ON L p. ESTIMATES FOR SQUARE ROOTS OF SECOND ORDER ELLIPTIC OPERATORS ON R n. Pascal Auscher

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1 Publ. Mat. 48 (2004), ON L p ESTIMATES FOR SQUARE ROOTS OF SECOND ORDER ELLIPTIC OPERATORS ON R n Pascal Auscher Abstract We prove that the square root of a unformly complex ellptc operator L = dv(a ) wth bounded measurable coeffcents n R n satsfes the estmate L 1/2 2n f p f p for sup(1, n+4 ε) < p < 2n + ε, whch s new for n 5 and p < 2 or for n 3 n 2 and p > 2n. One feature of our method s a Calderón-Zygmund n 2 decomposton for Sobolev functons. We make some further remarks on the topc of the converse L p nequaltes (.e. Resz transforms bounds), pushng the recent results of [BK2] and [HM] for 2n n+2 < p < 2 when n 3 to the range sup(1, 2n n+2 ε) < p < 2+ε. In partcular, we obtan that L 1/2 extends to an somorphsm from Ẇ 1,p (R n ) to L p (R n ) for p n ths range. We also generalze our method to hgher order operators. Contents 1. Introducton A Calderón-Zygmund lemma for Sobolev-L p functons Proof of Theorems 2 and Proof of techncal lemmata Comments on earler L p results Calderón-Zygmund lemma for W m,p -functons Results for hgher order operators Concludng remarks 184 References Mathematcs Subject Classfcaton. 42B20, 42B25, 35J15, 35J30, 35J45, 47F05, 47B44. Key words. Calderón-Zygmund decomposton, ellptc operators, square roots, functonal calculus.

2 160 P. Auscher 1. Introducton Let A = A(x) be an n n matrx of complex, L coeffcents, defned on R n, and satsfyng the ellptcty (or accretvty ) condton (1.1) λ ξ 2 Re Aξ ξ and Aξ ζ Λ ξ ζ, for ξ, ζ C n and for some λ, Λ such that 0 < λ Λ <. We defne a second order dvergence form operator (1.2) Lf dv(a f), whch we nterpret n the sense of maxmal accretve operators va a sesqulnear form. The accretvty condton (1.1) enables one to defne a square root L 1/2 (see [K]) agan n the sense of maxmal accretve operators. It s known that (1.3) L 1/2 f 2 f 2, n 1. Here s the equvalence n the sense of norms, wth mplct constants C dependng only on n, λ and Λ, and f p = ( R f(x) p n H dx)1/p denotes the usual norm for functons on R n valued n a Hlbert space H. Ths mples that the doman of L 1/2 s n all dmensons the Sobolev space H 1 (R n ), whch was known as Kato s conjecture. Indeed (1.3) s due to Cofman, McIntosh and Meyer [CMcM] when n = 1, to Hofmann and McIntosh [HMc] when n = 2 and to Hofmann, Lacey, McIntosh and Tchamtchan along wth the author [AHLMcT] for all dmensons. Although there s no explct connectons between square roots and Calderón-Zygmund operators (except f A s a constant matrx or f n=1) there are strong relatons: the L 2 -results are obtaned through refnements of standard technques n the theory (square functons, Carleson measures, T (b) theorem) so one can try to compare L 1/2 f and f n L p norms, p 2. Ths program was ntalsed n [AT1] for ths class of complex operators. It arose from a dfferent perspectve towards applcatons to boundary value problems n the works of Dahlberg, Jerson, Keng and ther collaborators (see [Ke, problem ]). At ths tme, the followng results are known (obtaned sometmes pror to the

3 L p Estmates for Square Roots 161 Kato conjecture by makng the L 2 -result an assumpton). (1.4) (1.5) (1.6) (1.7) (1.8) (1.9) (1.10) L 1/2 f p f p, n = 1, 1 < p <. L 1/2 f p f p, n = 2, 1 < p <. f p L 1/2 f p, n = 2, 1 < p < 2 + ε. L 1/2 f p f p, n = 3, 4, 1 < p < 2. f p L 1/2 f p, n 3, p n < p < 2. L 1/2 f p f p, n 3, 2 < p < p n. f p L 1/2 f p, n = 3, 4, 2 < p < 2 + ε. Here and subsequently we set p n = 2n n+2 for n 2, whch s the Sobolev exponent for the embeddng f 2 C f pn. Also, the nequaltes are stated for f n an approprate class for whch L 1/2 f s well-defned. In vew of the L 2 -results, C0 (R n ) works. The equvalence (1.4) s n [AT2]. In one dmenson, the theory of sngular ntegral operators s fully applcable but ths ceases n hgher dmensons. The equatons (1.5) and (1.6) are from [AT1]: they were proved assumng (1.3) and a techncal condton called the Gaussan property, whch s always vald n two dmensons from [AMcT]. Inequaltes of type (1.6) are known as L p -boundedness for the operator L 1/2, the (array of) Resz transforms assocated to L. Note that n (1.6), ε depends on the ellptcty constants only and the range of p s s sharp for the whole class of such operators from a counterexample of Keng (see [AT1, p. 119]) as ε can be as small as one wshes. The nequalty (1.7) s a consequence of [AHLMcT, Proposton 6.2], and an assumpton on the resolvent (1+t 2 L) 1 whch holds n these dmensons thanks to Sobolev nequaltes (see Secton 4). The nequalty (1.8) s due ndependently to Blunck and Kunstmann [BK2] and Hofmann and Martell [HM] takng (1.3) as startng pont. In fact, we shall explan how to lower p n to p n ε for some ε > 0 dependng on dmenson and the ellptcty constants only. Proposton 1. (1.11) f p L 1/2 f p, n 3, p n ε < p p n. The sharpness of the lower bound s not known for the whole class of operators L wth (1.1). We present n a subsequent work equvalent formulatons of L p -boundedness for Resz transforms allowng to conclude the exstence of operators for whch (1.11) fals for some p > 1 and also to dscuss ths sharpness ssue [A].

4 162 P. Auscher A cheap dualty argument gves (1.9) from (1.8) appled to the adjont L, and the range extends to p < (p n ε) thanks to Proposton 1. Lastly, (1.10) s a consequence of (1.3), (1.7) and (1.9) combned wth the perturbaton result n [AT1, p. 131]. Here, we wsh to complete ths study and prove results lke (1.7) n dmensons larger than or equal to 5. Theorem 2. We have for P n = npn p n+n = 2n n+4 (1.12) L 1/2 f p f p, n 5, sup(p n ε, 1) < p < 2. The sharpness of the lower bound s open. Note that P n 1 f n 4: n fact, our proof gves n lower dmensons a weak-type (1, 1) estmate: Theorem 3. We have the weak-type estmate (1.13) L 1/2 f 1, f 1, n 4. By nterpolaton, ths provdes us wth alternate proofs to the strong type (p, p) n dmensons n 4 for the range 1 < p < 2. Boundedness on the Hardy space H 1 s also known n these dmensons,.e. L 1/2 f H 1 C f H 1. The gan from p n n (1.11) to P n n (1.12) comes from a somewhat magcal use of Sobolev embeddngs. Let us return to large dmensons. As a consequence of the perturbaton method mentoned above, we have Corollary 4. (1.14) f p L 1/2 f p, n 5, 2 < p < 2 + ε. The upper bound s sharp as n dmenson 2. By repeatng the proof of [AT1, Theorem 21, p. 132], one obtans nvertblty results, whch were known f n 4. Corollary 5. (1.15) f p L 1/2 f p, n 2, sup(p n ε, 1) < p < 2 + ε. Furthermore, the operator L 1/2, a pror defned from C0 (R n ) nto L 2 (R n ), extends to a bounded an nvertble operator from Ẇ 1,p (R n ) onto L p (R n ) for p n the above range. We stress that these nequaltes only requre (1.1). More assumptons gve addtonal results. They wll be dscussed n Secton 5. The proof of Theorem 2 wll be completely dfferent from the ones n smaller dmensons that gave an estmate on the Hardy space H 1 (R n ) as none of them seem to extend to ths more general settng. In fact, we

5 L p Estmates for Square Roots 163 reprove those earler results n a unfed way. Our method s to obtan a weak type estmate for each p n the range. We shall use deas from Blunck and Kunstmann who ntroduced n [BK1] a crteron to obtan weak type estmates for p 1 whch apply to non-ntegral operators, generalzng methods and result from Duong-McIntosh for p = 1 [DMc]. These results, as the orgnal Calderón-Zygmund theorem, rely on the Calderón-Zygmund decomposton for L p functons. The novelty here (see Secton 2) s a Calderón-Zygmund decomposton for any Sobolev- L p functon, 1 p, namely any locally ntegrable functon whose gradent s n L p : t wrtes as the sum of a good part whch s Lpschtz and a locally fnte sum of bad functons whch are supported n cubes wth a control on the L p -average of ther gradents. In Secton 3, we prove Theorem 2. Secton 4 s concerned wth auxllary lemmata for ellptc operators used n Secton 3. In Secton 5, we make some more hstorcal remarks on earler L p results. In Sectons 6 and 7, we extend both the Calderón-Zygmund decomposton to Ẇ m,p functons and our results to hgher order operators: see there for statements. We make some fnal comments n Secton 8. Acknowledgements. We wsh to thank G. Davd for an nterestng dscusson on the Calderón-Zygmund decomposton. We are grateful to S. Blunck and to S. Hofmann for makng ther unpublshed work avalable to me. 2. A Calderón-Zygmund lemma for Sobolev-L p functons Theorem 6. Let n 1, 1 p and f D (R n ) be such that f p <. Let α > 0. Then, one can fnd a collecton of cubes (Q ), functons g and b such that (2.1) f = g + b and the followng propertes hold: (2.2) (2.3) (2.4) (2.5) g Cα, b W 1,p 0 (Q ) and b p Cα p Q, Q Q Cα R p f p, n 1 Q N, where C and N depends only on dmenson and p.

6 164 P. Auscher As usual, cubes are wth sdes parallel to the axes and E s the Lebesgue measure of a set E. The space W 1,p 0 (Ω) denotes the closure of C0 (Ω) n W 1,p (Ω). The pont s n the fact that the functons b are supported n cubes as the orgnal Calderón-Zygmund decomposton appled to f would not gve ths. Note that the assumpton on f mples by classcal regularty results that f s locally ntegrable. Proof: If p =, set g = f. Assume next that p <. Let Ω = {x R n ; M( f p )(x) > α p } where M s the uncentered maxmal operator over cubes of R n. If Ω s empty, then set g = f. Otherwse, the maxmal theorem gves us Ω Cα p R n f p. Let F be the complement of Ω. By the Lebesgue dfferentaton theorem, f α almost everywhere on F. We also have, Lemma 7. One can redefne f almost nowhere on F so that for all x F, for all cube Q centered at x, (2.6) f(x) m Q f Cαl(Q) where l(q) s the sdelength of Q and for all x, y F, (2.7) f(x) f(y) Cα x y. The constant C depends only on dmenson and p. Here m E f denotes the mean of f over E. It s well-defned f E s a cube as f s locally ntegrable. Let us postpone the proof of ths lemma and contnue the argument. Let (Q ) be a Whtney decomposton of Ω by dyadc cubes. Hence, Ω s the dsjont unon of the Q s, the cubes 2Q 1 are contaned n Ω and have the bounded overlap property, but the cubes 4Q ntersect F. As usual, λq s the cube co-centered wth Q wth sdelength λ tmes that of Q. Hence (2.4) and (2.5) are satsfed by the cubes 2Q. Let us now defne the functons b. Let (X ) be a partton of unty on Ω assocated to the coverng (Q ) so that for each, X s a C 1 functon supported n 2Q wth X + l X c(n), l beng the sdelength of Q. Pck a pont x 4Q F. Set b = (f f(x ))X. 1 Strctly speakng, 2Q may not have the bounded overlap property but cq do for some c > 1. The value of c does not play any role and we take t as 2 for smplcty.

7 L p Estmates for Square Roots 165 It s clear that b s supported n 2Q. Let us estmate 2Q b p. Introduce Q the cube centered at x wth sdelength 8l. Then 2Q Q. Set c = m 2Q f and c = m Q f and wrte b = (f c )X + (c c )X + ( c f(x ))X. By (2.6) and (2.5) for the cubes 2Q, c f(x ) Cαl, hence 2Q c f(x ) p X p Cα p 2Q. Next, usng the L p -Poncaré nequalty and the fact that Q F s not empty, c c 1 f c Cl 2Q Q ( 1 Q Q f p ) 1/p Cαl. Hence, 2Q c c p X p Cα p 2Q. Lastly, snce ( (f c )X ) = X f + (f c ) X, we have agan by the L p -Poncaré nequalty and the fact that the average of f p on 2Q s controlled by Cα p that 2Q ( (f c )X ) p Cα p 2Q. Thus (2.3) s proved. Set h(x) = f(x ) X (x). Note that ths sum s locally fnte and h(x) = 0 for x F. Note also that X (x) s 1 on Ω and 0 on F. Snce t s also locally fnte we have X (x) = 0 for x Ω. We clam that h(x) Cα. Indeed, fx x Ω. Let Q j be the Whtney cube contanng x and let I x be the set of ndces such that x 2Q. We know that I x N. Also for I x we have that C 1 l l j Cl and x x j Cl j where the constant C depends only on dmenson (see [St]). We have h(x) = (f(x ) f(x j )) X (x) C f(x ) f(x j ) l 1 CNα, I x I x by the prevous observatons. It remans to obtan (2.1) and (2.2). We easly have usng X (x)= 0 for x Ω, that f = ( f)1 F + h + b, a.e.. Now b s a well-defned dstrbuton on R n. Indeed, for a test functon u, usng the propertes of the Whtney cubes, ( ) b u C b (x) l 1 u(x) d(x, F ) dx

8 166 P. Auscher and the last sum converges n L p as a consequence of (2.4) and Lemma 8. Set p = np n p f p < n and p = otherwse, then for all real numbers r wth p r p, r (2.8) b l 1 Cα r Q. r Admt ths lemma and set g = f b. Then g = ( f)1 F + h n the sense of dstrbutons and, hence, g s a bounded functon wth g Cα. Proof of Lemma 8: By (2.5) and the Poncaré-Sobolev nequalty: r b l 1 N b l 1 r r NC l rθ b r p r where θ = n r n p. By (2.3), lrθ b r p αr l nr/p, hence r b l 1 CNα r l n. r Proof of Lemma 7: Let x be a pont n F. Fx such cube Q wth center x and let Q k be co-centered cubes wth l(q k ) = 2 k l(q) for k a negatve nteger. Then, by Poncaré s nequalty m Qk+1 f m Qk f 2 n m Qk+1 (f m Qk+1 f) C2 n l(q k )(m Qk+1 f p ) 1/p C2 k l(q)α snce Q k+1 contans x F. It easly follows that m Q f has a lmt as Q tends to 0. If, moreover, x s n the Lebesgue set of f, then ths lmt s equal to f(x). Redefne f on the complement of the Lebesgue set n F so that m Q f tends to f(x) wth Q centered at x wth Q 0. Morevover, summng over k the prevous nequalty gves us (2.6). To see (2.7), let Q x be the cube centered at x wth sdelength 2 x y and Q y be the cube centered at y wth sdelength 8 x y. It s easy to see that Q x Q y. As before, one can see that m Qx f m Qy f Cα x y. Hence by the trangle nequalty and (2.6), one obtans (2.7) readly.

9 L p Estmates for Square Roots 167 Remark. Note that ths argument can be adapted on a space of homogeneous type where a noton of gradent s avalable (see, e.g., [HaK]). Remark. Lemma 7 mples that f s Lpschtz on F wth Lpschtz constant α; t should be compared wth that of M. Wess (see [C, Lemma 1.4]) whch gves a slghtly stronger result but only for p > n. Remark. Note that f p > n then there s a sup norm estmate as follows p b l η Cα p Q, wth η = 1 n p. 3. Proof of Theorems 2 and 3 Let L be as n the Introducton. For 1 ρ, we say that L satsfes (S ρ ) f (3.1) C ρ 0 t > 0 f L 2 L ρ (R n ) e tl f ρ C ρ f ρ. Of course (S 2 ) holds by constructon wth C 2 = 1. The proof of the next lemma s defered to Secton 4. Lemma 9. If n = 1 or n = 2, then (S ρ ) holds for all ρ. If n 3, (S ρ ) holds for p n ε < ρ < (p n ε) for some ε > 0 dependng only on dmenson and the ellptcty constants. Theorem 2 and Theorem 3 are therefore a consequence of the next result combned wth Marcnkewcz nterpolaton. Theorem 10. Let n 1. Assume that (S ρ ) holds for some ρ [1, 2). Let ρ = nρ n+ρ. Then we have (3.2) (3.3) L 1/2 f p, f p, f 1 ρ < p < 2, L 1/2 f 1, f 1, f ρ < 1. Proof: By Lemma 9, one can always assume that ρ < p n. Let p = 1 f ρ < 1 and ρ < p < p n otherwse. Let f C0 (Rn ). We have to establsh the weak type estmate (3.4) {x R n ; L 1/2 f(x) > α} C α p f p, for all α > 0. We use the followng resoluton of L 1/2 : L 1/2 f = c 0 e t2l Lf dt

10 168 P. Auscher wth c = π 1/2 whch we omt from now on. It suffces to obtan the result for the truncated ntegrals R ε... wth bounds ndependent of ε, R, and then to let ε 0 and R. For the truncated ntegrals, all the calculatons are justfed. We gnore ths step and argue drectly on L 1/2. Apply the Calderón-Zygmund decomposton of Lemma 6 to f at heght α p and wrte f = g+ b. By constructon, g p c f p. Interpolatng wth (2.2) yelds g 2 cα 2 p f p. Hence { x R n ; L 1/2 g(x) > α } C 3 α 2 C α 2 L 1/2 g 2 g 2 C α p f p where we used the L 2 -estmate (1.3) for square roots. To compute L 1/2 b, let r = 2 k f 2 k l = l(q ) < 2 k+1 and set T = r 0 e t2l L dt and U = r e t2l L dt. It s enough to estmate A = {x R n ; T b (x) > α/3} and B = {x R n ; U b (x) > α/3}. Let us bound the frst term. Frst, { } A 4Q + x R n \ 4Q ; T b (x) > α 3, and by (2.4), 4Q C α p f p. To handle the other term, we need the followng lemma (not optmal but suffcent for our needs), whose proof s defered to Secton 4. Lemma 11. If (S ρ ) holds, for ρ < q < r < 2n n 2 (set 2n n 2 = f n 2) then for all closed sets E and F, all h L q (R n ) wth support n E and all t > 0, we have (3.5) e t2l t 2 Lh Lr (F ) C t cd(e,f ) 2 e t γ 2 h q wth γ = n q n r and d(e, F ) the dstance between E and F, where the constants C, c depend unquely on n, λ, Λ, C ρ, q, r.

11 L p Estmates for Square Roots 169 Let q = 2 f n = 1 and q = p = np n p, the Sobolev exponent for the embeddng Ẇ 1,p (R n ) L q (R n ), f n 2. Observe that ρ < q 2 by our choce of p. Now, { } x R n \ 4Q ; T b (x) > α 3 C q α q h wth h = 1 c T (4Q) b. To estmate the L q norm, we dualze aganst u L q (R n ) wth u q = 1 and follow the calculatons n [BK2]: u h = A j j=2 where A j = T b u, F j F j = 2 j+1 Q \ 2 j Q. Choose a number r wth q < r < 2n n 2. By Mnkowsk ntegral nequalty and Lemma 11 wth F = F j, E = Q and h = b T b Lr (F j ) c r 0 e t2l Lb Lr (F j) dt r c4 C j r 2 c e t 0 tγ+2 2 dt b q, where we used r l. By Poncaré-Sobolev nequalty and (2.3), b q cl 1 ( n p n q ) b p cαl 1+ n q, hence for some approprate constants C, c, T b L r (F j) Cαe c4j l n r. Now remark that for any y Q and any j 2, ( ) 1/r ( ) 1/r ( u r u r (2 n(j+1) Q ) 1/r M( u r )(y) F j 2 j+1 Q Applyng Hölder nequalty, one obtans ( A j Cα2 nj/r e c4j l n M( u r )(y) ) 1/r. ) 1/r.

12 170 P. Auscher Averagng over Q yelds A j Cα2 nj/r e c4j Q ( ) 1/r M( u r )(y) dy. Summng over j 2 and, we have u ( h Cα 1 Q (y) M( u r )(y) ) 1/r Applyng Hölder s nequalty wth exponent q, q and the maxmal theorem snce q > r, one obtans u h Cα 1 Q. q Hence { } x R n \ 4Q ; T b (x) > α 3 C q 1 Q C α p f p by (2.5) and (2.4). It remans to handlng the term B. Usng functonal calculus for L one can compute U as r 1 ψ(r 2 L) wth ψ the holomorphc functon on the sector arg z < π/2 gven by (3.6) ψ(z) = 1 e t2z z dt. It s easy to show that ψ(z) C z 1/2 e c z 2, unformly on subsectors arg z µ < π/2. We nvoke the followng lemma, whose proof s also defered to Secton 4. Lemma 12. If (S ρ ) holds then for ρ < q 2 ( ) 1/2 (3.7) ψ(4 k L)β k C β k 2, k Z q k Z q whenever the rght hand sde s fnte. The constant C depends on n, λ, Λ, C ρ, q. To apply ths lemma, observe that the defntons of r and U yeld U b = ψ(4 k L)β k k Z q dy.

13 L p Estmates for Square Roots 171 wth β k = b. r,r =2 k Usng the bounded overlap property (2.5), one has that ( ) 1/2 q β k 2 b q C r q. k Z By Lemma 8, together wth l r, b q r q Cα q Hence, by (2.4) { } x R n ; U b (x) > α 3 C q Q. Q C α p f p. 4. Proof of techncal lemmata Let L be as n the Introducton. There exsts ω [0, π/2) dependng only on the ellptcty constants such that L s of type ω (see, e.g. [AT1]) on L 2 (R n ). Ths mples that for some holomorphc functons f n the open sectors Σ µ = {z C ; arg z < µ}, f(l) can be defned as a bounded operator on L 2 (R n ). Snce L s maxmal-accretve, t has an H (Σ π/2 )-functonal calculus on L 2 (R n ). The semgroup (e tl ) has an analytc extenson to a complex semgroup (e zl ) of contractons on L 2 (R n ) for z Σ π/2 ω. Lemma 13. There s an r (1, 2) dependng on dmenson and the ellptcty constants only, such that L extends to a bounded and nvertble operator from Ẇ 1,p (R n ) onto Ẇ 1,p (R n ) and I +L extends to a bounded operator from W 1,p (R n ) onto W 1,p (R n ) for p < r. Recall that the homogenous Sobolev space Ẇ 1,p (R n ) s the closure of C0 (Rn ) for f p when 1 < p < and Ẇ 1,p (R n ) s ts dual space. Ths lemma s n [AT1]. It can be seen by two methods: ether by a drect comparson between L and the Laplacan operator after renormalsaton of the coeffcents of L and r can be estmated n terms of A I and the norms of the classcal Resz transforms j ( ) 1/2 actng on L p (R n ); or by an abstract nterpolaton method due to do I. Sneberg relyng on the Schwarz lemma for holomorphc functons.

14 172 P. Auscher Lemma 14. Let r = nr n+r. Let ρ 2 be such that ρ > r f r 1 or ρ = 1 f r < 1. For µ (ω, π/2) and z Σ π/2 µ, (4.1) e zl f 2 + e zl f 2 C z ( n ρ n 2 ) f ρ, f L 2 L ρ (R n ) where C dependng only on dmenson, ellptcty, ρ and µ. Proof: Assume frst that z = 1. By Lemma 13 and the Sobolev embeddng theorem, n a fnte number of steps (1+L) k extends to a bounded map from L ρ (R n ) nto L 2 (R n ). Note that k s depends only on r, hence ellptcty, and dmenson. Let f L 2 L ρ (R n ). Snce f s n L 2 (R n ), the equalty e zl f = e zl (I + L) k (I + L) k f s justfed. As e zl (I +L) k extends to bounded operator on L 2 (R n ), we have obtaned that e zl f ρ C f ρ, wth a constant C that depends only on ellpcty, dmenson, ρ and µ. If z 1, then the affne change of varable n R n defned by g(x) = f( z 1/2 x) gves us e zl f(x) = (e arg z L z g)( z 1/2 x) wth L z the second order operator wth coeffcents A( z 1/2 x). Snce L z has same ellptcty constants as L, the prevous bound apples and yelds the desred estmate. The same argument apples to L. Let us now recall the followng well-known results although not explctely stated as such n the lterature. Proposton 15. Let p [1, 2) and n If (S p ) holds then e tl : L p L 2 wth norm bounded by Ct γp/2, γ p = n p n If e tl : L p L 2 wth norm bounded by Ct γp/2, then for all q (p, 2) there s a sector Σ ν for whch we have the followng L q L 2 off-dagonal bounds: for all closed sets E and F, all h L q (R n ) wth support n E and all z Σ ν for some ν > 0, we have (4.2) e zl h L2 (F ) C cd(e,f ) 2 e z h z γq/2 q. 3. If L p L 2 off-dagonal bounds above hold then L satsfes (S p ). Proof: The proof of 1 s classcal from Nash type nequaltes. Brefly, we start from the Gaglardo-Nrenberg nequalty f 2 2 C f 2α 2 f 2β p

15 L p Estmates for Square Roots 173 wth α + β = 1 and (1 + γ p )α = γ p. By ellptcty e tl f 2 2 λ 1 R A e tl f, e tl f (2λ) 1 d dt e tl f 2 2. If f L 2 L p wth f p = 1, set ϕ(t) = e tl f 2 2. It s a non-ncreasng functon. Usng (S p ), one has ϕ(t) 1/α Cϕ (t). Integratng between t and 2t one fnds easly that ϕ(t) Ct α α 1, whch s the desred estmate. The proof of 2 conssts n nterpolatng by the complex method the L p L 2 boundeness assumpton wth the L 2 L 2 off dagonal bounds [Da], [AHLMcT]: for all closed sets E and F, all h L 2 (R n ) wth support n E and all z Σ µ, µ π 2 ω, we have (4.3) e zl cd(e,f )2 h L2 (F ) Ce z h 2. The proof of 3 can be seen by adaptng the one of Theorem 25 n [Da] to our stuaton. We now prove the lemmata stated n Secton 3. Proof of Lemma 9: The case of dmensons n = 1, 2 s n [AMcT]. For n 3, t suffces to combne Lemma 14 and Proposton 15. Proof of Lemma 11: The assumpton s that (S ρ ) holds. And we know (S p ) holds at least for p < 1 n. It follows from Proposton 15 that (4.4) e zl h Lr (F ) C n z q e n r 2n n 2 cd(e,f )2 z h q for any q, r wth ρ < q r < and arg z ν for some ν > 0 whenever h s supported n E. Then (3.5) follows by analytcty of the semgroup on L 2. Proof of Lemma 12: Dualzng, (3.7) s equvalent to the square functon estmate, ( ) 1/2 (4.5) ψ(4 k L )u 2 C u q k Z where L s the adjont of L. Now, t s proved n [BK1, Theorem 1.1], that off-dagonal estmates (4.4) mply that L has a bounded holomorphc functonal calculus on L q for ρ < q < 2 (see also Proposton 2.3 of ths same paper). By dualty, L has a bounded holomorphc functonal q

16 174 P. Auscher calculus on L q. But t s proved n [CDMcY] that ths mples (4.5) for a class of functons ψ contanng ours (see also [LeM] for an nformatve dscusson on ths). Ths proves Lemma 12. Proof of Proposton 1: Lemma 9 gves us a range (p n ε, 2) of ρ for whch (S ρ ) holds. Hence we obtan (4.4) and t suffces to apply [BK2, Theorem 1.1] to obtan L p -boundedness the Resz transforms for p (p n ε, 2), hence for p (p n ε, p n ]. 5. Comments on earler L p results Let us begn wth earler results on the Resz transform L p -boundedness. They hold n the range of 1 < p < 2 f the Gaussan property holds (see [AT1]), whch means that the kernel of the semgroup e tl satsfes a global (n tme) Gaussan upper bound together wth Hölder regularty n the space varables. In that case, the Resz transform s bounded on the Hardy space H 1. Ths apples for example to any real ellptc operator, symmetrc or not. The L p result holds under the weaker hypothess that the Gaussan upper bound hold, and n that case t s weak type (1, 1). Then, the next results are for a range ρ < p < for some ρ 1. Proposton 2.3 of [BK2] (see also the argument of Proposton 1 above) mples that f (S ρ ) s satsfed then weak type (p, p), hence strong type (p, p) holds n ths range. In partcular, the result apples to ρ = 1 whch gves an mprovement over [DMc] L p -boundedness result (whereas t does not yeld weak type (1, 1) as n [DMc]). Indeed, the Gaussan upper bound (assumed n [DMc]) mples (S 1 ) but the converse s not known n general. It s true f C 1 = 1,.e. the semgroup s contractng on L 1, as by [ABBO] ths s equvalent to the fact that L s real. But we are not aware f (S 1 ) wth C 1 > 1 mples the Gaussan upper bound. The L p -boundedness of Resz transforms for p > 2 cannot hold n general even for real and symmetrc operators. There s always an mprovement from L 2 -boundedness to L 2+ε -boundedness wth an ε dependng on the operator. We study related problems n a subsequent work [A]. Let us menton the results of [ERS] where t s proved that L p boundedness holds for all p (1, ) when the coeffcents are contnuous and perodc wth the same perod.

17 L p Estmates for Square Roots 175 Let us now turn to the reverse nequalty, that s L 1/2 f p f p. As we sad, t follows by dualty from the L p -boundedness of the Resz transform assocated to L (agan, we shall say more on ths n our forthcomng work). Ths gves one results for p > 2, but not for p < 2. In [AT1], t s proved that t holds for all p (1, ) provded the Gaussan property holds. In fact, there s a qute remarkable factorzaton of L 1/2 as T where T s a Calderón-Zygmund operator. Agan, ths apples to any real ellptc operator, symmetrc or not. The next avalable result s from [AHLMcT, Proposton 6.2]. There, a Hardy space H 1 estmate s obtaned, hence L p estmates for 1 < p < 2 by nterpolaton, under the hypothess of unform L ρ estmates for the resolvent (I +t 2 L) 1 for some ρ < n n 1. As mentoned n the Introducton, ths covers all dmensons up to and ncludng 4. But the method of proof seems lmted to such an hypothess. As we see the hypothess there s on the resolvent and here on the semgroup. They are essentally equvalent, up to changng the ρ s. The hypothess on the resolvent s mpled by (S ρ ) for the same ρ, usng the Laplace transform to compute resolvents from semgroups. Next by analytcty and complex nterpolaton, unform L ρ estmates for the resolvent (I + t 2 L) 1 for some ρ < 2 mples (S ρ ) for any ρ < ρ < 2. Thus, Theorem 10 s an extenson of Proposton 6.2 of [AHLMcT]. 6. Calderón-Zygmund lemma for W m,p -functons Lemma 16. Let n 1, 1 p and f D (R n ) be such that m f p <. Let α > 0. Then, one can fnd a collecton of cubes (Q ), functons g and b such that (6.1) f = g + b and the followng propertes hold: (6.2) (6.3) (6.4) (6.5) m g Cα, b W m,p 0 (Q ) and m b p Cα p Q, Q Q Cα R p m f p, n 1 Q N, where C and N depends only on dmenson and p.

18 176 P. Auscher Here m f s the array of all partal dervatves D γ f of f wth order m, m f p s the L p -norm of ts length, the latter beng computed wth any convenent norm on a fnte dmensonal space. The space W m,p 0 (Ω) denotes the closure of C0 (Ω) n W m,p (Ω). Note that the assumpton on f mples by classcal regularty results that f s locally ntegrable. Proof: If p =, set g = f. Assume next that p <. Let Ω = {x R n ; M( m f p )(x) > α p } where M s the uncentered maxmal operator over cubes of R n. If Ω s empty, then set g = f. Otherwse, the maxmal theorem gves us Ω Cα p R n m f p. Let F be the complement of Ω. Let (Q ) be a Whtney decomposton of Ω by dyadc cubes. Hence, Ω s the dsjont unon of the Q s, the cubes 2Q are contaned n Ω and have the bounded overlap property, but the cubes 4Q ntersect F. Hence (6.4) and (6.5) are satsfed by the cubes 2Q. Let us now defne the functons b. Let (X ) be a partton of unty on Ω assocated to the coverng (Q ) so that for each, X s a C m functon supported n 2Q wth l γ D γ X c(n, m) for all multndces γ wth γ m, l beng the sdelength of 2Q. If Q s a cube, let P Q f be the Poncaré polynomal of f wth degree less than or equal to m 1 assocated to Q. Set b = (f P 2Q f)x, and g = f b. It remans to establsh the desred propertes on b and g. Frst, we recall some propertes of P Q f, followng [Mor] (see [GM] for a short presentaton). It s unquely defned by the relatons Q Dγ (f P ) = 0 for all multndces γ wth γ m 1. For Q fxed, t s lnearly dependent on f. We have the unform estmates (6.6) sup D γ P Q f C(n, m)l n D β f Q β>γ, β m 1 Q and the Poncaré nequaltes (6.7) D γ (f P Q f) p C(n, m, p)l p(m γ ) Q Q m f p for all γ wth γ m 1, where C(n, m), C(n, m, p) are unversal constants ndependent of f, Q and l s the sdelength of Q.

19 L p Estmates for Square Roots 177 It s clear that b s supported n 2Q. Let us estmate 2Q m b p. It suffces to apply the Lebnz rule and to nvoke the Poncaré nequaltes to each term. Ths readly yelds (6.3). The next step s to verfy that b s a dstrbuton. Indeed, for a test functon u, usng the propertes of the Whtney cubes, ( ) b u C b (x) l m u(x) d(x, F ) m dx and the last sum converges n L p as a consequence of (6.4) and np n mp Lemma 17. Set p = f mp < n and p = otherwse. Then for all real numbers r wth p r p, r (6.8) b l m Cα r Q. r Proof of Lemma 17: By (6.5) and the Poncaré-Sobolev nequalty: r b l m N b l m r r NC l rθ m b r p r where θ = n r n p. By (6.3), lrθ m b r p αr l nr/p, hence r b l m CNα r l n. r Hence g = f b s well-defned. Let us compute m g. Recall that X (x) s 1 on Ω and 0 on F. Snce t s also locally fnte we have Dγ X (x) = 0 for x Ω for all γ wth 1 γ m. Hence, f γ = m, ths and the Lebnz rule yeld n the sense of dstrbutons on R n, D γ b = D γ f X + c β,γ D β (P f)d γ β X. β<γ, β 1 Fx a multndex β and set h = h β,γ = Dβ (P f)d γ β X. We show that h Cα. Admttng ths, we obtan D γ g s a bounded functon wth D γ g = (D γ f)1 F c β,γ h β,γ, β<γ, β 1 almost everywhere and ths gves us (6.2). Note that the sum defnng h s locally fnte on Ω and h(x) = 0 for x F. Let x Ω and Q j be the Whtney cube contanng x and let I x be the set of ndces such that x 2Q. We know that I x N. For I x we have that C 1 l l j Cl and z y Cl j for z Q

20 178 P. Auscher and y Q j, where the constant C depends only on dmenson (see [St]). Let x j be a pont n F 4Q j and let Q j be the smallest cube centered at x j contanng all of the cubes 2Q for I x. It s easy to see that ts lengthsde l j s comparable to l j. As γ β 0, we may wrte h(x) = I x D β (P 2Q f P Qj f)(x)d γ β X (x) so that the concluson wll follow from (6.9) D β (P 2Q f P Qj f)(x) C( l j ) m β ( Qj m f p ) 1/p wth a constant C ndependent of x, f, as Q j m f p M( m f p )(x j ) Q j α p Q j. Frst remark that by constructon and unqueness P 2Q (P Qj f) = P Qj f. Hence the left hand sde of (6.9) s domnated by sup 2Q D β (P 2Q (f P Qj f)). By (6.6), ths s controlled by constant tmes sums of l n D β (f P Qj f) 2Q wth β > β and β m 1. Usng that 2Q Q j, l l j, and the Poncaré nequaltes (6.7), ths last expresson s domnated by the rght hand sde of (6.9) as desred. 7. Results for hgher order operators Consder an homogeneous ellptc operator L of order m, m N, m 2, defned by (7.1) Lf = ( 1) m α (a αβ β f), α = β =m where the coeffcents a αβ are complex-valued L functons on R n, and we assume (7.2) a αβ (x) β f(x) α ḡ(x) dx R n Λ m f 2 m g 2 α = β =m

21 L p Estmates for Square Roots 179 and the Gårdng nequalty (7.3) Re a αβ (x) β f(x) α f(x) dx λ m f 2 2 R n α = β =m for some λ > 0 and Λ < + ndependent of f, g H m (R n )=W m,2 (R n ). Agan, L s constructed as before as a maxmal-accretve operator. It has a square root. The man result of [AHMcT] s (7.4) L 1/2 f 2 m f 2, n 1. If 2m n 1, t s a consequence of the method of [AT1] for second order operators that the followng L p a pror nequaltes hold: (7.5) (7.6) m f p L 1/2 f p, 1 < p < 2 + ε, L 1/2 f p m f p, 1 < p <. Ths s sharp n the range of p s. Further, f n = 1, (7.5) and (7.6) are true for 1 < p <. Let us now restrct ourselves to the case 2m < n. Frst the above nequaltes hold f the semgroup satsfes the Gaussan property. It s lkely that the method n [DMc] extends to gve us the Resz transform weak type (1, 1) estmate f only a Gaussan upper bound holds. In ther recent mentoned work [BK2], Blunck and Kunstmann establsh that (7.7) m f p L 1/2 f p, p(n, m) < p < 2 wth p(n, m) the Sobolev exponent for the Sobolev embeddng W m,p L 2 : p(n, m) = 2n 2m+n. Note that p(n, m) 1 s exactly the condton n 2m so that ther result recovers the part p < 2 of (7.5). In fact ther result, together wth the relaton between (S ρ ) and L p L q offdagonal estmates (whch are somehow abstract and apply as well to hgher order operators), states as follows: f (S ρ ) holds for some ρ [1, 2) then (7.7) s vald for ρ < p < 2. Next t s true that L p L q offdagonal estmates holds for p = p(n, m) and q = p(n, m), hence (S ρ ) for any ρ [p(n, m), p(n, m) ] [Da, Theorem 25]. In fact, as for second order operators, we observe that (S ρ ) always holds n an extended range p(n, m) ε < ρ < (p(n, m) ε) for some ε dependng only on dmenson and the ellptcty constants, hence the Resz transform L p -boundedness s vald n the range p(n, m) ε < ρ < 2. We are nterested n the reverse nequalty and we obtan the followng result.

22 180 P. Auscher Theorem 18. Let n 1 and m 2. We have for P (n, m) = 2n n+4m (7.8) L 1/2 f p m f p, n 1, sup(p (n, m) ε, 1) < p < 2. Furthermore, f P (n, m) 1 that s 1 n 4m, then we have the weak type (1, 1) estmate (7.9) L 1/2 f 1, m f 1. We observe that the proof works for 2m n and 2m < n as well. Note also that P (n, m) s the Sobolev exponent for the embeddng W m,p L p(n,m) f P (n, m) 1. The consequences are the same as for second order operators extendng what was known when 2m n. Corollary 19. (7.10) f p L 1/2 f p, n 2, sup(p(n, m) ε, 1) < p < 2 + ε. Furthermore, the operator L 1/2, a pror defned on C 0 (Rn ) wth values n L 2 (R n ), extends to a bounded an nvertble operator from Ẇ m,p (R n ) onto L p (R n ) for p n the above range. The method of proof of Theorem 18 parallels that for second order operators. Frst, we observe that (S ρ ) s vald when p(n, m) ε < ρ < (p(n, m) ε). So t suffces to show that f (S ρ ) holds for some ρ [1, 2), ρn ρm+n then (7.8) holds for nf(ρ, 1) < p < 2 where ρ = (ths s the Sobolev exponent p for W m,p L ρ when ρ 1) and furthermore (7.9) holds when ρ < 1. Next, one can always assume that ρ < p(n, m). Let p = 1 f ρ < 1 and ρ < p < p(n, m) otherwse. Let f C0 wth the L 2 -result, t suffces to establsh the weak type estmate (7.11) {x R n ; L 1/2 f(x) > α} C α p m f p, (Rn ). By nterpolatng for all α > 0. To take care of the parabolc homogenety, we resolve L 1/2 by L 1/2 f = c 0 e t2ml Lf d(t m ) wth c = π 1/2 whch we omt from now on. Agan, a rgorous argument would be to truncate the ntegral away from 0 and. Apply the Calderón-Zygmund decomposton of Lemma 16 to f at heght α p and wrte f = g + b. By constructon, m g p c m f p. Interpolatng

23 L p Estmates for Square Roots 181 wth (6.2) yelds m g 2 cα 2 p m f p. Hence { x R n ; L 1/2 g(x) > α } C 3 α 2 L 1/2 g 2 C α 2 m g 2 C α p m f p where we used the L 2 -estmate for square roots. To compute L 1/2 b, let r = 2 k f 2 k l = l(q ) < 2 k+1 and set T = r 0 e t2ml L d(t m ) and U = r e t2ml L d(t m ). It s enough to estmate A = {x R n ; T b (x) > α/3} and B = {x R n ; U b (x) > α/3}. Let us bound the frst term. Frst, { } A 4Q + x R n \ 4Q ; T b (x) > α 3, and by (2.4), 4Q C α p m f p. To handle the other term, we nvoke the off-dagonal estmates for hgher order operators whch can be obtaned as for second order operators usng the same arguments and [Da]: 2n n 2m Lemma 20. If (S ρ ) holds then for ρ < q < r < (f n 2m, then 2n set n 2m = ) for all closed sets E and F, all h Lq (R n ) wth support n E and all t > 0, we have ) h q (7.12) e t2ml t 2m Lh L r (F ) C t γ G ( cd(e, F ) t wth γ = n q n r and d(e, F ) the dstance between E and F, where the constants C, c depend unquely on n, λ, Λ, C ρ, q, r, and G(u) = exp( u 2m 2m 1 ). Let q = nf(2, p ) where p = np n mp s the Sobolev exponent for the embeddng Ẇ m,p (R n ) L p (R n ). Observe that ρ < q 2 by our choce of p. Now, { } x R n \ 4Q ; T b (x) > α 3 C q α q h

24 182 P. Auscher wth h = 1 (4Q) c T b. Let u L q (R n ) wth u q = 1, then u h = A j j=2 where A j = T b u, F j F j = 2 j+1 Q \ 2 j Q. Choose a number r wth q < r < n 2m, then by Mnkowsk ntegral nequalty and Lemma 20 wth q and r, F = F j, E = Q and h = b T b L r (F j) r 0 r 0 2n e t2ml Lb L r (F j) d(t m ) C t γ+2m G CG(c2 j )r γ+m b q, ( c2 j ) r d(t m ) b q t where we used r l. By Poncaré-Sobolev nequalty and (6.3), b q cl m ( n p n q ) m b p cαl m+ n q, hence T b Lr (F j) CαG(c2 j )l n r, for some approprate constants C, c. Now remark that for any y Q and any j 2, ( ) 1/r ( ) 1/r ( ) 1/r u r u r (2 n(j+1) Q ) 1/r M( u r )(y). F j 2 j+1 Q Applyng Hölder nequalty, one obtans ( A j Cα2 nj/r G(c2 j )l n M( u r )(y) Averagng over Q yelds A j Cα2 nj/r G(c2 j ) Q ) 1/r ( ) 1/r M( u r )(y) dy. Summng over j 2 and, we have u ( h Cα 1 Q (y) M( u r )(y). ) 1/r dy.

25 L p Estmates for Square Roots 183 Applyng Hölder nequalty wth exponent q, q and the maxmal theorem snce q > r, one obtans u h Cα 1 Q. q Hence { } x R n \ 4Q ; T b (x) > α 3 C q 1 Q C α p m f p by (6.4) and (6.5). It remans to handlng the term B. Agan, one has U = r m ψ(r 2m L) wth ψ gven by (3.6). We nvoke the followng lemma whch can be proved exactly as for second order operators usng recent results n [BK1]. Lemma 21. If (S ρ ) holds then for ρ < q 2 ( ) 1/2 (7.13) ψ(2 km L)β k C β k 2, k Z q k Z q whenever the rght hand sde s fnte. The constant C depends on n, λ, Λ, C ρ, q. wth To apply ths lemma, observe that the defntons of r and U yeld U b = ψ(2 km L)β k k Z β k = b. r,r =2 k Usng the bounded overlap property (6.5), one has that ( ) 1/2 q β k 2 b q C r q. k Z By Lemma 17, together wth l r, b q r q Cα q Q. q q

26 184 P. Auscher Hence, by (6.4) { } x R n ; U b (x) > α 3 C Q C α p m f p. 8. Concludng remarks For hgher order operators, (7.3) s often replaced by (8.1) Re a αβ (x) β f(x) α f(x) dx λ m f 2 2 κ f 2 2 R n α = β =m for some λ > 0 and κ 0 ndependent of f, g H m (R n ) = W m,2 (R n ). Now L + κ s constructed as before as a maxmal-accretve operator. It has a square root. By [AHMcT], f κ > κ, (8.2) (L + κ ) 1/2 f 2 m f 2 + f 2. The methods n [BK1] for the functonal calculus, n [BK2] or [HM] for the Resz transforms and n here can be adapted to ths stuaton. The Calderón-Zygmund decomposton s performed under the condton that f W m,p (R n ). To go back to an homogeneous stuaton, we set Df = (f, m f) and argue wth respect to the maxmal functon of Df p. We leave to the reader the care of statng the correspondng results, the condton (S ρ ) beng used only for small tmes t < 1. Fnally, all these methods can be appled to ellptc systems where ellptcty s n the sense of the valdty of the Gårdng nequalty. And the results are smlar. [A] [ABBO] References P. Auscher, On necessary and suffcent condtons for L p -estmates of Resz transforms assocated to ellptc operators on R n and related estmates, Preprnt 4, Unversté Pars XI-Orsay (2004). P. Auscher, L. Barthélemy, P. Bénlan and El M. Ouhabaz, Absence de la L -contractvté pour les semgroupes assocés aux opérateurs ellptques complexes sous forme dvergence, Potental Anal. 12(2) (2000), [AHLMcT] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and Ph. Tchamtchan, The soluton of the Kato square root problem for second order ellptc operators on R n, Ann. of Math. (2) 156(2) (2002),

27 L p Estmates for Square Roots 185 [AHMcT] [AMcT] [AT1] [AT2] [BK1] [BK2] [C] [CMcM] [CDMcY] [Da] [DMc] [ERS] [GM] P. Auscher, S. Hofmann, A. McIntosh and Ph. Tchamtchan, The Kato square root problem for hgher order ellptc operators and systems on R n, dedcated to the memory of Toso Kato, J. Evol. Equ. 1(4) (2001), P. Auscher, A. McIntosh and Ph. Tchamtchan, Heat kernels of second order complex ellptc operators and applcatons, J. Funct. Anal. 152(1) (1998), P. Auscher and Ph. Tchamtchan, Square root problem for dvergence operators and related topcs, Astérsque 249 (1998), 172 pp. P. Auscher and Ph. Tchamtchan, Calcul fontonnel précsé pour des opérateurs ellptques complexes en dmenson un (et applcatons à certanes équatons ellptques complexes en dmenson deux), Ann. Inst. Fourer (Grenoble) 45(3) (1995), S. Blunck and P. C. Kunstmann, Calderón-Zygmund theory for non-ntegral operators and the H functonal calculus, Rev. Mat. Iberoamercana 19(3) (2003), S. Blunck and P. C. Kunstmann, Weak-type (p, p) estmates for Resz transforms, Math. Z. (to appear). C. P. Calderón, On commutators of sngular ntegrals, Studa Math. 53(2) (1975), R. R. Cofman, A. McIntosh and Y. Meyer, L ntégrale de Cauchy défnt un opérateur borné sur L 2 pour les courbes lpschtzennes, Ann. of Math. (2) 116(2) (1982), M. Cowlng, I. Doust, A. McIntosh and A. Yag, Banach space operators wth a bounded H functonal calculus, J. Austral. Math. Soc. Ser. A 60(1) (1996), E. B. Daves, Unformly ellptc operators wth measurable coeffcents, J. Funct. Anal. 132(1) (1995), X. T. Duong and A. McIntosh, Sngular ntegral operators wth non-smooth kernels on rregular domans, Rev. Mat. Iberoamercana 15(2) (1999), A. F. M. ter Elst, D. W. Robnson and A. Skora, On second-order perodc ellptc operators n dvergence form, Math. Z. 238(3) (2001), M. Gaqunta and G. Modca, Regularty results for some classes of hgher order nonlnear ellptc systems, J. Rene Angew. Math. 311/312 (1979),

28 186 P. Auscher [HaK] P. Haj lasz and P. Koskela, Sobolev met Poncaré, Mem. Amer. Math. Soc. 145(688) (2000), 101 pp. [HMc] S. Hofmann and A. McIntosh, The soluton of the Kato problem n two dmensons, n: Proceedngs of the 6th Internatonal Conference on Harmonc Analyss and Partal Dfferental Equatons (El Escoral, 2000), Publ. Mat. Vol. extra (2002), [HM] S. Hofmann and J. M. Martell, L p bounds for Resz transforms and square roots assocated to second order ellptc operators, Publ. Mat. 47(2) (2003), [K] T. Kato, Fractonal powers of dsspatve operators, J. Math. Soc. Japan 13 (1961), [Ke] C. E. Keng, Harmonc analyss technques for second order ellptc boundary value problems, CBMS Regonal Conference Seres n Mathematcs 83, publshed for the Conference Board of the Mathematcal Scences, Washngton, DC; by the Amercan Mathematcal Socety, Provdence, RI, [LeM] C. Le Merdy, Square functons assocated to sectoral operators, Bull. Soc. Math. France (to appear). [Mor] C. B. Morrey, Jr., Multple ntegrals n the calculus of varatons, De Grundlehren der mathematschen Wssenschaften 130, Sprnger-Verlag New York, Inc., New York, [St] E. M. Sten, Sngular ntegrals and dfferentablty propertes of functons, Prnceton Mathematcal Seres 30, Prnceton Unversty Press, Prnceton, N.J., Laboratore de Mathématques CNRS UMR 8628 Unversté de Pars-Sud Orsay Cedex France E-mal address: pascal.auscher@math.u-psud.fr Prmera versó rebuda el 8 de mag de 2003, darrera versó rebuda el 27 de gener de 2004.