.2 yields‖∇f‖ L p (|x| α ) ‖L 1/2 f‖ L p (|x| α ).In the latter inequality, we have in fact three parameters: p ∈ (1, ∞), α ∈ (−n, ∞)(for w α ∈ A ∞ ) and L in the family of real elliptic operators. One can study sharpnessin various ways.Fix L as in Proposition 10.1 with n = 2. The remark following this result impliesthat the L p inequality can not hold for any (p, α) with −2 < α ≤ 2 ( p− 1) q +since(L)

**WEIGHTED** **NORM** **INEQUALITIES** AND ELLIPTIC OPERATORS 37in this case, one can produce an f(= v) where the left hand side is infinite and theright hand side finite.If we fix α = 0 and L, then the condition 1 < p < q + (L) is necessary (and sufficient)to obtain the L p estimate [Aus].If we fix p ∈ (1, ∞) and let L and α vary, then one can take L = −∆, in which casewe are looking at the L p power weight inequality for the usual Riesz transforms. Inthis case, it is known that this forces w α ∈ A p , hence α < n(p − 1).Let us consider the reverse inequalities. For a given weight w, Theorem 6.2 saysthat the range of exponents for the L p inequality contains W w (1, ∞), which is the setof p > 1 for which w ∈ A p . Hence, for w α we have‖L 1/2 f‖ L p (|x| α ) ‖∇f‖ L p (|x| α ) if − n < α < n(p − 1).This is the usual range for Calderón-Zygmund operators. This can also be seen fromthe fact proved in [AT1] that L 1/2 = T ∇ where T is a Calderón-Zygmund operator.Again for fixed p ∈ (1, ∞), this range of α is best possible by taking L = −∆.[Aus][ACDH]ReferencesP. Auscher, On necessary and sufficient conditions for L p estimates of Riesz transformassociated elliptic operators on R n and related estimates, Mem. Amer. Math. Soc., vol.186, no. 871, March 2007.P. Auscher, T. Coulhon, X.T. Duong & S. Hofmann, Riesz transforms on manifolds andheat kernel regularity, Ann. Scient. ENS Paris 37 (2004), no. 6, 911–957.[AHLMcT] P. Auscher, S. Hofmann, M. Lacey, A. M c Intosh & Ph. Tchamitchian, The solution ofthe Kato square root problem for second order elliptic operators on R n , Ann. Math. (2)156 (2002), 633–654.[AM1][AM2][AT1][AT2][BK1][BK2][DY][FPW][GR][Gra]P. Auscher & J.M. Martell, Weighted norm inequalities, off-diagonal estimates and ellipticoperators. Part I: General operator theory and weights, Preprint 2006. Available athttp://www.uam.es/chema.martellP. Auscher & J.M. Martell, Weighted norm inequalities, off-diagonal estimates and ellipticoperators. Part II: Off-diagonal estimates on spaces of homogeneous type, Preprint2006. Available at http://www.uam.es/chema.martellP. Auscher & Ph. Tchamitchian, Square root problem for divergence operators and relatedtopics, Astérisque Vol. 249, Soc. Math. France, 1998.P. Auscher & Ph. Tchamitchian, Calcul fonctionnel précisé pour des opérateurs elliptiquescomplexes en dimension un (et applications à certaines équations elliptiques complexesen dimension deux), Ann. Inst. Fourier 45 (1995), 721–778.S. Blunck & P. Kunstmann, Calderón-Zygmund theory for non-integral operators and theH ∞ -functional calculus, Rev. Mat. Iberoamericana 19 (2003), no. 3, 919–942.S. Blunck & P. Kunstmann, Weighted norm estimates and maximal regularity, Adv.Differential Equations 7 (2002), no. 12, 1513–1532.X.T. Duong & L. Yan, Commutators of BMO functions and singular integral operatorswith non-smooth kernels, Bull. Austral. Math. Soc. 67 (2003), no. 2, 187–200.B. Franchi, C. Pérez & R. Wheeden, Self-improving properties of John-Nirenberg andPoincaré inequalities on spaces of homogeneous type, J. Funct. Anal. 153 (1998), no. 1,108–146.J. García-Cuerva & J.L. Rubio de Francia, Weighted Norm Inequalities and RelatedTopics, North Holland Math. Studies 116, North Holland, Amsterdam, 1985.L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, New Jersey,2004.

38 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL[KW] N. Kalton & L. Weis, the H ∞ -calculus and sums of closed operators, Math. Ann. 321(2001), 319–345.[LeM][Mar][McI][PT][She][Ste][ST][Wei]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.A. M c Intosh, Operators which have an H ∞ functional calculus, Miniconference on operatortheory and partial differential equations, Vol. 14, Center for Math. and Appl.,210–231, Canberra, 1986. Australian National Univ.C. Pérez & R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators,J. London Math. Soc. (2) 65 (2002), 672–692.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 of functions, Princeton Univ. Press,1970.J.O. Strömberg & A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics1381, Springer-Verlag, 1989.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.p-sud.frJosé María Martell, Instituto de Matemáticas y Física Fundamental, Consejo Superiorde Investigaciones Científicas, C/ Serrano 123, 28006 Madrid, SpainandDepartamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid,SpainE-mail address: chema.martell@uam.es