Lp bounds for Riesz transforms and square roots associated to second order elliptic operators

Publ. Mat. 47 (2003), 497–515 LpBOUNDS FOR RIESZ TRANSFORMS AND SQUARE ROOTS ASSOCIATED TO SECOND ORDER ELLIPTIC OPERATORS Steve Hofmann∗and Jos´ e Mar´ ıa Martell† Abstract We consider the Riesz transforms ∇L−1/2, where L≡− divA(x)∇, and Ais an accretive, n×nmatrix with bounded measurable complex entries, defined on Rn.Weestablish boundedness of these operators on Lp(Rn), for the range pn<p≤2, where pn=2n/(n+ 2), n≥2, and we obtain a weak-type estimate at the endpoint pn. The case p=2was already known: it is equivalent to the solution of the square root problem of T. Kato. 1. Introduction Let A=A(x)=aj,k(x)j,k be an n×nmatrix where the coefficients aj,k are complex-valued L∞(Rn) functions. We assume that this matrix satisfies the following ellipticity (or “accretivity”) condition: there exist 0 <λ≤Λ<∞such that λ|ξ|2≤Re Aξ·¯ ξand |Aξ·¯ ζ|≤Λ|ξ||ζ|, for all ξ,ζ ∈Cn.Wehave used the notation ξ·ζ=ξ1ζ1+···+ξnζn and therefore ξ·¯ ζis the usual inner product in Cn. Note that then Aξ ·¯ ζ=j,k aj,k(x)ξk¯ ζj. Associated with this matrix we define the second order divergence form operator Lf =−div(A∇f), which is understood in the standard weak sense by means of a sesquilinear form. 2000 Mathematics Subject Classification. 42B20, 35J15. Key words. Riesz transforms, square roots of divergence form elliptic operators. ∗Partially supported by NSF. †Partially supported by MCYT Grant BFM2001-0189. 498 S. Hofmann, J. M. Martell The accretivity condition stated before allows one to define the square root L1 2=√L. Recently P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and P. Tchamitchian have obtained an affirmative answer to the so-called Kato square root problem for elliptic differential operators (see [AHLMT]). Namely, they have shown the following: Theorem 1.1 ([AHLMT, Theorem 1.4]).For any operator as before the domain of √Lcoincides with the Sobolev space H1 (Rn)and √Lf2∼ ∇f2. In [AT], assuming the result of Theorem 1.1, the authors obtain optimal Lpbounds for √Land for the associated Riesz transforms ∇L−1/2, under the extra hypothesis that one has a Gaussian upper bound for the heat kernel (along with “Nash-type” local H¨older continuity). In this paper, we consider the same problem without this extra hypothesis, i.e., we study the Lpboundedness of ∇L−1/2and √Lfor general second order elliptic operators Las above. Our results are new only in the case n≥3. Indeed, Gaussian bounds (and Nash’s estimates) always hold for the kernel of the semigroup e−tL in dimensions 1 and 2 [AMT], in which case the result of [AT] (or also that of [DM]) applies. In fact, the argument of [DM] requires only the Gaussian upper bounds, and not the H¨older continuity. We remark that in the presence of Gaussian bounds, the boundedness on Lqfor q≤2ofthe Riesz transforms associated to other operators (for example, the Laplace-Beltrami operator on a manifold) have been treated (see [CD]). For the operators under consideration in the present paper, it is known that the Gaussian bounds may fail in dimensions n≥5[AT, pp. 32–33] (also [MNP], where the example originates). Our main result says: Theorem 1.2. Let pn=2n n+2 . Then the Riesz transform ∇L−1 2is of weak type (pn,p n)and thus bounded on Lq(Rn)for pn<q≤2. Remarks. We learned during the preparation of this manuscript that the Lqboundedness of the Riesz transforms ∇L−1 2on the same range of q has also been obtained independently by S. Blunck and P. C. Kuntsmann [BK1], by essentially the same method as ours. Moreover, they have applied this technique to related matters, including Lpestimates for Riesz transforms associated to higher order elliptic operators [BK1], as well as to the existence of an H∞functional calculus [BK2]inLpspaces. We are grateful to Pascal Auscher and Alan McIntosh for bringing their work to our attention. LpBounds for Riesz Transforms 499 We should point out that p n=2 where 2=2n n−2is the Sobolev exponent of 2. On the other hand, we recall that, by a standard duality argument, the L2estimate of the Riesz transform is equivalent to √L∗f2≤C∇f2, where L∗is the adjoint operator of Lthat satisfies its same properties. In view of this fact, the L2estimate for the Riesz transform follows from Theorem 1.1 obtained in [AHLMT]. Let us also note that the problem is invariant with respect to the taking of adjoints, i.e., Land L∗are operators of the same nature and if one of them satisfies the required hypotheses then the other one also does. By using a standard duality argument, one has that from the boundedness of ∇L−1 2on some Lq(Rn)itfollows an Lq(Rn) domination of the square roots by the gradient. Then, as a consequence of our main result we get the following. Corollary 1.3. Let 2≤q< 2n n−2. Then, √Lfq≤C∇fq. The paper is organized as follows. In the next section we prove some technical estimates that will be used in the sequel. In Section 3 we prove our main theorem. 2. L2off-diagonal estimates Given E,Ftwo subsets of Rn,wewill denote by dist(E,F) the distance between them. We will use the notation  ffor n-tuples of functions. The identity operator will be written as I.Webegin by stating the fundamental average decay estimates satisfied by the heat kernels associated to our operator L. Lemma 2.1. Let Eand Fbe two closed sets of Rn. Then, for all t>0,  e−tLf L2(F)≤Ce −dist(E,F)2 ct fL2(E),supp f⊂E  tLe −tLf L2(F)≤Ce −dist(E,F)2 ct fL2(E),supp f⊂E  t1 2∇e−tLf L2(F)≤Ce −dist(E,F)2 ct fL2(E),supp f⊂E  t1 2e−tLdiv  f L2(F)≤Ce −dist(E,F)2 ct  fL2(E),supp  f⊂E where c>0depends only on λand Λ, and C>0on n,λand Λ. 500 S. Hofmann, J. M. Martell An analogous result for resolvent kernels (I+t2L)−1is proved in [AHLMT, Lemma 2.1], via an integration by parts argument similar to the proof of Cacciopoli’s inequality. The present lemma follows from the one for resolvent kernels by functional calculus. Alternatively, one can simply modify the integration by parts argument of [AHLMT], to obtain a direct proof of the present lemma, as in the proof of the parabolic Cacciopoli inequality. We omit the details. Lemma 2.2. Let m≥1be an integer (eventually, we shall choose m depending only upon n). Let Eand Fbe two closed sets of Rn. Then,   t1 2∇L−1 2I−e−tLmdiv  f  L2(F)≤Cdist(E,F)2 t−(m+1 2)  fL2(E), supp  f⊂E and   t1 2∇∇L−1 2I−e−tLm∗ f  L2(F)≤Cdist(E,F)2 t−(m+1 2)  fL2(E), supp  f⊂E where C>0depends only on n,m,λ,Λ, and ∇L−1 2I−e−tLm∗ f=−L−1 2I−e−tLm∗div  f.(1) To prove this result we need to establish an auxiliary lemma that says that the composition of two operators that satisfy two L2off-diagonal estimates as in Lemma 2.1 verifies a similar inequality. Lemma 2.3. Let {At}t>0and {Bt}t>0two families of linear operators. Assume that for all closed sets E,F, for all fsuch that supp f⊂Eand for all t>0we have the following estimates AtfL2(F)≤Ce −dist(E,F)2 ct fL2(E) and BtfL2(F)≤Ce −dist(E,F)2 ct fL2(E). Then, for all t, s > 0we have AtBsfL2(F)≤Ce −dist(E,F)2 cmax{t,s}fL2(E). LpBounds for Riesz Transforms 501 Proof: Set ρ= dist(E,F) and G={x: dist(x, F)<ρ/2}. Then, it is clear that dist(E,G)≥ρ/2 where we have written Gfor the topological closure of the set G. Besides, by definition dist(Rn\G, F)≥ρ/2. Then,  AtBsf·χG L2(F)≤ AtBsf·χG L2(Rn) ≤C Bsf L2(G) ≤Ce −dist(G,E)2 cs fL2(E) ≤Ce −ρ2 csfL2(E). Note that in the second inequality we have used that Atis uniformly bounded on L2(Rn), fact that follows from the hypotheses on Atby taking E=F=Rn. The third inequality is just the L2off-diagonal estimate assumed on Bs.Onthe other hand,  AtBsf·χRn\G L2(F)≤Ce −dist(Rn\G,F )2 ct BsfL2(Rn\G) ≤Ce −ρ2 ctBsfL2(Rn) ≤Ce −ρ2 ctfL2(E), where the first inequality is a consequence of the L2off-diagonal estimate for Atand the last one holds because, as before, Bsis uniformly bounded on L2(Rn). If we combine the two estimates we get: AtBsfL2(F)≤ AtBsf·χG L2(F)+ AtBsf·χRn\G L2(F) ≤Ce−ρ2 ct +e−ρ2 csfL2(E) ≤Ce −dist(E,F)2 cmax{t,s}fL2(E). Now we can give a proof of Lemma 2.2. Proof of Lemma 2.2: We prove the first estimate since the other one follows by duality. We use the following representation of the Riesz transform: ∇L−1 2h=1 2√π∞ 0∇e−sLhds √s=√m+2 2√π∞ 0∇e−(m+2) sLh√sds s 502 S. Hofmann, J. M. Martell which leads to ∇L−1 2I−e−tLmdiv  f=C∞ 0∇e−sLe−msLI−e−tLm e−sLdiv  f√sds s =Ct 0···+∞ t··· =C(It+IIt). Notice that we have used the commutation property of the semigroup. Now we study each operator separately. For the first one we have I−e−tLm= m  k=0 m j(−1)ke−ktL =I+ m  k=1 cke−ktL and then It=t 0∇e−sLe−msLe−sLdiv  f√sds s + m  k=1 ckt 0∇e−kt 2Le−(m+2) sLe−kt 2Ldiv  f√sds s =It,0+ m  k=1 ckIt,k. Then, It,0L2(F)≤t 0 ∇e−sLe−msLe−sLdiv  f L2(F)√sds s =t 0 s1 2∇e−sL◦e−msL◦s1 2e−sLdiv) f L2(F)s−1 2ds s ≤C fL2(E)t 0 e−dist(E,F)2 cms s−1 2ds s =C fL2(E)t−1 2∞ 1 e−dist(E,F)2 ct ss1 2ds s, where we have used Lemma 2.3 and noted that every operator satisfies the corresponding L2off-diagonal inequality due to Lemma 2.1. Now we LpBounds for Riesz Transforms 503 bound the integral: ∞ 1 e−dist(E,F)2 ct ss1 2ds s≤C∞ 1dist(E,F)2 ct s−(m+1 2) s1 2ds s =Cdist(E,F)2 t−(m+1 2) since m≥1. Then we have obtained It,0L2(F)≤Ct −1 2dist(E,F)2 t−(m+1 2)  fL2(E). Now, fix 1 ≤k≤m, then It,kL2(F)≤t 0 ∇e−kt 2Le−(m+2) sLe−kt 2Ldiv  f L2(F)√sds s =t 0    kt 2∇e−kt 2L◦e−(m+2) sL ◦kt 2e−kt 2Ldiv f    L2(F) 2 kt√sds s ≤C fL2(E)t−1t 0 e−dist(E,F)2 cmax{kt/2,(m+2) s}√sds s ≤C fL2(E)t−1e−dist(E,F)2 ct t 0 √sds s ≤Ct −1 2e−dist(E,F)2 ct  fL2(E) ≤Ct −1 2dist(E,F)2 t−(m+1 2)  fL2(E). Let us observe that because of Lemma 2.1 in the composition of the operators above each of them verifies an L2off-diagonal estimate. This fact allowed us to employ Lemma 2.3. We also used that 1 ≤k≤m. 504 S. Hofmann, J. M. Martell Collecting this estimate and the one proved for It,0we get ItL2(F)≤It,0L2(F)+ m  k=1 |ck|It,kL2(F) ≤Ct −1 2dist(E,F)2 t−(m+1 2)  fL2(E). Next, we proceed with the estimate of IIt. IItL2(F) ≤C∞ t √s∇e−sL◦e−sL −e−(s+t)Lm ◦√se −sLdiv  f L2(F)s−1 2ds s. We know that the first and the last operators satisfy an L2off-diagonal estimate. We are going to show not only that the one in the middle verifies a similar estimate but also that we gain some extra decay (t/s)m. Namely, let E,Fbe two closed sets and gsuch that supp g⊂E, then  e−sL −e−(s+t)Lg L2(F)=   −t 0 d dre−(s+r)Lgdr    L2(F) ≤t 0 (s+r)Le −(s+r)Lg L2(F) dr s+r ≤CgL2(E)t 0 e−dist(E,F)2 c(s+r)dr s+r ≤Ct se−dist(E,F)2 cs gL2(E), where in the last step we used that t≤s≤s+r≤s+t≤2s, and Lemma 2.1. Then we have proved that    s te−sL −e−(s+t)Lg  L2(F)≤Ce −dist(E,F)2 cs gL2(E) uniformly on t. Then this operator composed with itself mtimes will satisfy the same inequality in view of Lemma 2.3. We can use this LpBounds for Riesz Transforms 505 estimate to bound IItL2(F)as follows: IItL2(F)≤C fL2(E)∞ t e−dist(E,F)2 cs t sm s−1 2ds s =C fL2(E)t−1 21 0 e−dist(E,F)2 ct ssm+1 2ds s ≤C fL2(E)t−1 2∞ 0 e−dist(E,F)2 ct ssm+1 2ds s =C fL2(E)t−1 2dist(E,F)2 t−(m+1 2)∞ 0 e−ssm+1 2ds s ≤Ct −1 2dist(E,F)2 t−(m+1 2)  fL2(E). Collecting the estimates for Itand IItwe get  t1 2∇L−1 2I−e−tLmdiv  f L2(F)≤Ct1 2ItL2(F)+IItL2(F) ≤Cdist(E,F)2 t−(m+1 2)  fL2(E). 3. Proof of Theorem 1.2 The proof of this result is inspired by [DM], where a general method is developed to prove weak-type (1,1) estimates for singular integral operators without assuming explicit regularity on the space variables of the kernel. That method can be applied to derive the boundedness of the Riesz transforms if one assumes sufficient pointwise decay of the heat kernel. It is to be noted that in [DM], there is an extra assumption involving a pointwise estimate for the gradient of the heat kernel, but this hypothesis is superfluous, as observed in [CD]. Here, a further step is taken: we derive the bounds for the Riesz transforms from offdiagonal estimates on the heat semigroup that hold in an average, as opposed to a pointwise, sense. These average off-diagonal estimates hold for all operators of the type under consideration (this is the content of Lemma 2.1), in contrast to pointwise estimates which may fail. 512 S. Hofmann, J. M. Martell and we get the desired estimate for II3.Now we are concerned with II2. By Chebychev’s inequality we get (II2)1 2=   x∈E∗: j Djbj(x) >α    1 2 ≤1 α      j Djbj     L2(E∗) =1 αsup  hRn j Djbj(x), h(x)dx , (8) where ·,· denotes the inner product in Cnand the supremum is taken over all Cn-valued functions  h∈L2(E∗) with  hL2(E∗)=1. Let us fix such a function  h.Forevery j,wedefine as before  h(l,j)(x)=  h(x)χS(l,j)(x). Let us recall that E∗=Rn\∪ jQ∗ j.Inthis way, and since supp h⊂E∗⊂(2 Qj)cwe have Rn j Djbj(x), h(x)dx = j ∞  l=1 Rn!Djbj(x), h(l,j)(x)"dx = j ∞  l=1 Qj bj(x)D∗ j h(l,j)(x)dx = j ∞  l=1 Rn bj(x)D∗ j h(l,j)(x)−D∗ j h(l,j)Qjdx ≤ j ∞  l=1 bjLp(Qj)  D∗ j h(l,j)−D∗ j h(l,j)Qj  Lp(Qj) ≤Cα j ∞  l=1 |Qj|1 p  D∗ j h(l,j)−D∗ j h(l,j)Qj  Lp(Qj), (9) LpBounds for Riesz Transforms 513 where D∗ jis the operator given in (1) with t=tj, and we have used both properties in (3). We apply Poincar´e-Sobolev inequality and Lemma 2.2 to get   D∗ j h(l,j)−D∗ j h(l,j)Qj  Lp(Qj) ≤C  ∇D∗ j h(l,j)  L2(Qj) =C  ∇TI−e−tjLm∗ h(l,j)  L2(Qj) ≤Ct −1 2 jdist(S(l,j),Q j)2 tj−(m+1 2)  h(l,j)L2(S(l,j)) ≤C(Qj)−12−2(m+1 2)l hL2(S(l,j)), since tj=(Qj)2and for l≥1wehave dist(S(l, j),Q j)≥2l−2(Qj). We now plug this estimate into (9) and it follows that Rn j#Djbj(x), h(x)$dx ≤Cα j ∞  l=1 |Qj|1 p(Qj)−12−2(m+1 2)l hL2(S(l,j)) ≤Cα j ∞  l=1 |Qj|1 22−2(m+1 2)l|2l+1Qj|1 21 |2l+1Qj|2l+1Qj| h(y)|2dy1 2 ≤Cα j|Qj|ess inf y∈Qj M| h|2(y)1 2 ∞  l=1 2−l2(m+1 2)−n 2 ≤Cα {x∈Rn:M(fp)(x)1 p>α} 1 2, where in the last step we used that m>n−2 4and so 2 (m+1 2)−n 2>0, and we have proceeded as in (7). Then we can use this estimate, (8) and 514 S. Hofmann, J. M. Martell that Mis of weak type (1,1) to get II2≤C{x∈Rn:M(fp)(x)1 p>α}≤C αpRn f(x)pdx. The proof of the fact that Tis of weak type (p, p)isnow completed by collecting the estimates that we have obtained for I,II1,II2and II3. Note added in proof: P. Auscher has recently observed that the exponent pnmay be replaced by pn−,=(n, λ, Λ), n≥3[A]. References [A] P. Auscher,Onnecessary and sufficient conditions for Lp estimates of Riesz transforms associated to elliptic operators on Rn:asurvey, Preprint (2003). 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Plamenevski˘ ı, Absence of a De Giorgi-type theorem for strongly elliptic equations with complex coefficients, (Russian), Boundary value problems of mathematical physics and related questions in the theory of functions, 14, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 156–168, 309. Steve Hofmann: Department of Mathematics University of Missouri Columbia, MO 65211 U.S.A. E-mail address:[email protected] Jos´eMar´ıa Martell: Departamento de Matem´aticas, C-XV Universidad Aut´onoma de Madrid 28049 Madrid Spain E-mail address:[email protected] and Department of Mathematics University of Missouri Columbia, MO 65211 U.S.A. E-mail address:[email protected] Primera versi´o rebuda el 19 de desembre de 2002, darrera versi´o rebuda el 2 de maig de 2003.