Carleson measure estimates for the Green function
aa r X i v : . [ m a t h . A P ] F e b CARLESON MEASURE ESTIMATES FOR THE GREENFUNCTION
GUY DAVID, LINHAN LI, AND SVITLANA MAYBORODA
Abstract.
In the present paper we consider an elliptic divergence form op-erator in the half space and prove that its Green function is almost affine,or more precisely, that the normalized difference between the Green functionand a suitable affine function at every scale satisfies a Carleson measure esti-mate, provided that the oscillations of the coefficients satisfy the traditionalquadratic Carleson condition. The results are sharp, and in particular, it isdemonstrated that the class of the operators considered in the paper cannotbe improved.
Contents
1. Introduction 12. Preliminaries and properties of the weak solutions 83. Approximations and the main conditional decay estimate 113.1. A little more about orthogonality, J u , and β u L l316l316 mt1mt1 mt2mt2 cor maincor main Introduction
Let L = − div( A ∇ ) be a divergence form elliptic operator on the upper halfspace R d +1+ . In the present paper we show that if L is reasonably well-behavedthen the Green function for L is well approximated by multiples of the distanceto R d . There are many predecessors of these results which we will discuss below G. David was partially supported by the European Community H2020 grant GHAIA 777822,and the Simons Foundation grant 601941, GD. S. Mayboroda was partly supported by the NSFRAISE-TAQS grant DMS-1839077 and the Simons foundation grant 563916, SM. . ( kenig2001dirichlet, dindos2007lp, hofmann2017implies, hofmann2017uniform [KP01, DPP07, HMT17, HMM +
20] to mention only the closer ones). At this point,however, let us underline two important points. First, the class of the operators thatwe consider is of the nature of the best possible, as shown by the counterexamples inSection sec optmsec optm
6. The estimates themselves are sharp, and in fact, a weak version of themis equivalent to the uniform rectifiability
DM2020 [DM20]. We hope to ultimately show thatthe much stronger estimate proved here is also true for domains with a uniformlyrectifiable boundary, thus giving a strong and a weak characterization of uniformrectifiability in terms of approximation of the Green function (or more generallysolutions) by distance function. But this will have to be the subject of anotherpaper. Secondly, the method of the proof itself is quite unusual for this kind ofbounds. A typical approach is through integrations by parts, which, however, doesnot allow one to access the optimal class of the coefficients. Roughly speaking,we are working with the square of the second derivatives of the Green functionand given the roughness of the coefficients, there are too many derivatives in tocontrol to take advantage of the equation while integrating by parts. Here, instead,we make intricate comparisons with solutions of the constant coefficient operators,carefully adjusting them from scale to scale. We feel that the method itself is anovelty for this circle of questions and that it illuminates the nature of the Carlesonestimates in a completely different way, hopefully opening a door to many otherproblems.More generally, we are interested in the relations between an elliptic operator L on a domain Ω, the geometry of Ω, and the boundary behavior of the Green function.It is easy to see that the Green function with a pole at infinity for the Laplacian onthe upper half-space R d +1+ := (cid:8) ( x, t ) : x ∈ R d , t ∈ R + (cid:9) is a multiple of t , the distanceto the boundary, and more generally the Green function with a pole that is relativelyfar away is close to the distance function. There have been many efforts to generalizethis to more general settings. For instance, in caffarelli1981existence [AC81] the authors obtain flatnessof the boundary from local small oscillations of the gradient of the Green functionwith a pole sufficiently far away. Philosophically, similar considerations underpinthe celebrated results of Kenig and Toro connecting the flatness of the boundaryto the property that the logarithm of the Poisson kernel lies in VMO kenig1999free [KT99].Much more close to our setting is the study of the so-called Dahlberg-Kenig-Pipheroperators pioneered by Kenig and Pipher kenig2001dirichlet, dindos2007lp [KP01, DPP07] in combination with thestudy of the harmonic measure on uniformly rectifiable sets by Hofmann, Martell,Toro, Tolsa, and others (see hofmann2017harmonic, hofmann2017uniform [AHM +
17, HMM +
20] and many of their predecessors).Undoubtedly, the behavior of the harmonic measure is connected to the regularityof Green function G , yet the latter is different and surprisingly has been much lessstudied. In part, this is due to the fact that the harmonic measure is related to thegradient of G at the boundary while the estimates we target in this paper reachout to the second derivatives of G . One could say that the two are related by ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 3 an integration by parts, but in the world of the rough coefficients this is not so.Indeed, relying on these ideas, hofmann2017implies [HMT17] establishes second derivatives estimatesfor the Green function somewhat similar to ours under a much stronger conditionthat the gradient of the coefficients, rather than its square, is Carleson. It was clearalready then that the optimal condition must be a control of the square-Carlesonnorm, but their methods, using the aforementioned integration by parts, did notgive a possibility to overcome this restriction. In this paper we achieve the optimalresults and, indeed, demonstrate using the counterexamples that they are the bestpossible.In the present paper, we focus on Ω = R d +1+ , and show that for the operators sat-isfying a slightly weaker version of the Dahlberg-Kenig-Pipher condition describedbelow, the Green function is well approximated by multiples of t , in the sense thatthe gradient of normalized differences satisfies a square Carleson measure estimate.Notice that the class of coefficients authorized below is enough to treat the casewhen Ω is a Lipschitz graph domain, by a change of variables. As we mentionedabove, we plan to pursue more general uniformly rectifiable sets in the upcomingwork, which would give a much stronger version of our previous results in DM2020 [DM20]and would show that our estimates are equivalent to the uniform rectifiability of theboundary. At this point, restricting to the simple domain Ω = R d +1+ will have theadvantage of making the geometry cleaner and focusing on one of the tools of thispaper, concerning the dependence of G (or the solutions) on the coefficients. Evenin the “simple” case of the half-space, the question of good approximation of G bymultiples of t seems, to our surprise, to be widely open, and the traditional methodsof analysis break down brutally when trying to achieve such results. Perhaps onecould also say that this setting is more classical. Let us pass to the details.Consider an operator in divergence form L = − div( A ∇ ), where A = h a ij ( X ) i isan n × n matrix of real-valued, bounded and measurable functions on R n + , n = d +1.We say that L is elliptic if there is some µ > h A ( X ) ξ, ζ i ≤ µ | ξ | | ζ | and h A ( X ) ξ, ξ i ≥ µ − | ξ | for X ∈ R d +1+ and ξ, η ∈ R n . (1.1) cond ellp We use lower case letters for points in R d , e.g. x ∈ R d , and capital letters forpoints in R n = R d +1 , e.g. X = ( x, t ) ∈ R d +1 . We identify R d with R d × { } ⊂ R d +1 so, when t = 0, we may write x instead of ( x, ∈ R d +1 .For x ∈ R d and r >
0, we denote by ∆( x, r ) the surface ball B r ( x ) ∩{ t = 0 } ⊂ R d .Thus ∆( x, r ) is a ball in R d while B ( x, r ) is the ball of radius r in R d +1 . We denoteby T ( x, r ) := B r ( x ) ∩ R d +1+ and W ( x, r ) := ∆( x, r ) × (cid:16) r , r i ⊂ R d +1+ (1.2) TT the corresponding Carleson box and Whitney cube. Note that T ( x, r ) is a half ballin R d +1+ over ∆( x, r ). We may simply write T ∆ for a half ball over ∆ ⊂ R d . GUY DAVID, LINHAN LI, AND SVITLANA MAYBORODA d13
Definition 1.3 (Carleson measure) . We say that a nonnegative Borel measure µ is a Carleson measure in R d +1+ , if its Carleson norm k µ k C := sup ∆ ⊂ R d µ ( T ∆ ) | ∆ | is finite, where the supremum is over all the surface balls ∆ and | ∆ | is the Lebesguemeasure of ∆ in R d . We use C to denote the set of Carleson measures on R d +1+ .For any surface ball ∆ ⊂ R d , we use C (∆ ) to denote the set of Borel measuressatisfying the Carleson condition restricted to ∆ , i.e., such that k µ k C (∆ ) := sup ∆ ⊂ ∆ µ ( T ∆ ) | ∆ | < + ∞ . Next we want to define a (weaker) version of the Dahlberg-Kenig-Pipher condi-tions in the form which is convenient for the point of view taken in this paper. Wewould like to say that the matrix A = A ( X ) is often close to a constant coefficientmatrix. The simplest way to measure this is to use the numbers α ∞ ( x, r ) = inf A ∈ A ( µ ) sup ( y,s ) ∈ W ( x,r ) | A ( y, s ) − A | , (1.4) where the infimum is taken over the class A ( µ ) of (constant!) matrices A thatsatisfy the ellipticity condition ( cond ellpcond ellp A is allowed to dependon ( x, r ), so α ∞ ( x, r ) is a measure of the oscillation of A in W ( x, r ), similarly to dindos2007lp [DPP07]. We require A to satisfy ( cond ellpcond ellp A by one of the A ( y, s ), ( y, s ) ∈ W ( x, r ), which satisfies ( cond ellpcond ellp α ∞ ( x, r ) by at most 2. The same remark isvalid for the slightly more general numbers α q ( x, r ) = inf A ∈ A ( µ ) (cid:26) ( y,s ) ∈ W ( x,r ) | A ( y, s ) − A | q (cid:27) /q (1.5) where in fact q will be chosen equal to 2. d1a6 Definition 1.6 (Weak DKP condition) . We say that the coefficient matrix A sat-isfies the weak DKP condition with constant M >
0, when α ( x, r ) dxdrr is aCarleson measure on R d +1+ , with norm N ( A ) := (cid:13)(cid:13)(cid:13)(cid:13) α ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C ≤ M. (1.7) We may also say that α ( x, r ) satisfies a Carleson measure estimate. Recallthat this implies that α ( x, r ) is small most of the time (to the point of beingintegrable against the infinite invariant measure dxdrr ), but does not vanish at anyspecific speed given in advance.The name comes from a condition introduced by Dahlberg, Kenig, and Pipher,which instead demands that e α ( x, r ) satisfy a Carleson estimate, where e α ( x, r ) = r sup ( y,s ) ∈ W ( x,r ) |∇ A ( y, s ) | . (1.8) ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 5
In 1984, Dahlberg first introduced this condition, and conjectured that such aCarleson condition guarantees the absolute continuity of the elliptic measure withrespect to the Lebesgue measure in the upper half-space. In 2001, Kenig and Pipher kenig2001dirichlet [KP01] proved Dahlberg’s conjecture. Since it is obvious that α ( x, r ) ≤ α ∞ ( x, r ) ≤ e α ( x, r ), we see that our condition is weaker than the classical DKP condition, butimportantly they have the same homogeneity. A similar weakening of the DKPcondition, pertaining to the oscillations of the coefficients, has been considered, e.g.in dindos2007lp [DPP07]. We could also have chosen an exponent q ∈ (2 , ∞ ] for α q in Definition d1a6d1a6 R d +1+ , we would use other models than the function ( y, t ) t ,such as (functions of) the distance to the boundary, but here we are interested in(approximation by) the affine function ( y, t ) λt , with λ > u of Lu = 0 that vanish at the boundary.In addition, given such a solution u , when we are considering a given Carlesonbox T ( x, r ), we do not want to assume any a priori knowledge on the average sizeof u in T ( x, r ), so we just want to measure the approximation of u , in T ( x, r ), bythe best affine function a x,r than we can think of, and it is reasonable to pick a x,r ( z, t ) = λ x,r t, where λ x,r = λ x,r ( u ) = T ( x,r ) ∂ t u ( z, t ) dzdt (1.9) is the average on T ( x, r ) of the vertical derivative. See the beginning of Section sec constsec const L averageof the difference of the gradients (we seem to forget u but after all, it is easyto recuperate the functions from their gradients because they both vanish on theboundary), which we divide by the local energy of u because we do want the sameresult for u as for λu . That is, we set J u ( x, r ) = T ( x,r ) |∇ z,t ( u ( z, t ) − a x,r ( z, t )) | dzdt = T ( x,r ) |∇ z,t u ( z, t ) − λ x,r ( u ) e d +1 | dzdt, (1.10) where e d +1 = (0 , . . . ,
1) is the vertical unit vector, and then divide by E u ( x, r ) = T ( x,r ) |∇ u | (1.11) def E to get the number β u ( x, r ) = J u ( x, r ) E u ( x, r ) . (1.12) GUY DAVID, LINHAN LI, AND SVITLANA MAYBORODA
This number measures the normalized non-affine part of the energy of u in T ( x, r ).We want to say that u is often close a x,r , i.e., that β u ( x, r ) is often small, and thiswill be quantified by a Carleson measure condition on β u . We won’t need to square β u , because J u is already quadratic.The simplest version of our main result is the following. mt1 Theorem 1.13.
Let A be a ( d +1) × ( d +1) matrix of real-valued functions on R d +1+ satisfying the ellipticity condition ( cond ellpcond ellp . If A satisfies the weak DKP condition withsome constant M ∈ (0 , ∞ ) , and if we are given x ∈ R d , R > , and a positivesolution u of Lu = − div ( A ∇ u ) = 0 in T ( x , R ) , with u = 0 on ∆( x , R ) , thenthe function β u defined by ( satisfies a Carleson condition in T ( x , R/ , andmore precisely (cid:13)(cid:13)(cid:13)(cid:13) β u ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C (∆( x ,R/ ≤ C + C M (1.14) where C depends only on d and µ . That is, u is locally well approximated by affine functions in T ( x , R/ u on R d are well defined because solutions are locally H¨older continuousup to the boundary; this will be explained better in the next section.Notice that the constant M > u is the Green function for L ,with a pole anywhere in R d +1+ \ T ( x , R ), and even in the case of the Laplacian,the smallness of M does not guarantee the smallness of ( u is notnecessarily so close to an affine function at the scale R . This is natural (the impactof what happens outside of T ( x , R ) could be substantial), and this effect will beameliorated in the next statement, at the price of some additional quantifiers; thepoint is that the Green function with a pole at ∞ , or even a positive solution ina much larger box than T ( x , R ), behaves better and has a better approximation.The next theorem says that we can have Carleson norms for β u that are as smallas we want, provided that we take a small DKP constant and a large security boxwhere u is a positive solution that vanishes on the boundary. mt2 Theorem 1.15.
Let d , µ be given, let u and ∆( x , R ) satisfy the assumptions ofTheorem mt1mt1 A satisfy the weak DKP condition in ∆( x , R ) . Then for τ ≤ / we have the more precise estimate (cid:13)(cid:13)(cid:13)(cid:13) β u ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C (∆( x ,τR )) ≤ Cτ a + C (cid:13)(cid:13)(cid:13)(cid:13) α ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C (∆( x ,R )) , (1.16) where C and a > depends only on d and µ . This way the right-hand side can be made as small as we want. Notice that weonly need A to satisfy the weak DKP condition in ∆( x , R ); the values of A outside ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 7 of T ( x , R ) should be irrelevant anyway, because we do not know anything about u there.We observed earlier that this result applies to the Green function with a pole at ∞ (and then the Carleson norm of β is less than C N ( A ), with N ( A ) as in ( e α of ( cor main Corollary 1.17.
Let A be a ( d + 1) × ( d + 1) matrix of real-valued functions on R d +1+ satisfying the ellipticity condition ( cond ellpcond ellp . Suppose A satisfies the classical DKPcondition with constant C ∈ (0 , ∞ ) , that is, (cid:13)(cid:13)(cid:13)(cid:13)e α ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C ≤ C , (1.18) eq DKP where e α ( x, r ) is defined in ( . If we are given x ∈ R d , R > , and a positivesolution u of Lu = − div ( A ∇ u ) = 0 in T ( x , R ) , with u = 0 on ∆( x , R ) , thenthere exists some constant C depending only on d , µ and C such that ˆ T ∆ (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) u ( y, t ) t dydt ≤ C | ∆ | (1.19) for any ∆ ⊂ ∆( x , R/ . We state this corollary on the upper half-space for simplicity, but it can begeneralized to Lipschitz domains by a change of variables that preserves the DKPclass operators. In fact, the change of variables will be a bi-Lipcshitz mapping whosesecond derivatives satisfy a Carleson measure estimate. With such regularity of thechange of variables, as well as our estimates for β u in the main theorems, it reducesto the case of the upper half-space.In Section sec optmsec optm
6, we construct an operator that does not satisfy the DKP condition,for which the precise approximation estimates of Theorems mt1mt1 mt2mt2
DM2020 [DM20], but if we want precise approximation results forthe Green functions, the first significant step in the positive direction should be aversion of main results of the present paper in the higher co-dimensional context,and their extension to uniformly rectifiable sets.
GUY DAVID, LINHAN LI, AND SVITLANA MAYBORODA
The rest of this paper is organized as follows. In the next section we recall somenotation and the general properties of solutions that we need later. In Section sec constsec const J u and β u , prove some decay estimates for β u when u is aweak solution of a constant coefficient operator, and extend this to the general casewith a variational argument. The rest of the proof of our main theorems, whichconsists in Carleson measure estimates with no special relations with solutions, isdone in Section sec cmsec cm
4. We prove Corollary cor maincor main sec corsec cor mt1mt1 sec optmsec optm
6, we discuss the optimality of our results.2.
Preliminaries and properties of the weak solutions sec nota def
In this section we recall some classical results for solutions of elliptic operatorsin divergence form.Recall the notation B ( X, r ) for open balls centered at X ∈ R d +1 , ∆( x, r ) forsurface balls, T ( x, r ) for Carleson boxes, and W ( x, r ) for Whitney cubes (see near( TTTT ffl B f ( x ) dx := | B | ´ B f ( x ) dx the average of f on a set B .Let us collect some well-known estimates for solutions of L = − div( A ∇ ), where A is a matrix of real-valued, measurable and bounded functions, satisfying theellipticity condition ( cond ellpcond ellp def weak sol Definition 2.1 (Weak solutions) . Let Ω be a domain in R n . A function u ∈ W , (Ω) is a weak solution to Lu = 0 in Ω if for any ϕ ∈ W , (Ω), ˆ Ω A ( X ) ∇ u ( X ) · ∇ ϕ ( X ) dX = 0 . We will only be interested in the simple domains Ω = R d +1+ and Ω = R d +1+ ∩ B ( x, r ), with x ∈ R d and r >
0. The space W , (Ω) is the closure in W , (Ω) ofthe compactly supported smooth functions in Ω. Conventional or strong solutionsare obviously weak solutions as well. In this paper, our solutions are always takenin the sense of Definition def weak soldef weak sol u is a (weak) solution in Ω. When we say that u = 0 on somesurface ball ∆ = ∆( x, r ) ⊂ Ω, we mean this in the sense of W , ( T ∆ ). This meansthat u is a limit in W , ( T ∆ ) of a sequence of functions in C ( T ∆ \ ∆). We couldalso say that the trace of u , which is defined and lies in H / (∆), is equal to 0on ∆. Ultimately, the De Giorgi-Nash-Moser theory (cf. Lemma lem bdy reglem bdy reg u is in fact continuous in T r ∪ ∆ r , and,in particular, u vanishes on ∆. Hence, in the rest of this paper the distinction isimmaterial, but for now we will try to be precise.We refer the readers to kenig1994harmonic [Ken94] for proofs and references for the following lemmas. lem bdy cacio Lemma 2.2 (Boundary Caccioppoli Inequality) . Let u ∈ W , ( T ( x, r )) be a so-lution of L in T ( x, r ) , with u = 0 on ∆( x, r ) . There exists some constant C ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 9 depending only on the dimension and the ellipticity constant of L , such that T ( x,r ) |∇ u ( X ) | dX ≤ Cr T ( x, r ) | u ( X ) | dX. lem bdy reg Lemma 2.3 (Boundary De Giorgi-Nash-Moser inequalities) . Let u be as in Lemma lem bdy caciolem bdy cacio sup T ( x,r ) | u | ≤ C T ( x, r ) u ( X ) dX ! / , where C = C ( d, µ ) . Moreover, for any < ρ < r , we have, for some α = α ( d, µ ) ∈ (0 , , osc T ( x,ρ ) u ≤ C (cid:16) ρr (cid:17) α T ( x, r ) u ( X ) dX ! / , where osc Ω u := sup Ω u − inf Ω u . lem bdy Harnack Lemma 2.4 (Boundary Harnack Inequality) . Let u ∈ W , ( T ( x, r )) be a nonneg-ative solution of L in T ( x, r ) with u = 0 on ∆( x, r ) . Then u ( X ) ≤ Cu ( X r ) ∀ X ∈ T ( x, r ) , where C > depends only on the dimension and µ . Of course, each of these statements has an interior analogue where we wouldreplace T ( x, r ) by a ball B ( X, r ) such that B ( X, R ) ⊂ Ω and we would not have tospecify the boundary conditions. The interior Harnack inequality reads as follows.
Lemma 2.5 (Harnack Inequality) . There is some constant C , depending only onthe dimension and the ellipticity constant for A , such that if u ∈ W , (Ω) is anonnegative solution of Lu = 0 in B ( X, r ) ⊂ Ω , then sup B ( X,r ) u ≤ C inf B ( X,r ) u. We will also use the Comparison Principle.
Lemma 2.6 (Comparison Principle) . Let u, v ∈ W , ( T ( x, r )) be two nonnegativesolutions of L in T ( x, r ) , such that u = v = 0 on ∆( x, r ) and v is not identicallynull. Set X x,r = ( x, r ) (a corckscrew point for T ( x, r ) ). Then C − u ( X x,r ) v ( X x,r ) ≤ u ( X ) v ( X ) ≤ C u ( X x,r ) v ( X x,r ) for all X ∈ T ( x, r ) , where C = C ( n, µ ) ≥ . lem RH Lemma 2.7 (Reverse H¨older Inequality on the boundary) . We can find an expo-nent p > and a constant C ≥ , that depend only on d and the ellipticity constant µ for A , such that if u and T ( x, r ) are as in Lemma lem bdy caciolem bdy cacio T ( x,r ) |∇ u ( X ) | p dX ! /p ≤ C T ( x, r ) |∇ u ( X ) | dX ! / . See giaquinta1983multiple [Gia83], Chapter V for the proof of this Lemma.We prove the following simple consequence of the above for reader’s convenience. lem corkscrew
Lemma 2.8.
Let u ∈ W , ( T ( x, R )) be a nonnegative solution of L in T ( x, R ) ,with u = 0 on ∆( x, R ) . Then for all < r < R/ , T ( x,r ) |∇ u ( X ) | dX ≈ u ( X x,r ) r , (2.9) eqcs1 where X x,r = ( x, r ) as above and the implicit constant depends only on d and µ .Proof. By translation invariance, we may assume that x is the origin.To prove the & inequality in ( eqcs1eqcs1 lem bdy reglem bdy reg lem bdy Harnacklem bdy Harnack u ( X x,r ) ≤ C sup T x,r/ u ≤ C T x,r u ( X ) dX ≤ Cr T x,r |∇ u | . For the . inequality in ( eqcs1eqcs1 (cid:3) We now record a basic regularity estimate for constant coefficient operators. Thiswill be used in the next section to get decay estimates for J u , and then extendedpartially to our more general operators L , with comparison arguments. We shallsystematically use A to denote a constant real ( d + 1) × ( d + 1) matrix, which wealways assume to satisfy the ellipticity condition ( cond ellpcond ellp L := − div ( A ∇ ).Solutions to such operators enjoy additional regularity and in particular, we willuse the following result. We state it in T = T (0 ,
1) to simplify the notation. Moregenerally, set T r = T (0 , r ) for r > l2a10 Lemma 2.10.
Let u ∈ W , ( T ) be a solution to L u = 0 in T with u = 0 on ∆ .Then for any multiindex α , | α | ∈ Z + , sup T | D α u | ≤ C (cid:18) T |∇ u ( X ) | dX (cid:19) / , (2.11) eq reg solL0 where C = C ( d, µ , | α | ) . In particular, for any T ( x, r ) ⊂ T / , osc T ( x,r ) ∂ i u ≤ Cr (cid:18) T |∇ u ( X ) | dX (cid:19) / , i = 1 , , . . . , d + 1 , (2.12) eq osc in use where the constant C depends only on the dimension and µ .Proof. First we claim that the standard local estimates on solutions for constant-coefficient operators in R d +1+ ensure that k D α u k L ( T / ) . k∇ u k L ( T ) + k u k L ( T ) . (2.13) eqloc ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 11
This is due to the fact that any weak solution to Lu = f on a smooth boundeddomain Ω and with zero Dirichlet boundary data satisfies k u k W m +2 , (Ω) . k f k W m, (Ω) + k u k L (Ω) , m = 0 , , , ... ;see, e.g., evans2010partial [Eva10], § W m, (Ω) is the Sobolev space offunctions whose derivatives up to the order m lie in L (Ω). With this at hand, weobserve that for any smooth cutoff function η equal to 1 on B / and supported in B / we have L ( uη ) = − A ∇ η · ∇ u − A ∇ u · ∇ η + u L η, and hence the estimate above applied consecutively with m = 0 , , ... in somesmooth domain T / ⊂ Ω ⊂ T gives ( eqloceqloc k D α u k L ( T / ) . k∇ u k L ( T ) (2.14) eqloc2 for any multiindex α with | α | ∈ Z + . On the other hand, by the Sobolev embeddingtheorem ( adams2003sobolev [AF03] Theorem 4.12), for any multiindex α ,sup T / | D α u | ≤ C k u k W | α | + n, ( T / ) , where C depends on n and | α | . We combine this with ( eqloc2eqloc2 eq reg solL0eq reg solL0 eq osc in useeq osc in use eq reg solL0eq reg solL0 T ( x,r ) ∂ i u ≤ r sup T ( x,r ) |∇ ∂ i u | ≤ r sup T / |∇ ∂ i u | ≤ Cr (cid:18) T |∇ u | (cid:19) / , as desired. (cid:3) r2 Remark . Lemma l2a10l2a10 mt1mt1 mt2mt2 ∇ u isLipschitz in T / , so in particular ∇ u − ∇ u (0) is small near the origin. Notice that ∇ u (0) = (0 , ∂ t u (0)) because u vanishes on the boundary; with this and similarstatements for other surface balls, it would be rather easy to control β u and provethe theorems. We don’t do this here because we need more general estimatesanyway.3. Approximations and the main conditional decay estimate sec const
We observed in Remark r2r2 L is a constant coefficient operator. In this section, we use the results of the previoussection, together with an approximation argument, to prove some decay estimatefor β u in regions where A is nearly constant. See Corollary cor itrcor itr ||∇ u − ∇ u || , where u is a solu-tion for L in some Carleson box T ( x, r ), and u is a solution for a close enoughconstant coefficient operator L , with the same boundary values on ∂T ( x, r ). SeeLemma lem comp u u0lem comp u u0 subsec orth A little more about orthogonality, J u , and β u . First return to the ap-proximation of a solution u by the affine function a x,r ( z, t ) = λ x,r t of ( a x,r is the best affine approximation of this typein T ( x, r ). Recall from ( J u ( x, r ) = T ( x,r ) |∇ ( u ( z, t ) − a x,r ( z, t )) | dzdt = T ( x,r ) |∇ u − λ x,r ( u ) e d +1 | dzdt = T ( x,r ) |∇ z u ( z, t ) | dzdt + T ( x,r ) | ∂ t u ( z, t ) − λ x,r ( u ) | dzdt (3.1) where e d +1 = (0 , . . . ,
1) is the vertical unit vector, and we split the full gradient ∇ u into the horizontal gradient ∇ x u and the vertical part ∂ t u . Now λ x,r ( u ) = ffl T ( x,r ) ∂ t u by ( ∂ t u − λ x,r ( u ) is orthogonal to constants in L ( T ( x, r )),hence for any other λ , T ( x,r ) | ∂ t u − λ | = | λ − λ x,r ( u ) | + T ( x,r ) | ∂ t u − λ x,r ( u ) | , and, by the same computation as above, T ( x,r ) |∇ ( u − λt ) | = | λ − λ x,r ( u ) | + T ( x,r ) |∇ u − λ x,r ( u ) e d +1 | = | λ − λ x,r ( u ) | + J u ( x, r ) . (3.2) We may find it convenient to use the fact that, as a consequence, β u ( x, r ) = inf λ ∈ R ffl T ( x,r ) |∇ ( u − λt ) | ffl T ( x,r ) |∇ u | ≤ . (3.3) (compare with ( λ = 0).For most of the rest of this section, we concentrate on balls centered at the origin;to save notation, we set B r = B (0 , r ), T r = T (0 , r ) = B r ∩ R d +1+ , and W r = W (0 , r )(see ( TTTT J u ( r ) = J u (0 , r ) = T r |∇ ( u ( x, t ) − λ r ( u ) t ) | dxdt, where λ r ( u ) = λ ,r ( u ) = T r ∂ s u ( y, s ) dyds (see ( E u ( r ) = E u (0 , r ), β u ( r ) = β u (0 , r ) (see ( def Edef E Decay estimates for constant-coefficient operators.
We shall now provea few estimates on solutions of constant-coefficient equation, which will be usefulwhen we try to replace L by a constant-coefficient operator. We start with aconsequence of Lemma l2a10l2a10 ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 13 lem u0-lambda t
Lemma 3.4.
Let A be a constant matrix that satisfies the ellipticity condition ( cond ellpcond ellp , set L = − div ( A ∇ ) , and and let u be a solution to L u = 0 in T such that u = 0 on ∆ . There exists some constant C , depending only on the dimension and µ , such that for < r < / , J u ( r ) ≤ Cr J u (1) ≤ Cr E u (1) . (3.5) eq u0-lambda t Proof.
The second inequality follows at once from ( λ = 0) for u . Next let v ( x, t ) = u ( x, t ) − λ r ( u ) t . Since t is a solution for the constant coefficient operator L , v is a solution for L as well in the domain in T , with v ( x,
0) = 0 for all x ∈ ∆ .We claim thatthere exists some ( x ′ , t ′ ) ∈ T r for which ∂ t v ( x ′ , t ′ ) = 0 . (3.6) claim Dtv=0 To see this, we observe first that ∂ t v ( x, t ) = ∂ t u ( x, t ) − ffl T r ∂ t u ( x, t ) dxdt has meanvalue 0. Since u is a solution of the constant-coefficient equation L u = 0, ∂ t u is also a solution of the same equation. Therefore, by the De Giorgi-Nash-Mosertheory, ∂ t u is continuous in T r , and thus so is ∂ t v . Then ( claim Dtv=0claim Dtv=0 T r and the mean value theorem. Thanks to ( claim Dtv=0claim Dtv=0 T r | ∂ t v | ≤ osc T r ∂ t v , and thus by ( eq osc in useeq osc in use T r | ∂ t v | ≤ (cid:16) osc T r ∂ t v (cid:17) = (cid:16) osc T r ( ∂ t v + λ r ( u ) − λ ( u )) (cid:17) = (cid:16) osc T r ∂ t ( u − λ ( u ) t ) (cid:17) ≤ Cr T |∇ ( u ( x, t ) − λ ( u ) t ) | dxdt. For the rest of the gradient, notice that for 1 ≤ j ≤ d , ∂ j v ( x, t ) = ∂ j ( v ( x, t ) − λ r ( u ) t + λ ( u ) t ) , and therefore, T r | ∂ j v | ≤ (cid:16) osc T r ∂ j ( v ( x, t ) − λ r ( u ) t + λ ( u ) t ) (cid:17) ≤ Cr T |∇ ( u ( x, t ) − λ ( u ) t ) | dxdt = Cr J u (1) . Now ( eq u0-lambda teq u0-lambda t (cid:3)
Remark . The proof of Lemma lem u0-lambda tlem u0-lambda t J u ( r ) in ( eq u0-lambda teq u0-lambda t ffl T r |∇ x,t ( u ( x, t ) − λ s ( u ) t ) | , for any 0 < s ≤ r . That is, we also get that T r |∇ x,t ( u ( x, t ) − λ s ( u ) t ) | dxdt ≤ Cr J u (1) . (3.8) This may be a better estimate, since ( λ , J u ( r ) ≤ T r |∇ ( u ( x, t ) − λ t ) | dxdt. We will need a lower bound for the ratio E u ( r ) E u (1) for positive solutions of L u = 0. lem lw bd Lemma 3.9.
Let the matrix A be constant and satisfy the ellipticity condition ( cond ellpcond ellp , set L = − div ( A ∇ ) , and let u be a positive solution to L u = 0 in T suchthat u = 0 on ∆ . Then E u ( r ) ≥ C (1 − C ′ r ) E u (1) for < r < / , (3.10) eqlb where C and C ′ are positive constants depending only on the dimension and µ . Notice that when r is small, the lower bound ( eqlbeqlb r .This is better than what would we would get by simply applying Lemma lem corkscrewlem corkscrew u . The proof exploits the fact that t is a solution for the constant-coefficient operator L and the comparison principle. Proof.
Define λ = ∂ t u (0 , eq osc in useeq osc in use | λ r ( u ) − λ | ≤ osc T r ∂ t u ≤ Cr (cid:18) T |∇ u | (cid:19) / . Since t is a solution for L that vanishes on ∆ , the comparison principle andLemma lem corkscrewlem corkscrew X x,t = ( x, t )) u ( x, t ) t ≥ C − u ( X x,t ) t ≥ C − (cid:18) T |∇ u | (cid:19) / for ( x, t ) ∈ T / , which implies, by taking a limit and using the existence of ∇ u at 0, that λ = ∂ t u (0 , ≥ C − (cid:18) T |∇ u | (cid:19) / . Then E u ( r ) ≥ λ r ( u ) ≥ λ − ( λ r ( u ) − λ ) ≥ ((2 C ) − − C ′ r ) T |∇ u | (use the fact that a ≥ b − ( a − b ) ). This completes the proof of Lemma lem lw bdlem lw bd (cid:3) Extension to general elliptic operators L . We now return to a solutionof our original equation Lu = 0, and compare it with solutions u of L u = 0 of aconstant coefficient operator L = − div ( A ∇ ), with the same boundary data. Forthe moment we do not say who is the constant matrix A (except that we require itto satisfy the ellipticity condition ( cond ellpcond ellp A in T .Even though it does not look like much, the next lemma is probably the centralestimate of this paper. We do not need A to have constant coefficients here. lem comp u u0 Lemma 3.11.
Let L = − div ( A ∇ ) and L = − div ( A ∇ ) be two elliptic operators,and assume that A and A satisfy the ellipticity condition ( cond ellpcond ellp . Let u be a solutionto Lu = 0 in T , with u = 0 on ∆ , and let u be a solution of L u = 0 in T with ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 15 u = u on ∂T . Then there is some constant C > depending only on d and theellipticity constant µ , such that ˆ T (cid:12)(cid:12) ∇ u − ∇ u (cid:12)(cid:12) ≤ µ min (cid:26) ˆ T | A − A | |∇ u | dX, ˆ T | A − A | (cid:12)(cid:12) ∇ u (cid:12)(cid:12) dX (cid:27) . (3.12) eqmin Proof.
The solutions are in the space W , ( T ) by definition, and u = u on theboundary should be interpreted as u − u = 0 in the sense of W , ( T ), or equiva-lently, u − u ∈ W , ( T ). So the existence of u ∈ W , ( T ) as above is guaranteedby the Lax-Milgram Theorem. Alternatively, it is possible to find u because thetrace of u lies in H / ( ∂B ). In addition, u is nonnegative by the maximum prin-ciple.Since u − u lies in the set W , of test functions allowed in Definition def weak soldef weak sol µ ˆ T (cid:12)(cid:12) ∇ ( u − u ) (cid:12)(cid:12) ≤ ˆ T A ∇ ( u − u ) · ∇ ( u − u ) = − ˆ T A ∇ u · ∇ ( u − u )= ˆ T ( A − A ) ∇ u · ∇ ( u − u ) ≤ µ ˆ T | A − A | (cid:12)(cid:12) ∇ u (cid:12)(cid:12) + 12 µ ˆ T (cid:12)(cid:12) ∇ ( u − u ) (cid:12)(cid:12) , where we use ( cond ellpcond ellp u is a solution of div( A ∇ ) u = 0 in T (and u − u vanishes on the boundary), then the fact that u is a solution of div( A ∇ ) u = 0in T , followed by the inequality 2 ab ≤ µ a + µ − b . Then ˆ T (cid:12)(cid:12) ∇ ( u − u ) (cid:12)(cid:12) ≤ µ ˆ T | A − A | (cid:12)(cid:12) ∇ u (cid:12)(cid:12) . This gives the bound by one of the expressions in the minimum in ( eqmineqmin u and u , and A and A , we also obtain the other bound. (cid:3) A similar proof also gives the following (which can be applied even if A − A isnot small). lem u=u^0 Lemma 3.13.
Let A , A , u , and u be as in Lemma lem comp u u0lem comp u u0 µ − ˆ T (cid:12)(cid:12) ∇ u ( X ) (cid:12)(cid:12) dX ≤ ˆ T |∇ u ( X ) | dX ≤ µ ˆ T (cid:12)(cid:12) ∇ u ( X ) (cid:12)(cid:12) dX, (3.14) eq u=u^0 where µ still denotes the ellipticity constant. We shall immediately see that u being a solution is not necessary for the firstinequality to hold, and similarly, u being a solution is not necessary for the secondinequality. But the condition u − u ∈ W , ( T ) is essential. Proof.
We estimate µ − ˆ T |∇ u | ≤ ˆ T A ∇ u · ∇ u = ˆ T A ∇ u · ∇ ( u − u ) + ˆ T A ∇ u · ∇ u = ˆ T A ∇ u · ∇ u ≤ µ (cid:18) ˆ T |∇ u | (cid:19) / (cid:18) ˆ T (cid:12)(cid:12) ∇ u (cid:12)(cid:12) (cid:19) / . Hence, ˆ T |∇ u | ≤ µ ˆ T (cid:12)(cid:12) ∇ u (cid:12)(cid:12) . The left-hand side of ( eq u=u^0eq u=u^0 u and u , A and A , respectively. (cid:3) Let us announce how we intend to estimate the right-hand side of ( eqmineqmin | A − A | in L ∞ norm and use the L norm of ∇ u ,but if we do this we will get quantities that do not seem to be controlled even bythe α ∞ of ( γ ( x, r ) = inf A ∈ A ( µ ) (cid:26) ( y,s ) ∈ T ( x,r ) | A ( y, s ) − A | dyds (cid:27) / , (3.15) where as before the infimum is taken over the class A ( µ ) of constant matrices A that satisfy the ellipticity condition ( cond ellpcond ellp eqmineqmin γ ( x, r ). l316 Lemma 3.16.
If the matrix-valued function A satisfies the weak DKP conditionof Definition d1a6d1a6 ε > , then γ ( x, r ) dxdrr is Carleson measure on R d +1+ , with norm (cid:13)(cid:13)(cid:13)(cid:13) γ ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C ≤ C N ( A ) ≤ Cε, (3.17) where N ( A ) = (cid:13)(cid:13) α ( x, r ) dxdrr (cid:13)(cid:13) C as in ( , and γ ( x, r ) ≤ C N ( A ) ≤ Cε for ( x, r ) ∈ R d +1+ . (3.18) Here C depends only on d and µ . See the next section for the proof.Since we do not have a small L ∞ control on A , we need a better estimate on ∇ u .This will be achieved by reverse H¨older estimates (e.g. Lemma lem RHlem RH p > d and µ . We first state the neededestimate for the unit box T . l319 Lemma 3.19.
Let u be a positive solution to Lu = 0 in T , with u = 0 on ∆ ,choose a constant matrix A ∈ A ( µ ) that attains the infimum in the definition ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 17 ( of γ (0 , , and let u be as in Lemma lem comp u u0lem comp u u0 A ). Thenfor any δ > , ˆ T (cid:12)(cid:12) ∇ u − ∇ u (cid:12)(cid:12) dX ≤ (cid:0) δ + C δ γ (0 , (cid:1) E u (1) , (3.20) where C δ depends on d , µ , and δ .Proof. We discussed the existence of u when we proved Lemma lem comp u u0lem comp u u0 eqmineqmin ˆ T (cid:12)(cid:12) ∇ u − ∇ u (cid:12)(cid:12) ≤ C ˆ T | A − A | |∇ u | . (3.21) Let us cut off and consider first the set Z := n X ∈ T : |∇ u ( X ) | ≤ KE u (1) o , with K > ˆ Z | A − A | |∇ u | ≤ KE u (1) ˆ Z | A − A | ≤ Kγ (0 , E u (1) . (3.22) eq Z1 In the region T \ Z where |∇ u | > KE u (1), we see that |∇ u | = |∇ u | p |∇ u | − p ≤ |∇ u | p ( KE u (1)) − p , where p > lem RHlem RH ˆ T \ Z | A − A | |∇ u | ≤ µ ˆ T \ Z |∇ u | ≤ µ ( KE u (1)) − p ˆ T |∇ u | p dX. (3.23) eq Z2 We required u to be a nice solution in the larger set T , so that we can use thefollowing estimates from Section sec nota defsec nota def
2. First, (cid:8) T |∇ u | p dX (cid:9) p ≤ C T |∇ u | dX by Lemma lem RHlem RH lem corkscrewlem corkscrew T (with X = (0 , T (with X = (0 , T |∇ u | ≤ Cu ( X ) ≤ Cu ( X ) ≤ C T |∇ u | , where the intermediate inequality follows from Harnack’s inequality. From theseestimates and ( eq Z2eq Z2 T \ Z is ˆ T \ Z | A − A | |∇ u | ≤ CK − p E u (1) . Now we choose K so that CK − p = δ , and the desired estimate ( (cid:3) We now have enough information to derive the same sort of decay estimatesfor the non-affine part of our solution u that we proved, at the beginning of thissection, for solutions u of constant coefficient operators. We start with an analogueof Lemma lem u0-lambda tlem u0-lambda t lem Jur Lemma 3.24.
Let u be a solution to Lu = 0 in T with u = 0 on ∆ . Then for < r < / , J u ( r ) ≤ C (cid:16) r + K − p r − d − (cid:17) J u (1) + C K r d +1 γ (0 , E u (1) , (3.25) Jur where
K > is arbitrary, p = p ( d, µ ) > , C depends only on d , µ and p , and C K depends additionally on K . Notice that we do not require the positivity of u yet, which is why we don’t useLemma l319l319 Proof.
We write u as affine plus orthogonal on T , i.e. u ( x, t ) = v ( x, t ) + λ ( u ) t. Note that λ ( u ) ≤ E u (1), and E v (1) = J u (1).Choose a constant matrix A ∈ A ( µ ) that attains the infimum in the definition( γ (0 , L = − div A ∇ as usual. Now consider the L -harmonicextension to T / of the restriction of u to ∂T / , which can be written as u ( x, t ) = v ( x, t ) + λ ( u ) t, (3.26) ext u0 where we use the fact that t is a solution of the constant-coefficient equation, and v is the L -harmonic extension of v | ∂T / . These extensions are well-defined since u isH¨older continuous on T / , and the Lax-Milgram Theorem guarantees the existenceand uniqueness of the W , ( T / ) solution. In particular, L u = 0 in T / , with u = u on ∂T / .We claim that for any fixed 0 < r < / J u ( r ) ≤ Cr J u (1) + Cr d +1 T / | A ( x, t ) − A | |∇ u ( x, t ) | dxdt. (3.27) To see this, we use the inequality ( a + b + c ) ≤ a + b + c ) to write J u ( r ) = T r |∇ ( u − λ r ( u ) t ) | ≤ T r |∇ ( u − λ r ( u ) t ) | + 3 T r |∇ ( u − u ) | + 3 T r |∇ ( λ r ( u ) t − λ r ( u ) t ) | , (3.28) est u-lambda r where λ r ( u ) = ffl T r ∂ t u is defined as for u . Notice that T r |∇ ( λ r ( u ) t − λ r ( u ) t ) | = ( λ r ( u ) − λ r ( u )) = (cid:18) T r ( ∂ t u − ∂ t u ) dxdt (cid:19) ≤ T r |∇ ( u − u ) | ≤ Cr d +1 T / |∇ ( u − u ) | , (3.29) eq lambdas ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 19 simply enlarging the domain of integration. So by ( est u-lambda rest u-lambda r lem u0-lambda tlem u0-lambda t lem comp u u0lem comp u u0 J u ( r ) ≤ T r |∇ ( u − λ r ( u ) t ) | + Cr d +1 T (cid:12)(cid:12) ∇ ( u − u ) (cid:12)(cid:12) = 3 J u ( r ) + Cr d +1 T |∇ ( u − u ) | ≤ Cr J u (1 /
2) + Cr d +1 T |∇ ( u − u ) | ≤ Cr J u (1 /
2) + Cr d +1 T | A − A | |∇ u | . (3.30) eq u-lambda r cont However, the same sort of computation as above yields J u (1 /
2) = T (cid:12)(cid:12) ∇ ( u − λ / ( u ) t ) (cid:12)(cid:12) ≤ T |∇ ( u − u ) | + 3 T (cid:12)(cid:12) ∇ ( u − λ / ( u ) t ) (cid:12)(cid:12) + 3( λ / ( u ) − λ / ( u )) ≤ C T |∇ ( u − u ) | + 3 T (cid:12)(cid:12) ∇ ( u − λ / ( u ) t ) (cid:12)(cid:12) = C T |∇ ( u − u ) | + 3 J u (1 / . We plug this into ( eq u-lambda r conteq u-lambda r cont eq lambdaseq lambdas J u ( r ) ≤ Cr J u (1 /
2) + Cr d +1 T / | A ( x, t ) − A | |∇ u ( x, t ) | dxdt. Now the claim ( J u (1 / ≤ T / |∇ ( u ( x, t ) − λ ( u ) t ) | dxdt ≤ CJ u (1) , where in the first inequality we have used that λ / ( u ) t is the best affine approxi-mation in T / (see the discussion in Section subsec orthsubsec orth u is decomposed as in ( ext u0ext u0 T / | A − A | |∇ u | ≤ T / | A − A | |∇ v | + 2 λ ( u ) T / | A − A | |∇ t | ≤ T / | A − A | |∇ v | + 2 E u (1) γ (0 , . (3.31) We now estimate the first term on the right-hand side of (
K >
0, considerthe set Z K := n X ∈ T / : |∇ v ( X ) | ≤ KE u (1) o . The contribution of Z K to the integral is ˆ Z K | A − A | |∇ v | ≤ KE u (1) ˆ Z K | A − A | ≤ CKγ (0 , E u (1) . We are left with the complement of Z K . As in ( eq Z2eq Z2 l319l319 ˆ T / \ Z K | A − A | |∇ v | ≤ C ( KE u (1)) − p ˆ T / |∇ v | p (3.32) ZK cmpl where p > ´ T / |∇ v | p , we use thefollowing two reverse H¨older type estimates: for some p = p ( d, µ ) > ˆ T / |∇ v | p ! /p . ˆ T / |∇ v | ! / + ˆ T / |∇ v | p ! /p , (3.33) eq RH1 ˆ T / |∇ v | p ! /p . (cid:18) ˆ T |∇ v | (cid:19) / + | λ ( u ) | (cid:18) T | A − A | p (cid:19) /p , (3.34) eq RH2 where the implicit constants depend on d , µ and p . We postpone the proof ofthese two inequalities to the end of the proof of this lemma.Now by ( eq RH1eq RH1 eq RH2eq RH2 ˆ T / |∇ v | p . E v (1 / p/ + E v (1) p/ + | λ ( u ) | p T | A − A | p . Since v − v ∈ W , ( T / ) and v is L -harmonic, we have E v (1 / ≤ C µ E v (1 / ≤ CE v (1) = CJ u (1) , where the first inequality comes from Lemma lem u=u^0lem u=u^0 T | A − A | p ≤ C µ ,p T | A − A | = Cγ (0 , . So our estimate on ´ T / |∇ v | p can be simplified as ˆ T / |∇ v | p . J u (1) p/ + E u (1) p/ γ (0 , . Plugging this into (
ZK cmplZK cmpl ˆ T / \ Z K | A − A | |∇ v | ≤ CK − p E u (1) − p J u (1) p/ + CK − p γ (0 , E u (1) ≤ CK − p J u (1) + CK − p γ (0 , E u (1) , where in the last inequality we have used E u (1) ≥ J u (1), and thus E u (1) − p ≤ J u (1) − p . Combining this with the contribution on Z K , we get ˆ T / | A − A | |∇ v | ≤ CK − p J u (1) + C (cid:16) K + K − p (cid:17) γ (0 , E u (1) . ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 21
From this and (
JurJur (cid:3)
Proof of ( eq RH1eq RH1 . We will use L p boundary estimates for solutions. Recall that L v = 0 in T / , with v − v ∈ W , ( T / ). Set R = 10 − n − / . Then bythe boundary estimates in giaquinta1983multiple [Gia83] p.154, we have for any X ∈ T / , Q R / ( X ) ∩ T / |∇ v | p . Q R ( X ) ∩ T / |∇ v | ! p/ + Q R ( X ) ∩ T / |∇ v | p . T / |∇ v | ! p/ + T / |∇ v | p for some p >
2. Since T / can be covered by finitely many cubes Q R / ( X ), weobtain ( eq RH1eq RH1 (cid:3) Proof of ( eq RH2eq RH2 . Set R = 10 − n − / as before. For any X = ( x , t ) ∈ T / , any0 < R ≤ R , choose η ∈ C ( Q R ( X )), with η ≡ Q R/ ( X ), |∇ η | . /R .Here, Q R ( X ) is a cube centered at X with side length R , and we shall write Q R for Q R ( X ) when this does not cause confusion. Using Lu = 0 in T , v ( x, t ) = u ( x, t ) − λ ( u ) t , and L t = 0, we have for any w ∈ W , ( T ),0 = ˆ T A ∇ u · ∇ w dxdt = ˆ T A ∇ v · ∇ w dxdt + ˆ T A ∇ ( λt ) · ∇ w dxdt = ˆ T A ∇ v · ∇ w + ˆ T ( A − A ) ∇ ( λt ) · ∇ w, (3.35) RH2 eq1 where λ = λ ( u ).Now we choose w ( X ) = v ( X ) η ( X ) when t ≤ R , and w = (cid:16) v − ffl Q R v ( Y ) dY (cid:17) η when t > R . Notice that v ( x,
0) = 0, and thus w ∈ W , ( T ) (because Q R ⊂ B )as required. We plug w into ( RH2 eq1RH2 eq1
Case 1: t ≤ R . Here we obtain1 µ ˆ T |∇ v | η dX ≤ µ ˆ T |∇ v | η dX + C µ ˆ T v |∇ η | dX + C µ | λ | ˆ T | A − A | η dX. Extending v by zero below t = 0, this yields ˆ Q R/ |∇ v | dX ≤ C µ R ˆ Q R v dX + C µ | λ | ˆ Q R | A − A | dX. We apply the Poincar´e-Sobolev inequality to control ´ Q R v dX and deduce fromthe above that Q R/ |∇ v | dX ≤ C (cid:18) Q R |∇ v | nn +2 dX (cid:19) n +2 n + C | λ | Q R | A − A | dX. (3.36) RH2 eq2
Case 2: t > R . The same computation as in Case 1 gives ˆ Q R/ |∇ v | dX ≤ CR ˆ Q R (cid:12)(cid:12)(cid:12) v ( X ) − Q R v ( Y ) dY (cid:12)(cid:12)(cid:12) dX + C | λ | ˆ Q R | A − A | dX. Then by the Poinca´re-Sobolev inequality, (
RH2 eq2RH2 eq2 giaquinta1983multiple [Gia83] V. Proposition 1.1 to obtain Q R / |∇ v | p dX ≤ C Q R |∇ v | dX ! p + C | λ | p Q R | A − A | p dX for some p = p ( d, µ ) > eq RH2eq RH2 T / can be covered by finitely many Q R / . (cid:3) We now prove an analogue of Lemma lem lw bdlem lw bd Lu = 0. lem lwbd u Lemma 3.37.
Let u be a positive solution of Lu = − div( A ∇ ) u = 0 in T , with u = 0 on ∆ . Then for any δ > , < r < / , E u ( r ) ≥ − C ′ r C − C ′′ (cid:0) δ + C δ γ (0 , (cid:1) r d +1 ! E u (1) (3.38) where C , C ′ , C ′′ are positive constants depending only on d and µ .Proof. As before, we will only find this useful when the parenthesis is under control.Let A and u be as in Lemma l319l319 T r |∇ u | ≥ T r (cid:12)(cid:12) ∇ u (cid:12)(cid:12) − T r (cid:12)(cid:12) ∇ ( u − u ) (cid:12)(cid:12) ≥ T r (cid:12)(cid:12) ∇ u (cid:12)(cid:12) − r d +1 T (cid:12)(cid:12) ∇ ( u − u ) (cid:12)(cid:12) ≥ T r (cid:12)(cid:12) ∇ u (cid:12)(cid:12) − C (cid:0) δ + C δ γ (0 , (cid:1) r d +1 T |∇ u | . (3.39) eq lwbd u est1 Divide both sides of ( eq lwbd u est1eq lwbd u est1 ffl T |∇ u ( X ) | , and then observe that T (cid:12)(cid:12) ∇ u ( X ) (cid:12)(cid:12) ≈ T |∇ u ( X ) | by Lemma lem u=u^0lem u=u^0 ffl T r |∇ u | ffl T |∇ u | ≥ ffl T r (cid:12)(cid:12) ∇ u (cid:12)(cid:12) ffl T |∇ u | − C (cid:0) δ + C δ γ (0 , (cid:1) r d +1 ≥ C − ffl T r (cid:12)(cid:12) ∇ u (cid:12)(cid:12) ffl T |∇ u | − C (cid:0) δ + C δ γ (0 , (cid:1) r d +1 . Since u > T (by the maximum principle), we can apply Lemma lem lw bdlem lw bd u and obtain the desired estimate. (cid:3) ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 23
We are finally ready to prove the announced decay estimate for the quantity β u ( x, r ) = J u ( x, r ) E u ( x, r ) (3.40) (the proportion of non-affine energy) defined in ( r = τ which willbe chosen small enough, depending on d and µ , and then we will require that γ (0 , ≤ ε , (3.41) for some ε > r = τ , d , and µ .Our first requirement for r = τ is that C ′ r < in ( ε and δ so small (depending on τ ) that if( C ′′ (cid:0) δ + C δ γ (0 , (cid:1) r d +1 < C in ( E u ( r ) ≥ C E u (1) . (3.42) Let u be as in Lemma lem lwbd ulem lwbd u JurJur E u ( r ) and getthat β u (0 , r ) ≤ C (cid:16) r + K − p r − d − (cid:17) J u (1) E u ( r ) + C K r d +1 γ (0 , E u (1) E u ( r ) (3.43) Then we choose K to satisfy K − p = r d +3 = τ d +30 , assume that ( C ) β u (0 , τ ) ≤ Cτ β u (1) + C τ γ (0 , . (3.44) Finally we choose τ so small that (in addition to our earlier constraint) Cτ < in ( ε as above.We recapitulate what we obtained so far in the next corollary. Of course, bytranslation and dilation invariance, what was done with the unit box T can alsobe done with any other T ( x, R ), ( x, R ) ∈ R d +1+ . We use the opportunity to statethe general case, which of course can easily be deduced from the case of T byhomogeneity (or we could copy the proof). cor itr Corollary 3.45.
We can find constants τ ∈ (0 , − ) and C > which dependonly on d and µ , such that if u is a positive solution of Lu = − div( A ∇ ) u = 0 in T ( x, R ) , with u = 0 on ∆( x, R ) , then β u ( x, τ R ) ≤ β u ( x, R ) + Cγ ( x, R ) . (3.46) See ( β u ( x, τ R ) and γ ( x, R ). Proof.
The discussion above gives the result under the additional condition that γ ( x, R ) ≤ ε . But we now have chosen τ and ε , and if γ ( x, R ) > ε , ( β u ( x, τ R ) ≤ (cid:3) r339 Remark . As we remarked before, the complication of the decay estimate for J u ( r ) comes mainly from the lack of a small control of k A − A k L ∞ . If we knew γ ∞ ( x, R ) ≤ ε , where γ ∞ ( x, r ) = inf A ∈ A ( µ ) sup T ( x,r ) | A − A | , then we could simplify the proof of Corollary cor itrcor itr J u ( r ) ≤ Cr J u (1) + Cr d +1 T / | A ( x, t ) − A | |∇ u ( x, t ) | dxdt, (3.48) which can be obtained as ( ffl T | A − A | |∇ u | now becomesrather simple. We still choose A as to minimize in the definition of γ (0 , x, t ) ∈ T such that | A ( x, t ) − A | ≤ Cγ (0 , ≤ Cγ ∞ (0 , . Since | A ( y, s ) − A ( x, t ) | ≤ γ ∞ (0 ,
1) for ( y, s ) ∈ T , we see that | A − A | ≤ Cγ ∞ (0 , ≤ Cε on T . Then T | A − A | |∇ u | ≤ T | A − A | |∇ ( u − λ ( u ) t ) | + 2 λ ( u ) T | A − A | ≤ T | A − A | |∇ ( u − λ ( u ) t ) | + 2 E u (1) T | A − A | ≤ ε J u (1) + 2 γ (0 , E u (1)and by ( J u ( r ) ≤ C (cid:18) r + Cε r d +1 (cid:19) J u (1) + Cγ (0 , r d +1 E u (1) . This is our analogue of (
JurJur
Carleson measure estimates sec cm
In this section we complete the proof of our two theorems. We already haveour main decay estimate ( β u ( x, r ) tends to get smaller andsmaller, unless γ ( x, r ) is large. This is a way of saying that γ dominates β u , andit is not surprising that a Carleson measure estimate on the first function implies asimilar estimate on the second one. The fact that β u comes from a solution u willnot play any role in this argument. See the second part of this section. ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 25 subs41
Proof of Lemma l316l316
Before we deal with decay, let us prove Lemma l316l316 u plays no role.Let A be as in the statement. We want to show that γ ( x, r ) dxdrr is Carlesonmeasure on R d +1+ , and our first move is to estimate γ ( x, r ) in terms of the α ( y, s ).For each pair ( x, r ), we choose a constant matrix A x,r such that W ( x,r ) | A − A x,r | = α ( x, r ) . (4.1) The interested reader may check that we can choose the A x,r so that they dependon ( x, r ) in a measurable way, and in fact are constant on pieces of a measurablepartition of R d +1+ , maybe at the price of replacing α ( x, r ) in ( α ( x, r ) ,and making the W ( x, r ) a little larger first to allow extra room to move x and r .Let ∆ = ∆( x , r ) be given; we want to estimate γ ( x , r ), and we try theconstant matrix A = A x ,r . Thus γ ( x , r ) ≤ T | A − A | ≤ C Q | A − A | , (4.2) where we set T = T ( x , r ) and Q = ∆( x , r ) × (0 , r ]. We will cut this integralinto horizontal slices, using the radii r m = ρ m r , m ≥
0. Let us choose ρ = ,rather close to 1, to simplify the communication between slices.We first estimate how fast the A x,r change. We claim that | A x,r − A y,s | ≤ Cα ( x, r ) + Cα ( y, s ) when | x − y | ≤ r and 23 r ≤ s ≤ r . (4.3) Indeed, with these constraints there is a box R in W ( x, r ) ∩ W ( y, s ) such that | R | ≥ C − r d +1 , and then | A x,r − A y,s | = R | A x,r − A y,s | ≤ R | A x,r − A | + R | A − A y,s |≤ C W ( x,r ) | A x,r − A | + C W ( y,s ) | A − A y,s | ≤ Cα ( x, r ) + Cα ( y, s )by the triangle inequality, the fact that | R | ≃ | W ( x, r ) | ≃ | W ( y, s ) | , and H¨older’sinequality. We can iterate this and get that for y ∈ R d and m ≥ | A y,r m − A y,r | ≤ C m X j =0 α ( y, r j ) . (4.4) Now consider y ∈ ∆ ′ = ∆( x , r /
2) and notice that by ( | A y,r − A | ≤ Cα ( y, r ) + Cα ( x , r ), so ( | A y,r m − A | ≤ Cα ( x , r ) + C m X j =0 α ( y, r j ) . (4.5) Set H m = ∆ × ( r m +1 , r m ] for m ≥
0; thus Q is the disjoint union of the H m . Weclaim that ˆ H m | A − A | ≤ Cr m α ( x , r ) | ∆ | + Cr m ˆ ∆ ′ n m X j =0 α ( y, r j ) o dy. (4.6) We tried to discretize our estimates as late as possible, but this has to happen atsome point. Cover ∆ with disjoint cubes R i of sidelength (10 √ d ) − r m that meet∆ , and for each one choose a point x i ∈ R i such that α ( x i , r m ) is minimal. Thenset A i = A x i ,r m and W i = R i × ( r m +1 , r m ]; notice that the W i cover H m .The contribution of R i to the integral in ( ˆ W i | A ( y, t ) − A | dydt ≤ C ˆ W i | A ( y, t ) − A i | + | A i − A y,r m | + | A y,r m − A | dydt. (4.7) For the first term, ˆ W i | A ( y, t ) − A i | dydt ≤ C | W ( x i , r m ) | α ( x i , r m ) (4.8) because W i ⊂ W ( x i , r m ) and by definition of α . Next ˆ W i | A i − A y,r m | dydt ≤ C ˆ W i ( α ( x i , r m ) + α ( y, r m )) dydt ≤ Cr m ˆ R i α ( y, r m ) dy by ( α ( x i , r m ) is smaller. This integral is at least as large as theprevious one, again because α ( x i , r m ) is smaller. When we sum all these terms over i , we get a contribution bounded by Cr m ´ ∆ ′ α ( y, r m ) , which is dominated by theright hand side of ( | A y,r m − A | is majorized in ( i , is also dominated by the right-hand side of ( H m cover Q , we see that ( γ ( x , r ) ≤ C Q | A − A | ≤ C | Q | − X m ˆ H m | A − A | ≤ S + S , (4.9) where S = | Q | − X m r m α ( x , r ) | ∆ | ≤ Cα ( x , r ) , (4.10) and S = | Q | − X m r m ˆ ∆ ′ n m X j =0 α ( y, r j ) o dy ≤ C ∆ ′ X m ρ m n m X j =0 α ( y, r j ) o dy (4.11) because r m = ρ m r and | Q | ≃ r | ∆ ′ | . We are about to apply Hardy’s inequality,which says that for 1 < q < + ∞ , ∞ X m =0 n m + 1 m X j =0 a j o q ≤ C p X m a qm (4.12) ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 27 for any infinite sequence { a m } of nonnegative numbers. Here we take q = 2 and a j = a j ( y ) = ρ j α ( y, r j ). Then X m ρ m n m X j =0 α ( y, r j ) o ≤ X m ρ m/ n m X j =0 ρ m/ α ( y, r j ) o ≤ X m ρ m/ n m X j =0 ρ j/ α ( y, r j ) o = X m ( m + 1) ρ m/ n m + 1 m X j =0 a j o ≤ C X m a m (4.13) so that S ≤ C ∆ ′ X m a m ( y ) dy = C X m ρ m ∆ ′ α ( y, r m ) dy. (4.14) We return to ( γ ( x , r ) ≤ Cα ( x , r ) + C X m ρ m ∆ ′ α ( y, ρ m r ) dy (4.15) We kept the squares because our Carleson measure condition is in terms of squares.Recall that by assumption, α satisfies a Carleson measure condition, with norm N ( A ). At this stage, deducing that the same thing holds for γ will only be amatter of applying the triangle inequality. We write this because of the varyingaverage in the second term of ( x , r ). It is enough to bound I = ˆ ∆ ˆ r γ ( x, r ) dxdrr ≤ C ˆ ∆ ˆ r α ( x, r ) dxdrr + C X m ρ m I m , (4.16) where I m = ˆ x ∈ ∆ ˆ r r =0 y ∈ ∆( x, r/ α ( y, ρ m r ) dy dxdrr . (4.17) Since ˆ ∆ ˆ r α ( x, r ) dxdrr ≤ C N ( A ) r d (4.18) by definition, we may concentrate on I m . Of course we apply Fubini. First noticethat y ∈ ∆ ′ = ∆( x , r /
2) when y ∈ ∆( x, r/
2) and x ∈ ∆; since x ∈ ∆( y, r/ x cancels with the normalization in the average,and we get that I m = ˆ y ∈ ∆ ′ ˆ r r =0 α ( y, ρ m r ) dydrr = ˆ y ∈ ∆ ′ ˆ ρ m r t =0 α ( y, t ) dydtt , (4.19) where the second identity is a change of variable (and we used the invariance of dtt under dilations). The definition also yields I m ≤ C N q ( A ) r d . So we can sum theseries, and we get that I ≤ C N q ( A ) r d . This completes our proof of ( We still need to check the second statement ( γ is not expected to vary too much. Indeed, weclaim that γ ( x, r ) ≤ Cγ ( y, s ) whenever | x − y | ≤ r and 2 r ≤ s ≤ r . (4.20) This is simply because T ( x, r ) ⊂ T ( y, s ), so if A is well approximated by a constantcoefficient matrix A in T ( y, s ), this is also true in T ( x, r ). Now we square, average,and get that γ ( x, r ) ≤ C y ∈ ∆( x,r ) s ∈ (2 r, r ) γ ( y, s ) dyds ≤ Cr − d ˆ y ∈ ∆( x,r ) ˆ s ∈ (2 r, r ) γ ( y, s ) dydss ≤ C || γ ( y, s ) dydss || C ≤ C N ( A ) . (4.21) This completes our proof of Lemma l316l316 (cid:3) r425
Remark . There is also a local version of Lemma l316l316 α ( x, r ) dxdrr is Carleson measure relative to some surface ball 3∆ (see Definition d13d13 γ ( x, r ) dxdrr is Carleson measure on T ∆ , with norm (cid:13)(cid:13)(cid:13)(cid:13) γ ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C (∆ ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) α ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C (3∆ ) . (4.23) As usual, C depends only on d . For this the simplest is to observe that sincewe use nothing more than the estimate ( x , r ) ∈ T ∆ , we may replace α ( y, t ) with 0 when ( y, t ) / ∈ T . Then thereplaced function α satisfies a global square Carleson measure estimate and wecan conclude as above.The fact that γ ( x, r ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) α ( x, r ) dxdrr (cid:13)(cid:13)(cid:13)(cid:13) C (3∆ ) (4.24) for ( x, r ) ∈ T ∆ can be proved as ( ∆ . subs42 Proof of Theorems mt1mt1 mt2mt2
We will just need to prove Theo-rem mt2mt2 A be as in the statement ofboth theorems.We recently completed our proof of Corollary cor itrcor itr β u ( x, τ r ) ≤ β u ( x, r ) + Cγ ( x, r ) (4.25) whenever u is a positive solution of Lu = − div( A ∇ ) u = 0 in T ( x, r ), with u = 0on ∆( x, r ).In the statement of our theorems, u is assumed to be a positive solution of Lu = 0in T ( x , R ), with u = 0 on ∆( x , R ), so ( x, r ) ⊂ ∆( x , R ). ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 29
We pick such a pair ( x, r ) and iterate ( β u ( x, τ k r ) ≤ − k β u ( x, r ) + C k − X j =0 − j γ ( x, τ k − j − r ) . (4.26) Hence (writing r in place of τ − k r ) β u ( x, r ) ≤ − k β u ( x, τ − k r ) + C k − X j =0 − j γ ( x, τ − j − r ) (4.27) as soon as ∆( x, τ − k r ) ⊂ ∆( x , R ).We want to prove the Carleson bound ( β u in ∆( x , τ R ), so we giveourselves a surface ball ∆ = ∆( y, r ) ⊂ ∆( x , τ R ). We want to show that ˆ T ∆ β u ( x, s ) dxdss ≤ Cτ a r d + C N r d , (4.28) where we set N = (cid:13)(cid:13) α ( x, r ) dxdrr (cid:13)(cid:13) C (∆( x ,R )) .Let us first check that β u ( x, s ) ≤ Cτ a + C N when x ∈ ∆ and 0 < s ≤ r . (4.29) When τ ≥ − , this is true just because ( x, s ) ∈ T ( x , τ R ) and ( β u ( x, s ) ≤
1. Otherwise, let k be the largest integer such that τ − k r < − R (notice that k ≥ x, τ − k r ) ⊂ ∆( x , R ), so ( τ − j − r are also smaller than 10 − R , so γ ( x, τ − j − r ) ≤ C N by ( r425r425 β u ( x, s ) ≤ − k + C N ,and ( a that depends only on τ (which itself dependsonly on d and µ ). This is because our choice of k gives τ k +10 ≤ r/R ≤ τ .Call I the integral in ( I = P ∞ k = − I k , with I k = ˆ T ∆ τ k +20 r
0, wehave that P k,j ; k − j = ℓ − j ≤
2. Hence when we sum over k , we get that X k ≥ I k ≤ C X k ≥ − k ( τ a + N ) r d + C X ℓ ≥ ˆ ∆ ˆ τ ℓ rτ ℓ +10 r γ ( x, t ) dxdtt = C ( τ a + N ) r d + C ˆ ∆ ˆ τ r γ ( x, t ) dxdtt ≤ C ( τ a + N ) r d , by Lemma l316l316 r425r425 Proof of Corollary cor maincor main sec cor
Let us first prove a Caccioppoli type result for solutions on Whitney balls. Sinceit is an interior estimate, it holds on any domain Ω ⊂ R d +1 . For X ∈ Ω, denote by δ ( X ) the distance of X to ∂ Ω. lem CcpType Lemma 5.1.
Let A be a ( d + 1) × ( d + 1) matrix of real-valued functions on R d +1 satisfying the ellipticity condition ( cond ellpcond ellp , and for some C ∈ (0 , ∞ ) , |∇ A ( X ) | δ ( X ) ≤ C for any X ∈ Ω . (5.2) Adist
Let X ∈ Ω ⊂ R d +1 be given, and r = δ ( X ) . Let u ∈ W , ( B r ( X )) be a solutionof Lu = − div( A ∇ u ) = 0 in B r ( X ) . Then for any λ ∈ R , ˆ B r/ ( X ) (cid:12)(cid:12) ∇ u ( X ) (cid:12)(cid:12) dX ≤ Cr ˆ B r/ ( X ) |∇ u ( X ) − λ e d +1 | dX + Cλ ˆ B r/ ( X ) |∇ A ( X ) | dX, (5.3) where C depends only on d , µ and C .Proof. By (
AdistAdist |∇ A ( X ) | ≤ C /r for any X ∈ B r/ ( X ), which means A is Lipschitz in B r/ ( X ). So from gilbarg2015elliptic [GT01] Theorem 8.8, it follows that u ∈ W , ( B ( X )). Let ϕ ∈ C ∞ ( B r/ ( X )), with ϕ = 1 on B r/ ( X ), k∇ ϕ k L ∞ ≤ ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 31 Cr . Write “ ∂ ” to denote a fixed generic derivative. Since u ∈ W , ( B ( X )), ∂ ( u − λt ) ϕ ∈ W , ( B r/ ( X )) for any λ ∈ R . Therefore, there exists { v k } ⊂ C ∞ ( B r/ ( X )) such that v k converges to ∂ ( u − λt ) ϕ in W , ( B r/ ( X )). Set I = ´ |∇ ∂u ( X ) | ϕ ( X ) dX . Observe that for any λ ∈ R , I = ˆ R d +1 |∇ ∂ ( u ( x, t ) − λ t ) | ϕ ( x, t ) dxdt. By ellipticity, we have I ≤ µ ˆ R d +1 A ( x, t ) ∇ ∂ ( u ( x, t ) − λ t ) · ∇ ∂ ( u ( x, t ) − λ t ) ϕ ( x, t ) dxdt = µ ˆ R d +1 A ∇ ∂ ( u − λ t ) · ∇ (cid:0) ∂ ( u − λ t ) ϕ (cid:1) dxdt − µ ˆ R d +1 A ∇ ∂ ( u − λ t ) · ∇ ϕ ∂ ( u − λ t ) ϕdxdt =: µ I − µ I . For I , we use Cauchy-Schwarz to get | I | ≤ µ I / (cid:18) ˆ R d +1 | ∂ ( u − λ t ) | |∇ ϕ | dxdt (cid:19) / ≤ I + C µ r ˆ B r/ ( X ) |∇ ( u − λ t ) | dxdt. For I , we use the sequence { v k } and write I k := ˆ R d +1 A ∇ ∂ ( u − λ t ) · ∇ v k dxdt = ˆ R d +1 ∂ ( A ∇ ( u − λ t ) · ∇ v k ) dxdt − ˆ R d +1 A ∇ ( u − λ t ) · ∇ ∂v k dxdt − ˆ R d +1 ∂A ( x, t ) ∇ ( u − λ t )) · ∇ v k dxdt. Note that the first term on the right-hand side vanishes because it is a derivativeof a W , ( R d +1 ) compactly supported function. Moreover, since Lu = 0 and v k ∈ C ∞ ( B r/ ( X )) is a valid test function, we have I k = λ ˆ R d +1 A ∇ t · ∇ ∂v k dxdt − ˆ R d +1 ∂A ( x, t ) ∇ ( u − λ t )) · ∇ v k dxdt. Let a d +1 be the last column vector of A , then we have ˆ R d +1 A ∇ t · ∇ ∂v k dxdt = ˆ R d +1 a d +1 · ∇ ∂v k dxdt = − ˆ R d +1 div a d +1 ∂v k dxdt. Hence, | I | = (cid:12)(cid:12)(cid:12)(cid:12) lim k →∞ I k (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) λ ˆ R d +1 div a d +1 ∂ ( ∂uϕ ) dxdt (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˆ R d +1 ∂A ( x, t ) ∇ ( u − λ t )) · ∇ ( ∂ ( u − λt ) ϕ ) dxdt (cid:12)(cid:12)(cid:12)(cid:12) =: I + I . For I , we use Cauchy-Schwarz, | div a d +1 | ≤ ( d + 1) |∇ A | , and Young’s inequalityto get I ≤ | λ | ˆ R d +1 | div a d +1 | ∂ uϕ dxdt + 2 | λ | ˆ R d +1 | div a d +1 | ∂ ( u − λt ) ϕ∂ϕdxdt ≤ | λ | (cid:18) ˆ (cid:12)(cid:12) ∂ u (cid:12)(cid:12) ϕ dxdt (cid:19) / (cid:18) ˆ | div a d +1 | ϕ dxdt (cid:19) / + 2 | λ | (cid:18) ˆ | ∂ ( u − λt ) | |∇ ϕ | dxdt (cid:19) / (cid:18) ˆ | div a d +1 | ϕ dxdt (cid:19) / ≤ I + Cr ˆ B r/ ( X ) | ∂ ( u − λt ) | dxdt + Cλ ˆ B r/ ( X ) |∇ A | dxdt. For I , we have I ≤ ˆ R d +1 | ∂A ( x, t ) | |∇ ( u − λt ) | |∇ ( ∂u ) ϕ | dxdt + 2 ˆ R d +1 | ∂A ( x, t ) ∇ ( u − λt ) · ∇ ϕ∂ ( u − λt ) ϕ | dxdt ≤ I / ˆ B r/ ( X ) | ∂A | |∇ ( u − λt ) | dxdt ! / + Cr ˆ B r/ ( X ) | ∂A | |∇ ( u − λt ) | dxdt. By (
AdistAdist X ∈ B r/ ( X ), δ ( X ) ≥ r/
2, one sees I ≤ I + C ( d, C ) r ˆ B r/ ( X ) |∇ ( u − λt ) | dxdt. Collecting all the estimates, we can hide I to the left-hand side and obtain thedesired estimate. (cid:3) Let us point out that the assumption (
AdistAdist A in Lemma lem CcpTypelem CcpType eq DKPeq DKP cor maincor main Proof of Corollary cor maincor main
Observe that ( eq DKPeq DKP |∇ A ( x, t ) | t ≤ CC for any( x, t ) ∈ R d +1+ for some C depending only on the dimension. ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 33
Fix ∆ ⊂ ∆( x , R/ x, r ) ∈ T ∆ , and write X = ( x, r/ λ x, r = λ x, r ( u ) be defined as in ( lem CcpTypelem CcpType B r/ ( X ) (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) dydt ≤ Cr B r/ ( X ) |∇ ( u ( y, t ) − λ x, r t ) | dydt + Cλ x, r B r/ ( X ) |∇ A ( y, t ) | dydt. Notice that B r/ ( X ) ⊂ W ( x, r ) = ∆( x, r ) × ( r, r ]. Hence we can enlarge theregion of the integrals on the right-hand side and then multiply both sides by u ( x, r ) − r to get ffl B r/ ( X ) (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) dydtu ( x, r ) r ≤ Cr ffl W ( x, r ) |∇ ( u ( y, t ) − λ x, r t ) | dydtu ( x, r ) + Cr λ x, r u ( x, r ) W ( x, r ) |∇ A ( y, t ) | dydt. By Lemma lem corkscrewlem corkscrew e α ( x, r ), λ x, r and β u ( x, r ), ffl B r/ ( X ) (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) dydtu ( x, r ) r ≤ C ffl W ( x, r ) |∇ ( u ( y, t ) − λ x, r t ) | dydtr ffl T ( x, r ) |∇ u ( y, t ) | dydt + C (cid:16) ffl T ( x, r ) ∂ t u ( y, t ) dydt (cid:17) e α ( x, r ) r ffl T ( x, r ) |∇ u ( y, t ) | dydt ≤ Cβ u ( x, r ) r + C e α ( x, r ) r . Now we apply Theorem mt1mt1 eq DKPeq DKP ˆ T ∆ ffl B r/ ( X ) (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) dydtu ( x, r ) r dxdr ≤ C ( d, C , µ ) | ∆ | . (5.4) eq corpf1 We now use Fubini and Harnack’s inequality to obtain a lower bound for the left-hand side of ( eq corpf1eq corpf1 ˆ T ∆ ffl B r/ ( X ) (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) dydtu ( x, r ) r dxdr = C d ˆ ( y,t ) ∈ R d +1+ (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) ˆ ( x,r ) ∈ T ∆ B r/ ( X ) ( y, t ) r − d u ( x, r ) dxdrdydt. Observe that if | ( y, t ) − ( x, r/ | ≤ t , then t ≤ r , and thus B r/ ( X ) ( y, t ) = 1.So the right-hand side is bounded from below by C d ˆ ( y,t ) ∈ T ∆ (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) ˆ ( x,r );( x, r/ ∈ B t/ ( y,t ) r − d u ( x, r ) dxdrdydt. By Harnack, u ( x, r ) ≤ Cu ( y, t ) when ( x, r/ ∈ B t/ ( y, t ). Hence ˆ T ∆ ffl B r/ ( X ) (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) dydtu ( x, r ) r dxdr ≥ C d ˆ ( y,t ) ∈ T ∆ (cid:12)(cid:12) ∇ u ( y, t ) (cid:12)(cid:12) t u ( y, t ) dydt. From this and ( eq corpf1eq corpf1 (cid:3) Optimality sec optm
In this section, we construct an operator that does not satisfy the DKP con-dition and such that β G ∞ ( x, r ) dxdrr fails to be a Carleson measure. Moreover,we find a sequence of operators { L n } that satisfy the DKP condition with con-stants increasing to infinity as n goes to infinity, and for any fixed 1 < R < ∞ , (cid:13)(cid:13) β n ( x, r ) dxdrr (cid:13)(cid:13) C (∆ R ) ≥ C ( n − β n ( x, r ) = β G ∞ n ( x, r ), and G ∞ n is the Greenfunction with pole at infinity for L n . A similar construction is used in DM2020 [DM20] Re-mark 3.2 and
DFM2020 [DFM20]. As we shall see, it is very simple to get a bad oscillatingbehaviour for G ∞ in the vertical direction; it is typically harder to get oscillation inthe horizontal variables, as would be needed for bad harmonic measure estimates.Let A ( x, t ) = a ( t ) I for ( x, t ) ∈ R d +1+ , where I is the d + 1 identity matrix,and a ( t ) is a positive scalar function on R + . Let L = − div A ( x, t ) ∇ . We claimthat the Green function with pole at infinity for L in R d +1+ is (modulo a harmlessmultiplicative constant) G ( x, t ) = g ( t ) with g (0) = 0 , g ′ ( t ) = 1 a ( t ) . (6.1) Ginfty
In fact, it is easy to check that LG = 0 in R d +1+ , G ( x, ≡
0, and the uniquenessof G ∞ does the rest. The derivatives of G are simple. They are ∇ x G ( x, t ) = 0 , ∂ t G ( x, t ) = 1 a ( t ) . (6.2) de G Now we set a ( t ) = when t ≥ , k + c k − ≤ t ≤ k +1 − c k , k +1 + c k ≤ t ≤ k +2 − c k +1 , for all k ∈ Z with k ≤
49, and a ( t ) is smooth in the remaining strips S k =(2 k − c k − , k + c k − ), with | a ′ ( t ) | ≤ c k for t ∈ S k = (2 k − c k − , k + c k − ) . Here, c > a ( t ) = in a small neighborhood of t = 2 k to simplifyour computations. ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 35
We construct the approximation of a ( t ) as follows. Set a n ( t ) = a ( t ) when t ≥ − n , when 0 < t < − n . Then a n converges to a pointwise in R d +1+ .Let L n = − div A n ( x, t ) ∇ = − div ( a n ( t ) ∇ ), and let G n be the Green functionwith pole at infinity for L n , whose formula are given in ( GinftyGinfty A n . Notice that |∇ A n | 6 = 0 only in thestrips near 2 k with width c k for − n ≤ k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ( y,t ) ∈ W ( x,r ) |∇ A n ( y, t ) | rdxdr (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ≈ (cid:13)(cid:13)(cid:13) | a ′ n ( t ) | tdxdt (cid:13)(cid:13)(cid:13) C ≈ X k = − n k ( c k ) c k ≈ n + 100 c . Similarly, we can compute the DKP constant for A . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ( y,t ) ∈ W ( x,r ) |∇ A ( y, t ) | rdxdr (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ≈ (cid:13)(cid:13)(cid:13) | a ′ ( t ) | tdxdt (cid:13)(cid:13)(cid:13) C ≈ c −
10 100 X k = −∞ ∞ . Now we turn to β n . Recall the definition of β ( x, r ) ( G n ( de Gde G b n ( t ) = a n ( t ) and compute β n ( x, r ) with T ( x, r )replaced by ∆( x, r ) × (0 , r ) in the definition of β ( x, r ); then β n ( x, r ) = ´ y ∈ ∆( x,r ) ´ rt =0 (cid:12)(cid:12)(cid:12) ∂ t G n ( y, t ) − ˜ \ ∆( x,r ) × (0 ,r ) ∂ t G n ( y ′ , t ′ ) dy ′ dt ′ (cid:12)(cid:12)(cid:12) dtdy ´ y ∈ ∆( x,r ) ´ rt =0 |∇ G n ( y, t ) | dtdy = ´ r (cid:12)(cid:12) b n ( t ) − ffl r b n ( s ) ds (cid:12)(cid:12) dt ´ r | b n ( t ) | dt . (6.3) beta_n The estimates with our initial definition of T ( x, r ) would be very similar, or couldbe deduced from the estimates with ∆( x, r ) × (0 , r ) because T ( x, r/ ⊂ ∆( x, r ) × (0 , r ) ⊂ T ( x, β n ( x, r ) = 0 when r < − n . We estimate (cid:13)(cid:13) β n ( x, r ) dxdrr (cid:13)(cid:13) C (∆ R ) forsome fixed R ≥
1. For simplicity, we only do the calculation when R < .The main observation is that for any 2 − n +2 ≤ r ≤ R , (cid:12)(cid:12)(cid:12)(cid:12) b n ( t ) − r b n ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≥ t ∈ [2 − n , r ] \ ( ∪ k S k ) . (6.4) bn lwbd Once we have ( bn lwbdbn lwbd (cid:13)(cid:13) β n ( x, r ) dxdrr (cid:13)(cid:13) C (∆ R ) asfollows. First, observe that the total measure of those S k that intersects [2 − n , r ] is controlled. Namely, (cid:12)(cid:12) ∪ k S k ∩ [2 − n , r ] (cid:12)(cid:12) ≤ − n + j +1 X k = − n c k ≤ c − n + j +2 ≤ c r, where j is the integer that 2 − n + j ≤ r < − n + j +1 . Therefore, ˆ r (cid:12)(cid:12)(cid:12)(cid:12) b n ( t ) − r b n ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) dt ≥ ˆ [2 − n ,r ] \ ( ∪ k S k ) dt ≥ − c r =: C r. On the other hand, we have ´ r | b n ( t ) | dt ≤ r since | b n | ≤
1. Then by the formula( beta_nbeta_n β n , we obtain β n ( x, r ) ≥ C for r ∈ [2 − n +2 , R ] . So sup 2) ln 2 + ln R ) ≥ C d,c (2 n − . Now we justify ( bn lwbdbn lwbd ffl r b n ( s ) ds takesvalue strictly between 1 and , so when t is away from the strips S k , b n ( t ) shouldbe different than ffl r b n ( s ) ds . We just need to make sure that the lower bound doesnot depend on n in a way that would cancel the blow up.We first simplify our computation of ffl r b n ( s ) ds by observing that we can take c = 0. This is because if c = 0, we can always require the average of b n in (0 , r )to be the same as the case when b n is not smoothed out (i.e. c = 0), as long as r does not lie in any strip S k , by choosing our a n carefully. But if r ∈ S k , this shouldnot affect ffl r b n ( s ) ds much if we take c to be sufficiently small.Fix 2 − n +2 ≤ r ≤ R . If 2 k ≤ r < k +1 for some k ∈ Z , then a directcomputation shows r b n ( s ) ds = 1 + 2 − n r − k r . If 2 k ≤ r < k +1 for some k ∈ Z , then r b n ( s ) ds = 12 + 2 − n r + 2 k +1 r . Since b n is either 1 or 1 / , r ) \ S k , a case-by-case computation shows that forany 2 − n +2 ≤ r ≤ R , (cid:12)(cid:12) b n ( t ) − ffl r b n ( s ) ds (cid:12)(cid:12) ≥ for t ∈ [2 − n , r ] \ S k . Then with c > bn lwbdbn lwbd References caffarelli1981existence [AC81] Hans Alt and Luis Caffarelli. Existence and regularity for a minimum problem withfree boundary. Journal f¨ur die reine und angewandte Mathematik , 1981(325):105–144,1981. 2 adams2003sobolev [AF03] Robert A Adams and John JF Fournier. Sobolev spaces . Elsevier, 2003. 11 ARLESON MEASURE ESTIMATES FOR THE GREEN FUNCTION 37 hofmann2017harmonic [AHM + 17] Jonas Azzam, Steve Hofmann, Jos´e Mar´ıa Martell, Mihalis Mourgoglou, andXavier Tolsa. Harmonic measure and quantitative connectivity: geometric char-acterization of the L p -solvability of the Dirichlet problem. Part I. arXiv preprintarXiv:1907.07102v2, To appear in Invent. Math , 2017. 2 DFM2020 [DFM20] Guy David, Joseph Feneuil, and Svitlana Mayboroda. Green function estimateson complements of low-dimensional uniformly rectifiable sets. arxiv preprint,arXiv:2101.11646 , 2020. 34 DM2020 [DM20] Guy David and Svitlana Mayboroda. Approximation of green functions and do-mains with uniformly rectifiable boundaries of all dimensions. arxiv preprint,arXiv:2010.09793 , 2020. 2, 3, 7, 34 dindos2007lp [DPP07] Martin Dindos, Stefanie Petermichl, and Jill Pipher. The L p dirichlet problem forsecond order elliptic operators and a p-adapted square function. Journal of FunctionalAnalysis , 249(2):372–392, 2007. 2, 4, 5 evans2010partial [Eva10] Lawrence Evans. Partial Differential Equations , volume 19. American MathematicalSoc., second edition, 2010. 11 giaquinta1983multiple [Gia83] Mariano Giaquinta. Multiple integrals in the calculus of variations and nonlinearelliptic systems . Number 105. Princeton University Press, 1983. 10, 21, 22 gilbarg2015elliptic [GT01] David Gilbarg and Neil Trudinger. Elliptic partial differential equations of secondorder . Springer, 2001. 30 hofmann2017uniform [HMM + 20] Steve Hofmann, Jos´e Mar´ıa Martell, Svitlana Mayboroda, Tatiana Toro, and ZihuiZhao. Uniform rectifiability and elliptic operators satisfying a carleson measure con-dition. arXiv preprint arXiv:2008.04834 , 2020. 2 hofmann2017implies [HMT17] Steve Hofmann, Jos´e Mar´ıa Martell, and Tatiana Toro. A ∞ implies NTA for a class ofvariable coefficient elliptic operators. Journal of Differential Equations , 263(10):6147–6188, 2017. 2, 3 kenig1994harmonic [Ken94] Carlos Kenig. Harmonic analysis techniques for second order elliptic boundary valueproblems , volume 83. American Mathematical Soc., 1994. 8 kenig2001dirichlet [KP01] Carlos Kenig and Jill Pipher. The Dirichlet problem for elliptic equations with driftterms. Publicacions Matematiques , pages 199–217, 2001. 2, 5 kenig1999free [KT99] Carlos Kenig and Tatiana Toro. Free boundary regularity for harmonic measures andpoisson kernels. Annals of Mathematics , 150(2):369–454, 1999. 2Guy David, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay,91405 Orsay, France E-mail address : [email protected] Linhan Li, School of Mathematics, University of Minnesota, Minneapolis, MN 55455,USA E-mail address : [email protected] Svitlana Mayboroda, School of Mathematics, University of Minnesota, Minneapolis,MN 55455, USA E-mail address ::