The Dirichlet problem for elliptic equations with drift terms

Publ. Mat. 45 (2001), 199–217 THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH DRIFT TERMS Carlos E. Kenig and Jill Pipher Abstract We establish absolute continuity of the elliptic measure associated to certain second order elliptic equations in either divergence or nondivergence form, with drift terms, under minimal smoothness assumptions on the coefficients. 1. Introduction In this paper we prove some results on absolute continuity of the elliptic measure associated to a second order elliptic operator under certain natural, minimal conditions on the coefficients of these operators. Primarily, our operators Lare of divergence form; that is, L= div a∇ where a(X)=(aij(X)) is strongly elliptic in the sense that there exists a positive constant λsuch that λ|ξ|2≤ i,j aij(x)ξiξj<λ −1|ξ|2, for all Xand all  ξ∈Rn. However, thanks to some recent work of S. Hoffman and J. Lewis [HL], we can extend our results to operators with a drift term, i.e., L+b∇, under certain conditions on b. These conditions on bwill, in turn, yield information for non-divergence operators as well. One feature of these theorems is that we need not assume that the matrix (aij) is symmetric. Let us motivate the condition we shall place on the matrix aof L= div a∇by the following example. Consider the Laplacian ∆ = iDxixi, in a domain Ω above the graph of a Lipschitz function φ, and let dω denote the harmonic measure on the boundary of Ω, with respect to some fixed interior point. In [D1], Dahlberg proved the mutual absolute continuity of dω with respect to dσ, the surface measure on ∂Ω, and showed that the density k=dω/dσ Partially supported by the NSF. 200 C. E. Kenig, J. Pipher satisfies a reverse H¨older condition of order 2. One approach to studying this harmonic measure in domains with Lipschitz boundary is to map Ω to the upper half space Rn+1 +, and look at the resulting pull-back of ∆ under this transformation. If one simply “flattens” the domain Ω={(x, t)∈Rn×R|t≥φ(x)} by the map (x, t)→ (x, t−φ(x)), then the Laplacian is mapped to a symmetric divergence form operator L= div a∇whose coefficients a=(aij) are merely bounded and measurable, since they depend on the derivatives of the Lipschitz function φ. In general, the elliptic measures associated to the entire class of divergence structure elliptic operators, with bounded, measurable coefficients, may be singular with respect to the surface measure on the boundary. (See [CFK] for details.) But the operator Larising from the transformation above possesses additional structure —the coefficients aij are independent of the tvariable. It seemed reasonable to conjecture that this independence in the tvariable would imply that the general class of such operators would satisfy the same reverse H¨older estimates that the density for ∆ satisfies on ∂Ω. This was shown in [JK] and led to a series of works on exactly how to relax the requirement of independence in the tvariable. The paper [FKP] contains some sharp results in this direction, as well as many of the references to previous work. On the other hand, there is a more useful transformation discovered by B. Dahlberg, C. Kenig and E. Stein ([D2]) mapping Rn +to Ω of the form ρ(x, t)=(x, ct +θt∗φ(x)), where cis a constant that depends on ∇φ∞and can be chosen large enough to insure that ρis one-one. The function θ∈C∞ 0(Rn) is even, and θt(·)=t−nθ(·/t). The pullback of ∆ from Ω to Rn+1 +is also a symmetric elliptic operator, L= div a∇, where apossesses the properties: (i) |∇a(x, t)|≤C/t. (ii) t|∇a(x, t)|2dxdt is a Carleson measure. (For the definition of Carleson measure, see (2.3).) In 1984, Dahlberg posed two conjectures. The first conjecture concerned perturbation of operators. Suppose that, in the upper half space Rn+1 +, one has an elliptic operator L0= div A0∇for which the Dirichlet problem Dpwith data in Lp(Rn,dx) is solvable. Now suppose L1= div A1∇is a perturbation of L0in the sense that |A1(x, t)−A0(x, t)|2dxdt t Dirichlet Problem for Elliptic Equations 201 is a Carleson measure. Then, is the Dirichlet problem Dqfor Lqalso solvable, where qmay be larger than p? (The conjecture can be stated equivalently in terms of the weight condition dωL1∈A∞(dωL0) —see Definition (2.5).) This conjecture was solved affirmatively in [FKP], where references to earlier work (such as [D2]) may also be found. It is, however, the second conjecture that is the subject of this paper, and which concerns, not perturbations of operators, but classes of operators whose coefficients satisfy conditions (i) and (ii) above. Here the question is whether or not the Dirichlet problem Dpfor some pis solvable for such an L—equivalently, whether the A∞condition holds. Until recently, most positive results proving A∞estimates for a class of elliptic operators relied on L2identities, in the spirit of [JK], which in turn relied on symmetry assumptions. ([FJK] is one interesting exception to this.) But there are a variety of reasons for studying the non-symmetric situation. These include the connections with nondivergence form equations and the broader issue of obtaining estimates on elliptic measure in the absence of L2identities which relate tangential and normal derivatives. In [KKPT], the study of nonsymmetric divergence form operators with bounded measurable coefficients was initiated. In particular, some sharp A∞estimates were proven for certain operators in the class associated with the first transformation (flattening) mentioned above —that is, the operators on R2whose coefficients are independent of one of the variables. This result was an application of a new method of establishing mutual absolute continuity, and A∞([KKPT, Theorem 2.3]). In this paper, we show that this same technique can be used to prove A∞results for elliptic measures of operators satisfying the bounds and the Carleson measure conditions (i) and (ii) above. In the next section, we make some definitions and state our main results. We thank Steve Hofmann for helpful discussions on these topics and on his work [HL]. 2. Some definitions - Statements of the main theorems Definition 2.1. Z ⊂Rnis an M-cylinder of diameter dif there exists a coordinate system (x, t) such that Z={(x, t):|x|≤d, −2Md ≤t≤2Md} and for s>0, sZ={(x, t):|x|< sd, −2Md ≤t≤2Md}. 202 C. E. Kenig, J. Pipher Definition 2.2. Ω⊂Rnis a Lipschitz domain with Lipschitz ‘character’ (M,N,C0) if there exists a positive scale r0and at most Ncylinders {Zj}N j=1 of diameter d, with r0 c0≤d≤c0r0such that (i) 8Zj∩∂Ω is the graph of a Lipschitz function φj, φj∞≤M, φj(0) = 0. (ii) ∂Ω= j (Zj∩∂Ω). (iii) Zj∩Ω⊃(x, t):|x|<d,dist ((x, t),∂Ω) ≤d 2. If Q∈∂Ω and Br(Q)={x:|x−Q|≤r} then ∆r(Q) denotes the surface ball Br(Q)∩∂Ω and T(∆r)=Ω∩Br(Q) is the Cartesian region above ∆r(Q). Definition 2.3. A measure dµ defined in Ω ⊂Rnis a Carleson measure (denoted dµ ∈C) if there exists a constant C>0 such that for all r≤r0, µ(T(∆r)) ≤Cσ(∆r), where dσ is the surface measure of ∂Ω. Let ˜ L= div a∇+b·∇, where a=(aij) and b=(bj) are matrices with bounded measurable coefficients and (aij) satisfies the elliptic condition (1.1). We may assume that ais symmetric without any loss of generality for operators satisfying the hypothesis of Theorem 2.8 (see Remark 2.12). A function uin the Sobolev space W2 1,loc(Ω) is said to be a weak solution to Lin Ω if Ω a∇u·∇φ−(b·∇u)φdX =0(2.4) for all φ∈C∞ 0(Ω). When b∈L∞, weak solutions are in fact H¨older continuous, and satisfy an interior Harnack inequality, by the DiGiorgi-Nash-Moser theory (see [GT], for example). Our assumptions on bwill be weaker, but will imply that b∈L∞in the interior. Dirichlet Problem for Elliptic Equations 203 Associated to L= div a∇+b·∇, and a domain Ω ⊂Rn, is a family of elliptic measures dωX Lfor X∈Ω. These are the representing measures of solutions to Lin Ω which arise from the solvability of the continuous Dirichlet problem: Lu = 0 in Ω, u|∂D =g∈C(∂Ω). The unique solvability of this continuous Dirichlet problem was one of the results of [HL] for operators like Lwhich have drift terms b·∇, under certain conditions on b. We shall be more specific about this following the statements of Theorems 2.6 and 2.8 below. Fixing some X∈Ω we set dωL=dωX L, and refer to dωLas the elliptic measure for Lin Ω. The solvability of the Dirichlet problems for L, when the boundary data belongs to Lp(∂Ω,dσ), for some pand for dσ denoting surface measure on ∂Ω, depends upon the precise relationship between dωLand dσ. To help quantify this, we recall some definitions. Definition 2.5. If dµ and dν are doubling measures on ∂Ω, then dµ ∈ A∞(dν) if there exists a constant c>0 and δsuch that, for any ∆ ⊂∂Ω and any set E⊂∆, µ(E) µ(∆) ≤Cν(E) ν(∆)δ . As the constant in Definition 2.5 is independent of the sets Eand ∆, the A∞-condition is in fact a quantitative, or uniform, version of mutual absolute continuity. If dωL∈A∞(dσ) then it will follow from the general theory of weights ([CF], [M]) that there exists p<+∞for which the Dirichlet problem Dpfor Lon Ω with data in Lp(∂Ω,dσ) is uniquely solvable. (See [FKP] and the references therein for details.) The main results of this paper are the following. Theorem 2.6. Let L= div a∇be an elliptic operator and let Ω⊂Rn be a bounded Lipschitz domain. Let δ(X) = dist(X,∂Ω), and suppose that a=(aij)has distributional derivatives satisfying sup{δ(X)|∇aij(X)|2:X∈Bδ(Z)/2(Z)}(2.7) is a Carleson measure in Ω. Then, the elliptic measure dωLassociated to Lbelongs to A∞(dσ). Theorem 2.8. If ˜ L=L+b·∇, where Lis as in Theorem 2.6, and (2.9) sup{δ(X)| b(X)|2:X∈Bδ(Z)/2(Z)} is a Carleson measure in Ω. Then dω˜ L∈A∞(dσ). 204 C. E. Kenig, J. Pipher Remark 2.10.We do not assume, in Theorem 2.6, that the matrix (aij) is symmetric. It was observed in [KKPT], that the results of [CFMS] for operators L= div A∇with merely bounded measurable coefficients were valid even when Awas not symmetric. Thus, for operators of Theorem 2.6, the existence of elliptic measure, its relationship to the Green’s functions and various classical important properties of solutions (Harnack property, H¨older continuity of weak solutions, comparison principles) are all valid (see (1.3)–(1.14) of [KKPT]). Remark 2.11.The Carleson measure conditions 2.7 and 2.9 of the theorems are stated in terms of supremums so that they imply certain essential pointwise estimates on aij and bj. For example, (2.9) implies that | b(X)|≤Cδ(X)−1for x∈Ω, and (2.10) implies an analogous estimate on |∇aij|. Remark 2.12.In the statement of Theorem 2.8, the symmetry of the matrix a=(aij) is no longer a factor. That is, it suffices to assume that (aij) is symmetric, and the general case will follow. To see this, let L=Di(aij)Dj+b·∇, and let us assume (2.8) holds for a=(aij) symmetric, satisfying (2.7). Consider a non-divergence form operator L0= aijDiDj+b·∇where  bsatisfies (2.9), and where (aij) need not be symmetric. Let Aij =aij +aji 2, so that L0=AijDiDj+B·∇. Then L0may also be written as L0=Di(Aij)Dj+(B−(DiAij)j), and we see that the resulting lower order term B−(DiAij)jalso satisfies (2.9). Thus L0satisfies the hypotheses of Theorem 2.8. Now observe that L=Di(aijDj)+b·∇u=aijDiDj+(b+Diaij)j·∇, i.e., Lis an operator whose structure is the same as L0, to which the theorem applies. The above observations give rise to an interesting corollary of Theorem 2.8, namely that the same conclusion applies to operators in nondivergence form. Moreover, we also see that it now suffices to prove (2.6) assuming (aij) is symmetric, although the method of proof does not distinguish between the two cases. In light of the fact that the matrices (aij) for Lin the statement of Theorem 2.8 may be assumed symmetric, Theorem 2.8 follows from Theorem 2.6 by applying the perturbation Theorem 2.17 of [HL]. From A∞for the operator Lit will follow that A∞holds for ˜ L=L+b·∇ under conditions (2.9) on the drift terms. In [HL], the authors prove Dirichlet Problem for Elliptic Equations 205 A∞estimates for parabolic operators as well as elliptic ones, and must therefore deal with the non-trivial issues of the lack of a doubling condition on parabolic measures. But in the elliptic case, their conditions on the coefficients of the matrix (aij ) are more restrictive than ours, so Theorem 2.6 is not contained in their work. However, the perturbation methods they develop are powerful, and we did not find a simpler means of obtaining (2.8) from (2.6) without invoking their Theorem 2.17. Theorem 2.6 will follow from establishing the hypotheses of Theorem 2.9 of [KKPT], which gives a Littlewood-Paley criterion for A∞ estimates. To state the theorem, we recall the classical operators associated with the concepts of mathematical convergence of solutions at the boundary, and of uniqueness in the Dirichlet problem. Our domains will be assumed to be Lipschitz, and hence we may define the non-tangential approach regions, for each Q∈∂Ω, by Γα(Q)={X∈Ω:|X−Q|≤(1 + α) dist(X,∂Ω)}, which are compactly contained in Ω when αis sufficiently large. The parameter αis the aperture of the cone. Sometimes, we shall need to truncate the cones at height dand we denote by Γα,d(Q) this truncated cone Γα(Q)∩Bd(Q). The square function of u, defined in Ω ⊂Rn,ata point Q∈∂Ω relative to a family of cones {Γα,d}is defined by: Sα,du(Q)=Γα,d(Q)|∇u(X)|2dist(X,∂Ω)2−ndX 1 2 . The non-tangential maximal function of uis defined by Nα,du(Q) = sup{|u(X)|:X∈Γα,d(Q)}. Definition 2.13. The Dirichlet problem (Dp) with data in Lp(∂Ω,dσ) is solvable for Lif whenever f∈C(∂Ω), the solution uto the continuous Dirichlet problem with data fsatisfies the estimate: For fixed α,d, there exists C>0, depending on αand on ellipticity, such that Nα,d(u)Lp(dσ)≤CfLp(dσ).(2.14) 206 C. E. Kenig, J. Pipher The nontangential maximal function N(u), when u|∂Ω=fis comparable to a maximal function of Hardy-Littlewood type relative to the measure dωL: MωLf(Q) = sup ∆Q∆ f(P)dωL(P) ωL(∆) . Thus, by the theory of weights, Dpis solvable for Lif and only if dωL= kdσ and ksatisfies a reverse H¨older inequality of order p,1 p+1 p=1. Again, dωL∈A∞(dσ) if and only if a reverse H¨older inequality for the density is valid for some (possibly large) choice of exponent, and hence dωLbelongs to A∞(dσ) if and only if (Dp) is solvable for some p. One of the main results of [KKPT] is the following criterion for establishing A∞for operators Lin divergence form. Theorem 2.15 (2.9 of [KKPT]).Let L= div A∇be elliptic and A= (aij)be a matrix of bounded measurable functions. Let Ωbe a bounded Lipschitz domain. Suppose that for all Lipschitz subdomains D⊂Ωone has the Lpnorm equivalence, for some p>0, N(u)Lp(∂Ω,dσ)≤C1S(u)Lp(∂Ω,dσ) ≤C2N(u)Lp(∂Ω,dσ) (2.16) for constants C1,C2depending only on the Lipschitz character of Ω. Then dωL∈A∞(dσ)on Ω. We note that Theorem 2.15 was stated for p= 2, but once (2.16) holds for some choice of p, it holds for all, by a purely real variable argument. And finally, we showed that in [KKPT] that Theorem 2.15 was sharp in the sense that no conclusion stronger than A∞could be drawn. Theorems 2.6 and 2.8 are sharp as well, in the same sense. In [FKP], a class of divergence form operators (of Beurling-Ahlfors type) was shown to be sharp for the A∞condition. A routine computation verifies that their coefficients always satisfy the condition (2.7) of Theorem 2.6, and hence the conclusion cannot be improved. 3. Proof of Theorem 2.6 The main results of the paper are proven by establishing the hypothesis of Theorem 2.15 above. That is, on every sub-domain Ωof Ω we will prove the Lpnorm equivalence of the square function and the non-tangential maximal function with respect to surface measures on the boundary. The first lemma permits us to assume that the Carleson measure condition (2.7) holds on every Lipschitz subdomain of our original domain. Dirichlet Problem for Elliptic Equations 207 For Ω ⊂Rn+1, a Lipschitz domain, let 0Ω(X) = sup{|∇a(Z)|:Z∈B(X,δΩ(X)/2} where δΩ(X) = dist(X,∂Ω). If Ω = Rn+1 +, we omit the subscript and simply write 0(x, t) where X=(x, t)∈Rn×R+. Lemma 3.1. Suppose that Ω⊂Rn+1 is a Lipschitz domain and that δΩ(X)02 Ω(X)dX is a Carleson measure with norm C. Then, on every bounded Lipschitz subdomain Ωof Ω,δΩ(X)02 ΩdX is a Carleson measure with norm depending only on C0and on the Lipschitz character of Ω. Proof: It suffices to prove the lemma in the case where Ω = Rn+1 +. Let Ω denote the Lipschitz subdomain of Rn+1 +and fix a Carleson box T(∆) associated to a surface ball ∆ ⊂∂Ω. Let X0=(x0,t 0) be the center of T(∆). We consider two cases. In case 1, we assume that the diameter of T(∆) is smaller than 1 10 t0. Then if (x, t)∈Rn+1 +belongs to T(∆), t0≤10 9t, since t0=(t0−t)+t≤ diam T(∆)+t. Because t02(x, t)dxdt is a Carleson measure, |∇a(x, t)|≤ C t, and hence 0Ω(x, t)≤C t0, by our assumption on diamT(∆). This gives T(∆) δΩ(X)02 Ω(X)dX ≤C t02 [diam T(∆)]n+2 ≤C(diam T(∆))n ≤Cσ(∆), which is the Carleson condition on Ω. In case 2, when diam T(∆) >1 10 t0, we let Qbe the cube in Rnwhose center (x0,0) is the projection of X0and whose diameter is (M+1)t0, where Mdepends only on the Lipschitz character of Ω and is chosen so that diam T(∆) ≤Mt0. With this choice, it is easy to verify that T(∆) ⊂T(Q). Thus, since δΩ(X)≤tand 0Ω(x, t)≤0(x, t), T(∆) δΩ(X)02 Ω(X)dX ≤CT(Q) t02(x, t)dxdt ≤C|Q| ≤Cσ(∆). Let now L= div a∇be an elliptic operator satisfying condition (2.7) of Theorem 2.6 defined in Ω. We wish to establish the equivalence on 214 C. E. Kenig, J. Pipher in [KKPT]. There, it was necessary to prove the Lp-inequalities on graphs, so as to use the G. David arguments building on the case of small Lipschitz constant. The need to ‘build up’ from the small constant case makes the arguments much more technical, requiring a more elaborate localization scheme. Proof of Theorem 3.8: (We omit those details which are standard, and can be found in many references.) Let {∆l(Ql,r l)}be a Whitney decomposition of {Na(u)>λ 32 }. That is, {Na(u)>λ 32 }⊂∪ l∆l,∆ l= Bl(Ql,r l)∩∂Ω with lχBl(X)≤C(M), and each 8n∆l=Bl(Ql,8rl)∩ ∂Ω is contained in the graph of a Lipschitz function. Moreover, there exists a parameter r0=r0(M,N,C0) such that if rl≤r0, then B(Ql,2nrl)∩ ∂Ω contains a point Q∗ lwhere Na(u)(Q∗ l)≤λ 32 . Since we make the a priori assumption that S4a(u)Lp<+∞, interior estimates together with the normalization on uhandles the estimates on the ‘large’ Whitney balls —those with radius larger that the parameter r0(see [D3]). Thuswefixa∆=∆ lwith rl≤r0in the Whitney decomposition and let F=∆∩{Na(u)>2λ, M(S4a(u)) ≤λγ, M(S2 4a(u))1 4·M(N2 a(u))1 4≤γλ}. By choosing γsufficiently small, we can ensure that Na,βl(u)>λ 2in Fwhere βlis a truncation of the cones which equals c1rl, where rlis the radius of ∆l, and which satisfies the conditions of Lemma 3.6, rescaled to rl. Now set λ=2 j, and define the Lipschitz function hj(x) as in Lemma 3.5 relative to the Lipschitz function φ, where 8∆lis contained in the graph of φ. The function v(x, t) in the definition of hjwill be supported in T(2∆l) since the cut-off functions are tailored to scale rl. Let ∆∗ l= {(x, hj(x)):(x, φ(x)) ∈4∆l}. Then by 3.6, for γ<ρ, for all (x, φ(x)) ∈ F,wehaveMj(uχ∆∗ l)(x, hj(x)) >2j 16 , where Mjdenotes the HardyLittlewood maximal function on the graph hj: MjF(x, hj(x)) = sup I F(z,hj(z))dσj(z):x∈I. Let ArlbeapointinOlsatisfying inequality 3.7. Since ∆lis one of the ‘small’ Whitney balls, the existence of the point Q∗ lwhere Na(u)(Q∗ l)≤ 2j 32 means that Mj(˜uX∆∗ l)(x, hj(x)) >2j 32 if ˜u(X)=u(X)−u(Al). Dirichlet Problem for Elliptic Equations 215 Thus, by replacing uby ˜u, we may assume that uvanishes at Al. (This means that the constant γin the definition of Fshould be replaced by √γ.) Therefore, by the weak type inequality for the maximal function, σ(F)≤C2−2j∆∗ l u2(x, hj(x))dσ. We now apply inequality (3.7) to bound the above expression by two terms, one of which is C2−2j2∆∗ l S2 4a,2βl,hj(u)dσj.(3.10) The indices on the square function show that this is defined with respect to the graph hj(x) using truncated cones (2βl) of aperture a. The cones Γ4a,2βl,hj(x, hj(x)) are always contained in the cones Γ4a(x, φ(x)) so, in dimension n= 2, the square functions S4a,2βl,hju(x, hj(x)) are dominated pointwise by the square functions S4a(u)(x, φ(x)). Then the quantity (3.10) would be bounded by M(S4a(u)(x, φ(x))·σ(8∆l), for (x, φ(x)) ∈F, which is, in turn, bounded by (γ2j)2σ(8∆l). In dimension n>2, the desired upper bound can be obtained by carrying out the integration: 2∆∗ l S2 4a,2βl,hjudσjT(3∆l) δj(X)|∇u(X)|2dX(3.11) where δj(X) = dist(X, graph of hj). Then, since δj(X)≤δ(X),we recover 4∆lS2 4a(u)(Q)dσ(Q), and the upper bound (γ2j)2σ(∆l). The other term arising from 3.7 in the bound for 3.10 is handled similarly. Summing our estimates on lproves 3.8. The arguments for the converse inequality to that of 3.8, Sa(u)Lp(dσ)≤CN4a(u)p(dσ), are similar, but simpler. Here we do not need the stopping time argument, nor the introduction of the hj’s. We omit the details. We also note that parabolic analogs of Theorem 2.6 hold, with similar proofs. The same applies to Theorem 2.8 if the Carleson measure norm in (2.9) is sufficiently small, for then the parabolic measure is a doubling measure ([HL]). Whether or not doubling is true for parabolic measures when the norm in (2.9) is large remains an interesting open problem. 216 C. E. Kenig, J. Pipher Bibliography [CFK] L. A. Caffarelli, E. B. Fabes and C. E. Kenig, Completely singular elliptic-harmonic measures, Indiana Univ. Math. J. 30(6) (1981), 917–924. [CFMS] L. A. Caffarelli, E. B. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30(4) (1981), 621–640. [CF] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. [D1] B. E. J. Dahlberg, On the Poisson integral for Lipschitz and C1-domains, Studia Math. 66(1) (1979), 13–24. [D2] B. E. J. Dahlberg, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc. 97(1) (1986), 41–48. [D3] B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67(3) (1980), 297–314. [FJK] E. B. Fabes, D. S. Jerison and C. E. Kenig, Multilinear Littlewood-Paley estimates with applications to partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 79(18) (1982), 5746–5750. [Fe-St] C. Fefferman and E. M. Stein,Hpspaces of several variables, Acta Math. 129(3–4) (1972), 137–193. [FKP] R. A. Fefferman, C. E. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134(1) (1991), 65–124. [GT] D. Gilbarg and N. S. Trudinger,“Elliptic partial differential equations of second order”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224, Springer-Verlag, Berlin-New York, 1983. [HL] S. Hofmann and J. Lewis, The Dirichlet problem for parabolic operators with singular drift terms, to appear in Trans. Amer. Math. Soc. [JK] D. S. Jerison and C. E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113(2) (1981), 367–382. Dirichlet Problem for Elliptic Equations 217 [KKPT] C. Kenig, H. Koch, J. Pipher and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153(2) (2000), 231–298. [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. Department of Mathematics Brown University Box 1917 Providence, Rhode Island 02912 U.S.A. E-mail address:[email protected] E-mail address:Jill−[email protected] Rebut el 6 de juny de 2000.