Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

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1 Weighed norm inequaliies, off-diagonal esimaes and ellipic operaors. Par III: Harmonic analysis of ellipic operaors Pascal Auscher, José Maria Marell To cie his version: Pascal Auscher, José Maria Marell. Weighed norm inequaliies, off-diagonal esimaes and ellipic operaors. Par III: Harmonic analysis of ellipic operaors. 38 pages. Third of 4 papers. 26. <hal-21878> HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 28 Mar 26 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 WEIGHTED NORM INEQUALITIES, OFF-DIAGONAL ESTIMATES AND ELLIPTIC OPERATORS PART III: HARMONIC ANALYSIS OF ELLIPTIC OPERATORS PASCAL AUSCHER AND JOSÉ MARÍA MARTELL ccsd-21878, version 1-28 Mar 26 Absrac. This is he hird par of a series of four aricles on weighed norm inequaliies, off-diagonal esimaes and ellipic operaors. For L in some class of ellipic operaors, we sudy weighed norm L p inequaliies for singular non-inegral operaors arising from L ; hose are he operaors ϕl) for bounded holomorphic funcions ϕ, he Riesz ransforms L 1/2 or /2 L 1/2 ) and is inverse L 1/2 /2, some quadraic funcionals g L and G L of Lilewood-Paley-Sein ype and also some vecor-valued inequaliies such as he ones involved for maximal L p -regulariy. For each, we obain sharp or nearly sharp ranges of p using he general heory for boundedness of Par I and he off-diagonal esimaes of Par II. We also obain commuaor resuls wih MO funcions. Conens 1. Inroducion 1 2. General crieria for boundedness and he se W w, q ) 3 3. Off-diagonal esimaes 5 4. Funcional Calculi 8 5. Riesz ransforms Reverse inequaliies for square roos Square funcions Some vecor-valued esimaes Commuaors wih bounded mean oscillaion funcions Real operaors and power weighs 36 References Inroducion In his par, we consider divergence form uniformly ellipic complex operaors L = diva ) in R n and we are ineresed in weighed L p esimaes for: a) ϕl) wih bounded holomorphic funcions ϕ on secors Secion 4). b) The square roo L 1/2 compared o he ones for and, in paricular, he Riesz ransforms L 1/2 Secions 5, 6). Dae: March 21, Mahemaics Subjec Classificaion. 422, 4225, 47A6. Key words and phrases. Muckenhoup weighs, ellipic operaors in divergence form, singular noninegral operaors, holomorphic funcional calculi, square funcions, square roos of ellipic operaors, Riesz ransforms, maximal regulariy, commuaors wih bounded mean oscillaion funcions. This work was parially suppored by he European Union IHP Nework Harmonic Analysis and Relaed Problems 22-26, Conrac HPRN-CT HARP). The second auhor was also suppored by MEC Programa Ramón y Cajal, 25 and by MEC Gran MTM

3 2 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL c) Typical square funcions à la Lilewood-Paley-Sein: one, g L, using only funcions of L, and he oher, G L, combining funcions of L and he gradien operaor Secion 7). d) Vecor-valued inequaliies for he operaors above and he so-called R-boundedness of he analyic semigroup {e z L } which is linked o maximal regulariy Secion 8). Le us sress ha hose operaors may no be represenable wih usable kernels: hey are non-inegral. u hey sill are singular in he sense ha hey are of order. Hence, usual mehods for singular inegrals have o be srenghened. The unweighed L p esimaes are described in [Aus] for he operaors in a) c), wih emphasis on he sharpness of he ranges of p. The insrumenal ools are wo crieria for L p boundedness, valid in spaces of homogeneous ype: one was a sharper and simpler version of a heorem by lunck and Kunsmann [K1] in he spiri of Hörmander s crierion via he Calderón-Zygmund decomposiion, and he oher one a crierion of he firs auhor, Coulhon, Duong and Hofmann [ACDH] in he spiri of Fefferman and Sein s sharp maximal funcion via a good-λ inequaliy. The main ineres of hose resuls were ha hey yield L p boundedness of sub)linear operaors on spaces of homogeneous ype for p in an arbirary inerval. Such heorems are exended in Par I of our series [AM1] o obain weighed L p bounds for he operaor iself, is commuaors wih a MO funcion and also vecor-valued expressions. In Par II [AM2], we sudied one-parameer families of operaors saisfying local L p L q esimaes called off-diagonal esimaes on balls he seing is ha of a space of homogeneous ype). Among oher hings, such esimaes imply uniform L p - boundedness and are sable under composiion. In case of one-parameer semigroups, we showed ha as soon as here exiss one pair p, q) of indices wih p < q for which hese local L p L q esimaes hold, hen hey hold for all pairs of indices aken in he inerior of he range of L p boundedness. This fac is of umos imporance for applicaions as we ofen need o play wih exponens. We showed ha such esimaes pass from he unweighed case o he weighed case. Evenually, we made a horough sudy of weighed off-diagonal esimaes on balls for he semigroup arising from he operaor L above. Our sraegy here has he same wo seps in each of he four siuaions described above. The firs sep consiss in obaining a firs range of exponens p depending on he weigh) by applying he absrac machinery from Par I. This range urns ou o be he bes possible for boh classes of operaors and weighs. However, given one operaor and one weigh, he range of p obained above may no be sharp, and his leads us o he second sep. The sharp range is in fac relaed o he one for weighed off-diagonal esimaes esablished in Par II. A his poin, we use he main resuls of Par I in he Euclidean space bu now equipped wih he doubling measure wx). We wish o poin ou ha some of our resuls can be obained by differen mehods essenially from geomeric heory of anach spaces) once he bounded holomorphic funcional calculus is esablished in a). We give he references in he ex.

4 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 3 We wish o say ha our proofs are echnically simpler han he ones in [Aus] even for he unweighed case, because he noion of off-diagonal esimaes used here is more appropriae. Finally, hanks o he general resuls in Par I, he same echnology allows us o prove in passing weighed L p esimaes for commuaors of he operaors in a) c) wih MO funcions in he same ranges of exponens see Secion 9). 2. General crieria for boundedness and he se W w, q ) The underlying space is he Euclidean seing R n equipped wih Lebesgue measure or a doubling measure obained from an A weigh. We sae wo resuls used in his work, referring o [AM1] for saemens in sronger form and for references o earlier works. Given a ball, we wrie h = 1 hx). Le us inroduce some classical classes of weighs. Le w be a weigh ha is a non negaive locally inegrable funcion) on R n. We say ha w A p, 1 < p <, if here exiss a consan C such ha for every ball R n, ) p1 w w p C. For p = 1, we say ha w A 1 if here is a consan C such ha for every ball R n, w C wy), for a.e. y. The reverse Hölder classes are defined in he following way: w RH q, 1 < q <, if here is a consan C such ha for any ball, )1 w q q C w. The endpoin q = is given by he condiion w RH whenever here is a consan C such ha for any ball, wy) C w, for a.e. y. The following facs are well-known see for insance [GR, Gra]). Proposiion 2.1. i) A 1 A p A q for 1 p q <. ii) RH RH q RH p for 1 < p q. iii) If w A p, 1 < p <, hen here exiss 1 < q < p such ha w A q. iv) If w RH q, 1 < q <, hen here exiss q < p < such ha w RH p. v) A = A p = RH q 1 p< 1<q vi) If 1 < p <, w A p if and only if w 1p A p.

5 4 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL vii) If w A, hen he measure dw = w is a orel doubling measure. If he Lebesgue measure is replaced by a orel doubling measure µ, hen all he above properies remain valid wih he noaion change [ST]. Given 1 < q and w A wih respec o a orel doubling measure µ) we define he se W w, q ) = { p : < p < q, w A p RH q p ) }. If w = 1, hen W 1, q ) =, q ). As i is shown in [AM1], if no empy, we have q ) W w, q ) = r w, s w ) where r w = inf{r 1 : w A r }, s w = sup{s > 1 : w RH s }. We use he following noaion: if is a ball wih radius r) and λ >, λ denoes he concenric ball wih radius rλ ) = λ r), C j ) = 2 j+1 \ 2 j when j 2, C 1 ) = 4, and h dµ = C j ) 1 µ2 j+1 ) C j ) h dµ. Theorem 2.2. Le µ be a doubling orel measure on R n and 1 < q. Le T be a sublinear operaor acing on L µ), {A r } r> a family of operaors acing from a subspace D of L µ) ino L µ) and S an operaor from D ino he space of measurable funcions on R n. Assume ha TI A r) )f dµ gj) Sf p dµ, 2.1) j 1 2 j+1 and T A r) f q q dµ gj) j 1 2 j+1 Tf dµ, 2.2) for all f D, all ball where r) denoes is radius for some gj) wih gj) < wih usual changes if q = ). Le p W w, q ), ha is, < p < q and w A p RH q p. ) There is a consan C such ha for all f D Tf L p w) C Sf L p w). 2.3) An operaor acing from A o is jus a map from A o. Sublineariy means Tf +g) Tf + Tg and Tλf) = λ Tf) for all f, g and λ R or C alhough he second propery is no needed in his secion). Nex, L p w) is he space of complex valued funcions in L p dw) wih dw = w dµ. However, all his exends o funcions valued in a anach space. Remark 2.3. In he applicaions below, we have, eiher Sf = f wih f L c he space of compacly suppored bounded funcions on R n, or Sf = f wih f S he Schwarz class on R n see Secion 6).

6 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 5 Le us recall ha he doubling order D of a doubling measure µ is he smalles number κ such ha here exiss C for which µλ ) C µ λ κ µ) for every ball and for any λ > 1. The oher crierion we are going o use is he following. Theorem 2.4. Le µ be a doubling orel measure on R n, D is doubling order and 1 < q. Suppose ha T is a sublinear operaor bounded on L q µ) and ha {A r } r> is family of linear operaors acing from L c ino L q µ). Assume ha for j 2, and for j 1, C j ) TI A r) )f p dµ gj) C j ) A r) f q q dµ gj) f dµ 2.4) f dµ 2.5) for all ball wih r) is radius and for all f L c suppored in. If j gj) 2D j < hen T is of weak ype, ) and hence T is of srong ype p, p) for all < p < q. More precisely, here exiss a consan C such ha for all f L c Tf L p µ) C f L p µ). Again, he saemen has a vecor-valued exension for linear operaors acing on and ino L p funcions valued in a anach space. Remark 2.5. Noice he symmery beween 2.1) and 2.4). 3. Off-diagonal esimaes We firs inroduce he class of ellipic operaors considered in his work. Le A be an n n marix of complex and L -valued coefficiens defined on R n. We assume ha his marix saisfies he following ellipiciy or accreiviy ) condiion: here exis < λ Λ < such ha λ ξ 2 ReAx) ξ ξ and Ax) ξ ζ Λ ξ ζ, for all ξ, ζ C n and almos every x R n. We have used he noaion ξ ζ = ξ 1 ζ ξ n ζ n and herefore ξ ζ is he usual inner produc in C n. Noe ha hen Ax) ξ ζ = j,k a j,kx) ξ k ζj. Associaed wih his marix we define he second order divergence form operaor Lf = diva f), which is undersood in he sandard weak sense as a maximal-accreive operaor on L 2 R n, ) wih domain DL) by means of a sesquilinear form. The operaor L generaes a C -semigroup {e L } > of conracions on L 2 R n, ). Define ϑ [, π/2) by, ϑ = sup{ arg Lf, f : f DL)}. Then he semigroup has an analyic exension o a complex semigroup {e z L } z Σπ/2ϑ of conracions on L 2 R n, ). Here we have wrien for < θ < π, Σ θ = {z C : arg z < θ}.

7 6 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL Le w A. Here and hereafer, we wrie L p w) for L p R n, w) and if w = 1, we drop w in he noaion. We define J w L) and K w L) as he possibly empy) inervals of hose exponens p [1, ] such ha {e L } > is a bounded se in LL p w)) and { e L } > is a bounded se in LL p w)) respecively where LX) is he space of linear coninuous maps on a anach space X). We exrac from [Aus, AM2] some definiions and resuls someimes in weaker form) on unweighed and weighed off-diagonal esimaes. See here for deails and more precise saemens. Se de, F) = inf{ x y : x E, y F } where E, F are subses of R n. Definiion 3.1. Le 1 p q. We say ha a family {T } > of sublinear operaors saisfies L p L q full off-diagonal esimaes, in shor T F L p L q), if for some c >, for all closed ses E and F, all f and all > we have F )1 T χ E f) q q 1 2 n p n q ) e c d2 E,F) E f p p. 3.1) We se Υs) = max{s, s 1 } for s >. Given a ball, recall ha C j ) = 2 j+1 \ 2 j for j 2 and if w A we use he noaion h dw = 1 1 h dw, h dw = h dw. w) w2 j+1 ) C j ) C j ) Definiion 3.2. Given 1 p q and any weigh w A, we say ha a family of sublinear operaors {T } > saisfies L p w) L q w) off-diagonal esimaes on balls, in shor T O L p w) L q w) ), if here exis θ 1, θ 2 > and c > such ha for every > and for any ball wih radius r and all f, )1 T χ f) q q dw and, for all j 2, )1 T χ Cj ) f) q q dw and C j ) )1 T χ f) q q dw Υ r ) θ2 ) 2 j θ 1 2 j θ2 r Υ e c 4j r 2 ) 2 j θ 1 2 j θ2 r Υ e c 4j r 2 )1 f p p dw ; 3.2) C j ) f p p dw 3.3) )1 f p p dw. 3.4) Le us make some relevan commens for his work see [AM2] for furher deails). In he Gaussian facors he value of c is irrelevan as long as i remains non negaive. We will freely use he same leer from line o line even if is value changes. These definiions exend o complex families {T z } z Σθ wih replaced by z in he esimaes. In boh definiions, T may only be defined on a dense subspace D of L p or L p w) 1 p < ) ha is sable by runcaion by indicaor funcions of measurable ses for example, L p L 2, L p w) L 2 or L c ). Here and hereafer, for wo posiive quaniies A,, by A we mean ha here exiss a consan C > independen of he various parameers) such ha A C.

8 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 7 If q =, one should adap he definiions in he usual sraighforward way. L 1 w)l w) off-diagonal esimaes on balls are equivalen o poinwise Gaussian upper bounds for he kernels of T. oh noions are sable by composiion: T O L q w) L r w) ) and S O L p w) L q w) ) hen T S O L p w) L r w) ) when 1 p q r and similarly for full off-diagonal esimaes. When w = 1, L p L q off-diagonal esimaes on balls are equivalen o L p L q full off-diagonal esimaes. Noice he symmery beween 3.3) and 3.4). If I is a subinerval of [1, ], In I denoes he inerior in R of I R. Proposiion 3.3. Fix m N and < µ < π/2 ϑ. a) There exiss a non empy maximal inerval in [1, ], denoed by J L), such ha if p, q J L) wih p q, hen {zl) m e z L } z Σµ saisfies L p L q full offdiagonal esimaes and is a bounded se in LL p ). Furhermore, J L) J L) and In J L) = In J L). b) There exiss a non empy maximal inerval of [1, ], denoed by KL), such ha if p, q KL) wih p q, hen { z zl) m e z L } z Σµ saisfies L p L q full offdiagonal esimaes and is a bounded se in LL p ). Furhermore, KL) KL) and In KL) = In KL). c) KL) J L) and, for p < 2, we have p KL) if and only if p J L). d) Denoe by p L), p + L) he lower and upper bounds of J L) hence, of In J L) also) and by q L), q + L) hose of KL) hence, of In KL) also). We have p L) = q L) and q L)) p + L). e) If n = 1, J L) = KL) = [1, ]. f) If n = 2, J L) = [1, ] and KL) [1, q + L)) wih q + L) > 2. g) If n 3, p L) < 2n n+2, p +L) > 2n n2 and q +L) > 2. We have se q = q n nq, he Sobolev exponen of q when q < n and q = oherwise. Proposiion 3.4. Fix m N and < µ < π/2 ϑ. Le w A. a) Assume W w p L), p + L) ) Ø. There is a maximal inerval of [1, ], denoed by J w L), conaining W w p L), p + L) ), such ha if p, q J w L) wih p q, hen {zl) m e z L } z Σµ saisfies L p w)l q w) off-diagonal esimaes on balls and is a bounded se in LL p w)). Furhermore, J w L) J w L) and In J w L) = In J w L). b) Assume W w q L), q + L) ) Ø. There exiss a maximal inerval of [1, ], denoed by K w L), conaining W w q L), q + L) ) such ha if p, q K w L) wih

9 8 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL p q, hen { z zl) m e z L } z Σµ saisfies L p w) L q w) off-diagonal esimaes on balls and is a bounded se in LL p w)). Furhermore, K w L) K w L) and In K w L) = In K w L). c) Le 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 wl). d) If n = 1, he inervals J w L) and K w L) are he same and conain r w, ] if w / A 1 and are equal o [1, ] if w A 1. We have se qw = q n rw when q < n r n r wq w and qw = oherwise. Recall ha r w = inf{r 1 : w A r } and also ha s w = sup{s > 1 : w RH s }. Noe ha W w p L), p + L) ) Ø means p +L) > r p L) ws w ). This is a compaibiliy condiion beween L and w. Similarly, W w q L), q + L) ) Ø means q +L) > r q L) ws w ), which is a more resricive condiion on w since q L) = p L) and q + L) p + L). In he case of real operaors, J L) = [1, ] in all dimensions because he kernel e L saisfies a poinwise Gaussian upper bound. Hence W w p L), p + L) ) = r w, ). If w A 1, hen one has ha J w L) = [1, ]. If w / A 1, since he kernel is also posiive and saisfies a similar poinwise lower bound, one has J w L) r w, ]. Hence, In J w L) = W w p L), p + L) ). The siuaion may change for complex operaors. u we lack of examples o say wheher or no J w L) and W w p L), p + L) ) have differen endpoins. Remark 3.5. Noe ha by densiy of L c in he spaces L p w) for 1 p <, he various exensions of e z L and e z L are all consisen. We keep he above noaion o denoe any such exension. Also, we showed in [AM2] ha as long as p J w L) wih p, {e L } > is srongly coninuous on L p w), hence i has an infiniesimal generaor in L p w), which is of ype ϑ. From now on, L denoes an operaor as defined in his secion wih he four numbers p L) = q L) and p + L), q + L). We ofen drop L in he noaion: p = p L),.... For a given weigh w A, we se W w p, p + ) = p, p + ) when i is no empy) and In J w L) = p, p + ). We have p p < p + p +. Similarly, we se W w q, q + ) = q, q + ) when i is no empy) and In K w L) = q, q + ). We have q q < q + q Funcional Calculi Le µ ϑ, π) do no confuse wih he measure µ used in Secion 2) and ϕ be a holomorphic funcion in Σ µ wih he following decay ϕz) c z s 1 + z ) 2 s, z Σ µ, 4.1) for some c, s >. Assume ha ϑ < θ < ν < µ < π/2. Then we have ϕl) = e z L η + z) dz + e z L η z) dz, 4.2) Γ + Γ where Γ ± is he half ray R + e ±i π/2θ), η ± z) = 1 e ζ z ϕζ) dζ, z Γ ±, 4.3) 2 π i γ ±

10 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 9 wih γ ± being he half-ray R + e ±i ν he orienaion of he pahs is no needed in wha follows so we do no pay aenion o i). Noe ha η ± z) min1, z s1 ), z Γ ±, 4.4) hence he represenaion 4.2) converges in norm in LL 2 ). Usual argumens show he funcional propery ϕl) ψl) = ϕ ψ)l) for wo such funcions ϕ, ψ. Any L as above is maximal-accreive and so i has a bounded holomorphic funcional calculus on L 2. Given any angle µ ϑ, π): a) For any funcion ϕ, holomorphic and bounded in Σ µ, he operaor ϕl) can be defined and is bounded on L 2 wih where C only depends on ϑ and µ. ϕl)f 2 C ϕ f 2 b) For any sequence ϕ k of bounded and holomorphic funcions on Σ µ converging uniformly on compac subses of Σ µ o ϕ, we have ha ϕ k L) converges srongly o ϕl) in LL 2 ). c) The produc rule ϕl) ψl) = ϕ ψ)l) holds for any wo bounded and holomorphic funcions ϕ, ψ in Σ µ. Le us poin ou ha for more general holomorphic funcions such as powers), he operaors ϕl) can be defined as unbounded operaors. Given a funcional anach space X, we say ha L has a bounded holomorphic funcional calculus on X if for any µ ϑ, π), for any ϕ holomorphic and saisfying 4.1) in Σ µ one has ϕl)f X C ϕ f X, f X L 2, 4.5) where C depends only on X, ϑ and µ bu no on he decay of ϕ). If X = L p w) as below, hen 4.5) implies ha ϕl) exends o a bounded operaor on X by densiy. Tha a), b) and c) hold wih L 2 replaced by X for all bounded holomorphic funcions in Σ µ, follow from he heory in [McI] using he fac ha on hose X, he semigroup {e L } > has an infiniesimal generaor which is of ype ϑ see he las remark of previous secion). We skip such classical argumens of funcional calculi. Theorem 4.1. [K1, Aus] The inerior of he se of exponens p 1, ) such ha L has a bounded holomorphic funcional calculus on L p is equal o In J L) defined in Proposiion 3.3. Our firs resul is a weighed version of his heorem. We menion [Mar] where similar weighed esimaes are proved under kernel upper bounds assumpions. Theorem 4.2. Le w A be such ha W w p L), p + L) ) Ø. Le p In J w L) and µ ϑ, π). For any ϕ holomorphic on Σ µ saisfying 4.1), we have ϕl)f L p w) C ϕ f L p w), f L c, 4.6) wih C independen of ϕ and f. Hence, L has a bounded holomorphic funcional calculus on L p w).

11 1 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL Remark 4.3. Fix w A wih W w p L), p + L) ) Ø. If 1 < p < and L has a bounded holomorphic funcional calculus on L p w), hen p J w L). Indeed, ake ϕz) = e z. As In J w L) = In J w L) by Proposiion 3.3, his shows ha range obained in he heorem is opimal up o endpoins. Proof. I is enough o assume µ < π/2. Noe ha he operaors e z L are uniformly bounded on L p w) when z Σ µ, hence, by 4.4), he represenaion 4.2) converges in norm in LL p w)). Of course, his simple argumen does no yield he righ esimae, 4.6), which is our goal. I is no loss of generaliy o assume ha ϕ = 1. We spli he argumen ino hree cases: p p, p + ), p p, p + ), p p, p + ). Case p p, p + ). y iii) and iv) in Proposiion 2.1, here exis, q such ha p < < p < q < p + and w A p RH q p ). The desired bound 4.6) follows on applying Theorem 2.2 for he underlying measure and weigh w o T = ϕl) wih, q, A r = I I e r2 L ) m where m 1 is an ineger o be chosen and S = I. As ϕl) and I e r2 L ) m are bounded on L uniformly wih respec o r for he laer) by Proposiion 3.3 and Theorem 4.1, i remains o checking boh 2.1) and 2.2) on D = L c. We sar by showing 2.2). We fix f L c and a ball. We will use several imes he following decomposiion of any given funcion h: h = j 1 h j, h j = h χ Cj ). 4.7) Fix 1 k m. Since q and, q J L), we have e L O L L ) q we are using he equivalence beween he wo noions of off-diagonal esimaes for he Lebesgue measure), hence e k r2 L h j q q 2 j θ 1+θ 2 ) e α4j and by Minkowski s inequaliy e k r2 L h q q j 1 gj) 2 j+1 2 j+1 h p h p 4.8) wih gj) = 2 j θ 1+θ 2 ) e α4j for any h L. This esimae wih h = ϕl)f L yields 2.2) since, by he commuaion rule, ϕl)e k r2 L f = e k r2 L h. We nex show 2.1). Le f L c and be a ball. Wrie f = j 1 f j as before. For j = 1, we use he L boundedness of ϕl) and I e r2 L ) m, hence ϕl)i e r2 L ) m f 1 p f p. 4.9) For j 2, he funcions η ± associaed wih ψz) = ϕz e r2 z ) m by 4.3) saisfy 4 η ± z) r2m z m+1, z Γ ±.

12 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 11 Since J L), {e z L } z Γ± O L L ) p and so η + z) e z L f j dz p p e z L f j Γ + Γ + ) 2 j θ 2 j θ2 r 1 Υ e α 4j r 2 r 2 m z Γ + z z dz m+1 2 j θ 12m) f p C j ) C j ) η + z) dz f p provided 2 m > θ 2. We have used, afer a change of variable, ha Υs) θ 2 e c s2 s 2 m ds s <. The same is obained when one deals wih he erm corresponding o Γ. Plugging boh esimaes ino he represenaion of ψl) given by 4.2) one obains ϕl)i e r2 L ) m f j p 2 j θ 12 m) f p, 4.1) C j ) herefore, 2.1) holds when 2m > max{θ 1, θ 2 } since C j ) 2 j+1. Case p p, p + ): Take, q such ha p < < p + and < p < q < p +. Le A r = I I e r2 L ) m for some large enough m 1. Remark ha by he previous case, ϕl) has he righ norm in LL w)) and so does A r by Proposiion 3.4. We apply Theorem 2.2 wih he orel doubling measure dw and no weigh. Thus, i is enough o see ha ϕl) saisfies 2.1) and 2.2) for dw on D = L c L w). u his follows by adaping he preceding argumen replacing everywhere by dw and observing ha e z L O L w) L q w) ) since, q In J w L) and q. We skip deails. Case p p, p + ): Take, q such ha p < q < p + and p < < p < q. Se A r = I I e r2 L ) m for some ineger m 1 o be chosen laer. Since q p, p + ), by he firs case, ϕl) has he righ norm in LL q w)) and so does A r by Proposiion 3.4. We apply Theorem 2.4 wih underlying measure dw. I is enough o show 2.4) and 2.5). Fix a ball and f L c suppored in. We begin wih 2.5) for A r. I is enough o show i for e k r2 L wih 1 k m. Since, q J w L) and q we have e L O L w) L q w) ), hence e k r2 L f q q dw 2 j θ 1+θ 2 ) e c 4j f p dw. 4.11) C j ) This implies 2.5) wih gj) = C 2 j θ 1+θ 2 ) e c4j and j 1 gj) 2D j < holds where D is he doubling order of dw. We urn o 2.4). Le j 2. The argumen is he same as he one for 4.1) by reversing he roles of C j ) and, and using dw and e z L O L w) L w) ) since J w L)) insead of and e z L O L L ) p. We obain C j ) ϕl)i e r2 L ) m f dw 2 j θ 12 m) f dw

13 12 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL provided 2 m > θ 2 and i remains o impose furher 2 m > θ 1 + D o conclude. Remark 4.4. If W w p, p + ) Ø, he las par of he proof yields weighed weak-ype p, p ) of ϕl) provided p J w L). To do so on only has o ake = p. 5. Riesz ransforms The Riesz ransforms associaed o L are j L 1/2, 1 j n. Se L 1/2 = 1 L 1/2,..., n L 1/2 ). The soluion of he Kao conjecure [AHLMcT] implies ha his operaor exends boundedly o L 2 we ignore he C n -valued aspec of hings). This allows he represenaion L 1/2 f = 1 π e L f d, 5.1) in which he inegral converges srongly in L 2 boh a and when f L 2. Noe ha for an arbirary f L 2, h = L 1/2 f makes sense in he homogeneous Sobolev space Ḣ1 which is he compleion of C R n ) for he semi-norm h 2 and becomes he exension of he gradien o ha space. This consrucion can be forgoen if n 3 as Ḣ1 L 2 S ε = 1 1/ε π ε and converge srongly in L 2. This defines L 1/2. bu no if n 2. To circumven his echnical difficuly, we inroduce L d e for < ε < 1 and, in fac, S ε are uniformly bounded on L 2 Theorem 5.1 [Aus]). The maximal inerval of exponens p 1, ) for which L 1/2 has a bounded exension o L p is equal o In KL) defined in Proposiion 3.3 and for p In KL), f p L 1/2 f p for all f DL 1/2 ) = H 1 he Sobolev space). Again, he operaors S ε are uniformly bounded on L p and converge srongly in L p as ε. Indeed, for f L c, S εf DL) DL 1/2 ) and S ε f p L 1/2 S ε f p. Observe ha L 1/2 S ε = ϕ ε L), where ϕ ε is a bounded holomorphic funcion in Σ µ for any < µ < π/2 wih sup ε ϕ ε < and {ϕ ε } ε converges uniformly o 1 on compac subses of Σ µ as ε. The claim follows by Theorem 4.1 and densiy. We urn o weighed norm inequaliies. Remark ha by Proposiion 3.4, for all p J w L), S ε is bounded on L p w) he norm mus depend on ε) and for all p K w L), S ε is bounded on L p w) wih no conrol ye on he norm wih respec o ε. Theorem 5.2. Le w A be such ha W w q L), q + L) ) Ø. For all p In K w L) and f L c, Hence, L 1/2 has a bounded exension o L p w). L 1/2 f L p w) f L p w). 5.2) We noe ha for a given p In K w L), once 5.2) is esablished, similar argumens using Theorem 4.2 imply convergence in L p w) of S ε f o L 1/2 f for f L c. Proof. We spli he argumen in hree cases: p q, q + ), p q, q + ), p q, q + ). Case p q, q + ): y iii) and iv) in Proposiion 2.1, here exis, q such ha q < < p < q < q + and w A p RH q p ).

14 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 13 The desired esimae 5.2) is obained by applying Theorem 2.2 wih underlying measure and weigh w. Hence, i suffices o verify 2.1) and 2.2) on D = L c for T = L 1/2, S = I and A r = I I e r2 L ) m, wih m a large enough ineger. These condiions will be proved as in [Aus], bu here we use he whole range of exponens for which he Riesz ransforms are bounded on unweighed L p spaces, ha is, q, q + ). Lemma 5.3. Fix a ball. For f L c and m large enough, L 1/2 I e r2 L ) m f p g 1 j) j 1 and for f L such ha f L and 1 k m, e k r2 L f q g 2 j) q j 1 2 j+1 C j ) f p 5.3) f p, 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 α4l for some θ >. Assume his is proved. Noe ha if 2m > θ hen j 1 g 1j) < and he firs esimae is 2.1). Nex, expanding A r = I I e r2 L ) m, he laer esimae applied o S ε f in place of f since S ε f L and S ε f L ) and he commuing rule A r S ε = S ε A r give us S ε A r f q q g 2 j) j 1 2 j+1 S ε f p. y leing ε go o he jusificaion uses he observaions made a he beginning of his secion and is lef o he reader), we obain 2.2) using j 1 g 2j) <. Therefore, by Theorem 2.2, 5.2) holds for f L c. Proof of Lemma 5.3. We begin wih he firs esimae. Decomposing f as in 4.7), L 1/2 I e r2 L ) m f p L 1/2 I e r2 L ) m f j p. j 1 For j = 1, since q < < q +, L 1/2 and e r2 L are bounded on L by Theorem 5.1 and Proposiion 3.3. Hence, L 1/2 I e r2 L ) m f 1 p f p. For j 2, we use a differen approach. If h L 2, by 5.1) L 1/2 I e r2 L ) m h = 1 π 4 ϕl, )h d, where ϕz, ) = e z 1 e r2 z ) m. The funcions η ±, ) associaed wih ϕ, ) by 4.3) saisfy η ± z, ) r 2m z + ) m+1, z Γ ±, >.

15 14 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL Since z e z L O L L ) p noe ha p KL) and we are using he equivalence beween he wo noions of off-diagonal esimaes for he Lebesgue measure), η + z) e z L f j dz p p Γ + ) 1 z e z L f j p η + z) dz z Γ + 2 j θ 1 2 j θ 1 Υ Γ + ) 2 j θ2 r e α 4j r 2 z z ) 2 j θ2 r Υ e α 4j r 2 s s Observing ha when 2 m > θ 2 ) 2 j θ2 r Υ e α 4j r 2 s s z η + z) dz s s r 2m s + ) ds m+1 C j ) C j ) r 2m dsd s + ) m+1 = C 4j m, f p f p. and plugging his, plus he corresponding erm for Γ, ino he represenaion 4.2), we obain e L I e r2 L ) m f j p ϕl, )f j p d 2 j θ 12 m) f p. 5.5) C j ) This readily yields he firs esimae in he lemma. Le us ge he second one. Fix 1 k m. Le f L such ha f L. We wrie h = f f 4 where f λ is he -average of f on λ. Then by he conservaion propery see [Aus]) e L 1 = 1 for all >, we have e k r2 L f = e k r2 L f f 4 ) = e k r2 L h = j 1 e k r2 L h j, wih h j = h χ Cj ). Hence, e k r2 L f q q j 1 e k r2 L h j q q. Since q and, q KL), e L O L L ) q. This and he L -Poincaré inequaliy for yield e k r2 L h j q q 2j θ 1+θ 2 ) e ) α4j 1 h j p r C j ) 2j θ 1+θ 2 ) α 4j e f f 4 p r 2j θ 1+θ 2 ) α 4j e r 2 j+1 2 j+1 f f 2 j+1 p + j l=2 ) f 2 l f 2 l+1

16 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 15 2j θ 1+θ 2 ) α 4j e r j 2 j θ 1+θ 2 ) α 4j e j l=1 l=1 2 l+1 2 l 2 l+1 which is he desired esimae wih θ = θ 1 + θ 2. f f 2 p l+1 f p, 5.6) Case p q, q + ): Take, q such ha q < < q + and < p < q < q +. Le A r = I I e r2 L ) m for some m 1 o be chosen laer. As q, q + ), boh L 1/2 and A r are bounded on L w) we have jus shown i for he Riesz ransforms and A r are bounded uniformly in r by Proposiion 3.4). y Theorem 2.2 wih underlying doubling measure dw and no weigh, i is enough o verify 2.1) and 2.2) on D = L c for T = L 1/2, S = I and A r. To do so, i suffices o copy he proof of Lemma 5.3 in he weighed case by changing sysemaically o dw, off-diagonal esimaes wih respec o by hose wih respec o dw given he choice of, q. Also in he argumen wih we used a Poincaré inequaliy. Here, since W w q, q + ), w A p /q and in paricular w A p since q 1). Therefore we can use he L w)-poincaré inequaliy see [FPW]): f f,w p dw r) f p dw for all f L 1 loc w) such ha f L loc w), where f,w is he dw-average of f over. We leave furher deails o he reader. Case p q, q + ): Take, q such ha q < q < q + and q < < p < q. Se A r = I I e r2 L ) m for some ineger m 1 o be chosen laer. Since q q, q + ), i follows ha L 1/2 is already bounded on L q w) and so is A r. Tha L 1/2 is bounded on L p w) will follow on applying Theorem 2.4 wih underlying measure w. Hence i is enough o check boh 2.4) and 2.5). We begin wih 2.5). y Proposiion 3.4, inf J w L) = inf K w L) = q. Since > q and q W w q, q + ) W w p, p + ) J w L), we have, q J w L) and so e L O L w) L q w) ). This yields 4.11), hence 2.5) wih gj) = C 2 j θ 1+θ 2 ) e c4j which clearly saisfies j gj) 2D j <, wih D he doubling order of dw. We nex show 2.4). Le f L c be suppored on a ball and j 2. The argumen is he same as he one for 5.5) by reversing he roles of C j ) and, and using dw and z e z L O L w) L w) ) since K w L)) insead of and z e z L O ) L L. Hence, we obain C j ) L 1/2 I e r2 L ) m f p dw 2 j θ 12 m) f p dw provided 2 m > θ 2 and i remains o impose furher 2 m > θ 1 + D o conclude. Remark 5.4. If W w q, q + ) Ø, he las par of he proof yields weighed weak-ype q, q ) of L 1/2 provided q K w L), one only needs o ake = q.

17 16 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL Remark 5.5. Theorem 5.1 assers ha In KL) is he exac range of L p boundedness for he Riesz ransforms when w = 1. When w 1, we canno repea he same argumen as i used Sobolev embedding which has no simple counerpar in he weighed siuaion. However, if we inser in he inegral of 5.1) a funcion m) wih m L, ), hen 5.2) holds wih a consan proporional o m. Indeed, Le ϕ m z) = z 1/2 e z m) d for z Σ µ, ϑ < µ < π/2. Then, ϕ m is holomorphic in Σ µ and bounded wih ϕ m c µ m. Now for f L c, e L f m) d = L 1/2 ϕ m L)f hence, combining Theorems 4.2 and 5.2, we obain for p In K w L), e L f m) d m f L p w). L p w) Conversely, given an exponen p 1, ), assume ha his L p w) esimae holds for all m L. Using randomizaion echniques which we skip see Secion 8 for some accoun on such echniques), his implies e L f 2 d 2 f L p w). L p w) This square funcion esimae is proved direcly in Secion 7 and we indicae a he end of ha secion why his inequaliy implies p K w L). Thus, he range in p is sharp up o endpoins see Proposiion 3.4). 6. Reverse inequaliies for square roos We coninue on square roos by sudying when he inequaliy opposie o 5.2) hold. Firs we recall he unweighed case. Theorem 6.1. [Aus] If max { 1, n p L) n+p L)} < p < p+ L) hen for f S, L 1/2 f p f p. 6.1) To sae our resul, we need a new exponen. For p >, define p w, = n r w p n r w + p, where r w = inf{r 1 : w A r }. Se also p w = n rw p n r wp for p < n r w and p = oherwise. Noe ha p w, ) w = p. Theorem 6.2. Le w A wih W w p L), p + L) ) Ø. If max { r w, p ) w, } < p < p + hen for f S, L 1/2 f L p w) f L p w). 6.2) Remark 6.3. Recall ha p = p L) r w and we have p ) w, < p p. If p L) = 1, hen p ) w, r w, so max { r w, p ) w, } = rw = p. This happens for example when L is real or when n = 1, 2. Define Ẇ 1,p w) as he compleion of S under he semi-norm f L p w). Arguing as in [AT1] see [Aus]) combining Theorems 5.2 and 6.2, we obain he following consequence.

18 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 17 Corollary 6.4. Assume W w q L), q + L) ) Ø. If p In K w L) wih p > r w, hen L 1/2 exends o an isomorphism from Ẇ 1,p w) ino L p w). Proof of Theorem 6.2. We spli he argumen in hree cases: p p, p + ), p p, p + ), p max { r w, p ) w, }, p+ ). Case p p, p + ): I relies on he following lemma. Lemma 6.5. Le In J L) and q J L) wih < q. Le be a ball and m 1 an ineger. For all f S, we have L 1/2 I e r2 L ) m f g 1 j) f p 6.3) j 1 for m large enough depending on and q, and L 1/2 I I e r2 L ) m )f q g 2 j) q j 1 2 j+1 2 j+1 L 1/2 f p, 6.4) where g 1 j) = C m 2 j θ 4 m j and g 2 j) = C m 2 j θ e α4j for some θ >, and he implici consans are independen of and f. Admi his lemma for a momen. Since p p, p + ) = W w p, p + ), by iii) and iv) in Proposiion 2.1, here exis, q such ha p < < p < q < p + and w A p RH q p ). Noe ha 6.3) and 6.4) are respecively he condiions 2.1) and 2.2) of Theorem 2.2 wih underlying measure and weigh w, T = L 1/2, A r = I I e r2 L ) m, wih m large enough, and Sf = f. Hence we obain 6.2). Proof of Lemma 6.5. We firs show 6.4). Using he commuaion rule and expanding I e r2 L ) m i suffices o apply 4.8) as, q J L) o h = L 1/2 f. We urn o 6.3). If ϕz) = z 1/2 1 e r2 z ) m, hen ϕl)f = L 1/2 I e r2 L ) m f. y he conservaion propery ϕl) f = ϕl) f f 4 ) = j 1 ϕl) h j, 6.5) where h j = f f 4 ) φ j. Here, φ j = χ Cj ) for j 3, φ 1 is a smooh funcion wih suppor in 4, φ 1 1, φ 1 = 1 in 2 and φ 1 C/r and, evenually, φ 2 is aken so ha j 1 φ j = 1. We esimae each erm in urn. For j = 1, since p < < p +, by he bounded holomorphic funcional calculus on L Theorem 4.1) and ϕl) h 1 = I e r2 L ) m L 1/2 h 1, one has uniformly in r, ϕl) h 1 p L 1/2 h 1 p. Nex, Theorem 6.1, L -Poincaré inequaliy and he definiion of h 1 imply Therefore, L 1/2 h 1 p h 1 p f L 4 ). ϕl) h 1 p 4 f p.

19 18 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL For j 3, he funcions η ± associaed wih ϕ by 4.3) saisfy η ± z) r2m z m+3/2, z Γ ±. Since J L), {e z L } z Γ± O L L ) p and so η + z) e z L h j dz p p e z L h j Γ + Γ + ) 2 j θ 2 j θ2 r 1 e α 4j r 2 r 2 m z z z dz m+3/2 Γ + Υ 2 j θ 12 m1) j l=1 2 l f p, 2 l+1 C j ) η + z) dz h j p provided 2 m + 1 > θ 2, where he las inequaliy follows by repeaing he calculaions made o derive 5.6). The erm corresponding o Γ is conrolled similarly. Plugging boh esimaes ino he represenaion of ϕl) given by 4.2) one obains ϕl)h j p 2 j θ 12m1) The reamen for he erm j = 2 is similar using j 2 l l=1 2 l+1 f p. h 2 f f 4 χ 8 \2 f f 2 χ 8 \2 + f 4 f 2 χ 8 \2. Applying Minkowski s inequaliy and 6.5), we obain 6.3). The lemma is proved. Case p p, p + ): Take, q such ha p < < p + and < p < q < p +. Observe ha W w p, p + ) In Jw L) and q J w L). The proof of Lemma 6.5 exends muais muandis wih dw replacing since here is an L -Poincaré inequaliy for dw see Secion 5). I suffices o apply Theorem 2.2 wih underlying measure dw and no weigh. We leave furher deails o he reader. Case p max { r w, p ) w, }, p+ ): I follows a mehod in he unweighed case by [Aus] using an adaped Calderón-Zygmund decomposiion. Lemma 6.6. Le n 1, w A and 1 p < such ha w A p. Assume ha f S is such ha f L p w) <. Le α >. Then, one can find a collecion of balls { i } i, smooh funcions {b i } i and a funcion g L 1 loc w) such ha and he following properies hold: f = g + i b i 6.6) gx) Cα, for µ-a.e. x 6.7) supp b i i and b i p dw Cα p w i ), 6.8) i w i ) C f α R p dw, 6.9) p n i

20 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 19 χ i N, 6.1) i where C and N depends only on he dimension, he doubling consan of µ and p. In addiion, for 1 q < p w, we have )1 b i q q dw α r i ). 6.11) i Proof. Since w A p, we have an L p w) Poincaré inequaliy see [FPW]). On he oher hand, as w A p and 1 q < p w if p = 1, i.e. w A 1, i holds also a q = 1 w = n when n 2) we can apply [FPW, Corollary 3.2] when checking he n1 balance condiion in ha reference we have used ha w A r implies E / ) r we)/w) for any ball and any E ). Thus here is an L p w)l q w) Poincaré inequaliy: )1 f f,w q q dw r) f p p dw 6.12) for all locally Lipschiz funcions f and all balls. These are all he ingrediens needed o invoke [AM1, Proposiion 9.1]. We use he following resoluion of L 1/2 : L 1/2 f = 1 π Le L f d. I suffices o work wih R..., o obain bounds independen of ε, R, and hen o ε le ε and R : indeed, he runcaed inegrals converge o L 1/2 f in L 2 when f S and a use of Faou s lemma concludes he proof. For he runcaed inegrals, all he calculaions are jusified. We wrie L 1/2 where i is undersood ha i should be replaced by is approximaion a all places. Take q so ha p < q < p +. y he firs case of he proof, L 1/2 f L q w) f L q w). 6.13) We may assume ha max{ r w, p ) w, } < p < p, oherwise here is nohing o prove. We claim ha i is enough o show ha L 1/2 f L p, w) f L p w). 6.14) Assuming his esimae we wan o inerpolae. To his end, we use he following lemma. Lemma 6.7. Assume r > r w. Then D = { /2 f : f S, supp f R n \ {} } is dense in L r w), where f denoes he Fourier ransform of f. Proof. I is easy o see ha D S hence D L r w). As in [Gra, p. 353], using ha he classical Lilewood-Paley series converges in L r w) since w A r, i follows ha he se D = { g S : supp ĝ R n \ {}, supp ĝ is compac } is dense in L r w). We see ha D D and so D is dense in L r w). For g D, f = /2 g is well-defined in S as fξ) = c ξ 1/2 ĝξ) and supp f R n \ {}. Hence, g = /2 f D.

21 2 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL If r > r w, he usual Riesz ransforms, /2, are bounded on L r w) his can be reobained from he resuls in Secion 5). Also, for g L r w), one has g L r w) /2 g L r w) using he ideniy I = R R2 n where R j = j /2. Thus, for g D, L 1/2 /2 g = L 1/2 f if f = /2 g and f L r w) g L r w) for r > r w. As r w < p < q, 6.13) and 6.14) reformulae ino weighed srong ype q, q ) and weak ype p, p) of T = L 1/2 /2 a priori defined on D. Since D is dense in all L r w) when r > r w by he above lemma, we can exend T by densiy in boh cases and heir resricions o he space of simple funcions agree. Hence, we can apply Marcinkiewicz inerpolaion and conclude again by densiy ha 6.13) holds for all q wih p < q < q which leads o he desired esimae. Our goal is hus o esablish 6.14), more precisely: for f S and α >, w{ L 1/2 f > α} = w{x R n : L 1/2 fx) > α} C α p R n f p dw. 6.15) Since p > r w, we have w A p. From he condiion p ) w, < p, we have p < p w. Therefore, here exiss q p, p + ) = In J w L) such ha p < q < p w. Thus, we can apply he Calderón-Zygmund decomposiion of Lemma 6.6 o f a heigh α for he measure dw and wrie f = g + i b i. Using 6.13), 6.7) and q > p, we have { w L 1/2 g > α } 3 1 α q 1 α p α q R n L 1/2 g q dw 1 R n f p dw + 1 α p R n g q dw 1 g p dw R α n p R n p 1 b i dw f p dw, α p i R n where he las esimae follows by applying 6.1), 6.8), 6.9). To compue L 1/2 i b i), le r i = 2 k if 2 k r i ) < 2 k+1, hence r i r i ) for all i. Wrie and hen { w i L 1/2 = 1 π r 2 i L d Le + 1 L d Le = T i + U i, π ri 2 L 1/2 2 α } ) b i > w 4 i + w 3 i + w R n \ i { 1 α p R n f p dw + I + II, α } U i b i > 3 i ) { α }) 4 i T i b i > 3 where we have used 6.9). We esimae II. Since q J w L), i follows ha L e L O L q w) L q w) ) by Proposiion 3.4, hence II 1 α i j 2 C j i ) T i b i dw 1 α w2 j i ) i j 2 r 2 i C j i ) i L e L b i dw d 3/2

22 1 α i WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 21 2 j D w i ) i j 2 r 2 i 2 j D e c4j w i ) j 2 i ) 2 j θ 1 2 j θ2 r i Υ e c 4j ri 2 d 3/2 w i ) 1 α p R n f p dw, 1 i b i q dw where we have used 6.11) and 6.9), and D is he doubling order of dw. I remains o handling he erm I. Using funcional calculus for L one can compue U i as r 1 i ψri 2 L) wih ψ he holomorphic funcion on he secor Σ π/2 given by z d ψz) = c z e. 6.16) I is easy o show ha ψz) C z 1/2 e c z, uniformly on subsecors Σ µ, µ < π. 2 We claim ha, since q In J w L), ψ4 k L )1 L) β k β k ) q w) k Z k Z L w). q The proof of his inequaliy is posponed unil he end of Secion 7. We se β k = b i i : r i =2 k r i. Then, U i b i = ) ψ4 k b i L) = ψ4 k L)β k. r i i k Z i : r i =2 k k Z Using 6.17), he bounded overlap propery 6.1), 6.11), r i r i ) and 6.9), one has I 1 q U α q i b i 1 )1 q β L q w) α q k b i q α q R r q dw n i i k Z L q w) i )1 q i w i ) 1 α p R n f p dw. Collecing he obained esimaes, we conclude 6.14) as desired. Remark 6.8. If w A 1, W w p L), p + L) ) Ø and p ) w, < 1 hen for all f S L 1/2 f L 1, w) f L 1 w). This ha is 6.15) wih p = 1) uses a similar argumen lef o he reader) once we have chosen an appropriae q for which L 1 w) L q w) Poincaré inequaliy holds: since w A 1, one needs q n. As r n1 w = 1, he assumpion p ) w, < 1 means ha p < n and so we pick q In J n1 wl) wih p < q < n. n1 7. Square funcions We define he square funcions for x R n and f L 2, g L fx) = L/2 e L fx) 2 d G L fx) = e L fx) 2 2 d. )1 2,

23 22 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL They are represenaive of a larger class of square funcions and we resric our discussion o hem o show he applicabiliy of our mehods. They saisfy he following L p esimaes. Theorem 7.1 [Aus]). and In { 1 < p < : g L f p f p, f L p L 2} = p L), p + L) ) In { 1 < p < : G L f p f p, f L p L 2} = q L), q + L) ). In his saemen, can be replaced by : he square funcion esimaes for L wih ) auomaically imply he reverse ones for L. The par concerning g L can be obained using an absrac resul of Le Merdy [LeM] as a consequence of he bounded holomorphic funcional calculus on L p. The mehod in [Aus] is direc. We remind he reader ha in [Se], hese inequaliies for L = were proved differenly and he boundedness of G follows from ha of g and of he Riesz ransforms j /2 or vice-versa) using he commuaion beween j and e. Here, no such hing is possible. We have he following weighed esimaes for square funcions. Theorem 7.2. Le w A. a) If W w p L), p + L) ) Ø and p In J w L) hen 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 In K w L) hen for all f L c G L f L p w) f L p w). we have Noe ha he operaors L/2 e L and e L exend o L p w) when p In J w L) and p In K w L) respecively. y seeing g L and G L as linear operaors from scalar funcions o H-valued funcions see below for definiions), he above inequaliies exend o all f L p w) by densiy see he proof). We also ge reverse weighed square funcion esimaes as follows. Theorem 7.3. Le w A. a) If W w p L), p + L) ) Ø and p In J w L) hen 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 resricion ha f L 2 can be removed provided g L and G L are appropriaely inerpreed: see he proofs. We add a commen abou sharpness of he ranges of p a he end of he secion. As a corollary, g L resp. G L ) defines a new norm on L p w) when p In J w L) resp. p In K w L) and p > r w ). Again, Le Merdy s resul cied above [LeM] also gives such a resul for g L, bu no for G L. The resricion p > r w in par b) comes from he argumen. We do no know wheher i is necessary for a given weigh non idenically 1.

24 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 23 efore we begin he argumens, we recall some basic facs abou Hilber-valued exensions of scalar inequaliies. To do so we inroduce some noaion: by H we mean L 2, ), d) and denoes he norm in H. Hence, for a funcion h: R n, ) C, we have for x R n hx, ) = hx, ) 2 d /2. In paricular, wih ϕz, ) = z/2 e z and g L fx) = ϕl, )fx) G L fx) = ϕl, )fx) wih ϕz, ) = e z. Le L p H w) be he space of H-valued Lp w)-funcions equipped wih he norm h L p H w) = hx, ) p p dwx). R n Lemma 7.4. Le µ be a orel measure on R n for insance, given by an A weigh). Le 1 p q <. Le D be a subspace of M, he space of measurable funcions in R n. Le S, T be linear operaors from D ino M. Assume here exiss C > such ha for all f D, we have Tf L q µ) C α j Sf L p F j,µ), j 1 where F j are subses of R n and α j. Then, here is an H-valued exension wih he same consan: for all f : R n, ) C such ha for almos) all >, f, ) D, Tf L q H µ) C α j Sf L p H F j,µ). j 1 The exension of a linear operaor T on C-valued funcions o H-valued funcions is defined for x R n and > by Th)x, ) = T h, ) ) x), ha is, can be considered as a parameer and T acs only on he variable in R n. This resul is essenially he same as he Marcinkiewicz-Zygmund heorem and he fac ha H is isomeric o l 2. Tha he norm decreases uses p q. We refer o, for insance, [Gra, Theorem 4.5.1] for an argumen ha exends sraighforwardly o our seing. Proof of Theorem 7.2. Par a). We spli he argumen in hree cases: p p, p + ), p p, p + ), p p, p + ). Case p p, p + ): y Proposiion 2.1, here exis, q such ha p < < p < q < p + and w A p RH q p ). We are going o apply Theorem 2.2 wih T = g L, S = I, A r = I I e r2 L ) m, m large enough, underlying measure and weigh w. We firs see ha 2.2) holds for all f L c. Here, we could have used he approach in [Aus], bu he one below adaps o he oher wo cases wih minor changes.

25 24 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL As, q J L) and q, we know ha e L O L L ) q. If is a ball, j 1 and g L wih supp g C j ) we have e k r2 L g q q C 2 j θ 1+θ 2 ) e α4j g p. 7.1) C j ) Lemma 7.4 applied o S = I, T : L = L R n, ) L q = L q R n, ) given by yields Tg = C 2 j θ 1+θ 2 ) e α4j 2 j q χ e k r2 L χ Cj ) g) e k r2 L gx, ) q q C 2 j θ 1+θ 2 ) e α4j C j ) for all g L H wih supp g, ) C j) for each >. As in 4.7), for h L H wrie hx, ) = j 1 h j x, ), x R n, >, gx, ) p 7.2) where h j x, ) = hx, ) χ Cj )x). Using 7.2), we have for 1 k m, e k r2 L hx, ) q q e k r2 L h j x, ) q q j 2 j θ 1+θ 2 ) e α4j hx, ) p. 7.3) j 1 2 j+1 Take hx, ) = L/2 e L fx). Since g L fx) = hx, ) and f L c, h L H Theorem 7.1 and g L e k r2 L f)x) = Thus 7.3) implies g L e k r2 L f) q L/2 e L e k r2 L fx) 2 d )1 2 = e k r2 L hx, ). q j 1 2 j θ 1+θ 2 ) e α4j g L f p 2 j+1 by and i follows ha g L saisfies 2.2). I remains o show ha 2.1) wih Sf = f holds for all f L c. Wrie f = j 1 f j as before. If j = 1 we use ha boh g L and I e r2 L ) m are bounded on L see Theorem 7.1 and Proposiion 3.3): g L I e r2 L ) m f 1 p f p. 7.4) For j 2, we observe ha g L I e r2 L ) m f j x) = 4 L/2 e L I e r2 L ) m f j x) 2 d 2 = ϕl, )f j x)

26 WEIGHTED NORM INEQUALITIES AND ELLIPTIC OPERATORS 25 where ϕz, ) = z/2 e z 1 e r2 z ) m. As in [Aus], he funcions η ±, ) associaed wih ϕ, ) by 4.3) verify Thus, η ± z, ) 1/2 z + ) 3/2 η ± z, ) Γ + z + ) 3 r 2m z + ) m, z Γ ±, >. r 4 m )1 d 2 z + ) 2 m r2 m z m ) Nex, applying Minkowski s inequaliy and e z L O L L ) p, since p J L), we have e z L f j x) η + z, )dz p p Γ + ) e z L p p f j x) η + z, ) dz Γ + e z L f j p r 2m dz z m+1 2 j θ 1 2 j θ 12 m) ) 2 j θ2 r Υ e α 4j r 2 s s C j ) f p r 2 m ds s m s C j ) f p provided 2 m > θ 2. This plus he corresponding erm for Γ yield g L I e r2 L ) m f j p 2 j θ 12 m) f C j ). 7.6) Collecing he laer esimae and 7.4), we obain ha 2.1) holds whenever 2 m > max{θ 1, θ 2 }. Case p p, p + ): Take, q such ha p < < p + and < p < q < p +. Le A r = I I e r2 L ) m for some m 1 o be chosen laer. Remark ha by he previous case, g L is bounded in L w) and so does A r by Proposiion 3.4. We apply Theorem 2.2 o T = g L and S = I wih underlying measure dw and no weigh: i is enough o see ha g L saisfies 2.1) and 2.2) on L c. u his follows by adaping he preceding argumen replacing everywhere by dw and observing ha e z L O L w) L q w) ). We skip deails. Case p p, p + ): Take, q such ha p < q < p + and p < < p < q. Se A r = I I e r2 L ) m for some ineger m 1 o be chosen laer. Since q p, p + ), by he firs case, g L is bounded on L q w) and so does A r by Proposiion 3.4. y Theorem 2.4 wih underlying orel doubling measure dw, i is enough o show 2.4) and 2.5). Fix a ball, f L c suppored on. Observe ha 2.5) follows direcly from 4.11) since, q J w L) and q. We urn o 2.4). Assume j 2. The argumen is he same as he one for 7.6) by reversing he roles of C j ) and, and using dw and e z L O L w) L w) )

27 26 PASCAL AUSCHER AND JOSÉ MARÍA MARTELL since J w L)) insead of and e z L O L L ) p. We obain g L I e r2 L ) m f dw 2 j θ 12 m) f dw C j ) provided 2 m > θ 2 and i remains o impose furher 2 m > θ 1 + D o conclude, where D is he doubling order of w. Proof of Theorem 7.2. Par b). We spli he argumen in hree cases: p q, q + ), p q, q + ), p q, q + ). Case p q, q + ): y Proposiion 2.1, here exis, q such ha q < < p < q < q + and w A p RH q p ). We are going o apply Theorem 2.2 wih underlying measure and weigh w o T = G L, S = I, A r = I I e r2 L ) m, m large enough. We begin wih 2.2). Fix 1 k m and a ball. Combining 5.4) and Lemma 7.4 wih T = e k r2 L and S =, we obain e k r2 L hx, ) q g 2 j) hx, ) p q j 1 2 j+1 wih g 2 j) = C m 2 j l j 2l θ e α4l for some θ > whenever h: R n, ) C is such ha h and h belong o L our space D). Seing hx, ) = e L fx) for f L c, we noe ha h, ) L and h, ) L for each >. Hence, he above esimae applies. Since hx, ) = G L fx) and e k r2 L hx, ) = G L e k r2 L f)x), we obain G L e k r2 L f) q g 2 j) G L f p, q j 1 2 j+1 which is 2.2) afer expanding A r. I remains o checking 2.1) for G L and S = I for f L c. Fix a ball. As before, wrie f = j 1 f j where f j = f χ Cj ). Since In KL), boh G L and I e r2 L ) m are bounded on L by Theorem 7.1 and Proposiion 3.3. Then for j = 1 we have G L I e r2 L ) m f 1 p f p. 7.7) For j 2, we observe ha G L I e r2 L ) m f j x) = 4 e L I e r2 L ) m f j x) 2d 2 = ϕl, )f j x) where ϕz, ) = e z 1 e r2 z ) m. As in [Aus], he funcions η ±, ) associaed for each > wih ϕ, ) by 4.3) verify r 2 m η ± z, ) z + z + ) m, z Γ ±, >, and so η ± z, ) z + ) 2 r 4 m d 2 z + ) 2 m r2 m z m+1/2. 7.8)