LITTLEWOOD-PALEY ESTIMATES: SOME APPLICATIONS

TO ELLIPTIC BOUNDARY VALUE PROBLEMS

Jill C. Pipher

§0 Introduction.

The square function has played an important role in many areas related to harmonic analysis - it has a long and rich history in complex analysis, the theory of singular integrals, differentiation theory, probability (martingale theory), and partial differential equations. We shall focus, in this article, on some applications of estimates involving square functions (estimates of Littlewood-Paley type) to certain elliptic boundary value problems. Some of these applications are fairly standard in this theory; others are more surprising. Square functions can be introduced in a variety of ways − associated with a given function f on R^{n−1} is really a family of such functions S(f) defined by means of some extension of f to R_{+}^{n} . A classical example is the harmonic extension u(x, t) = f ∗ P_{t}(x), where P_{t}(x) is the Poisson kernel on R^{n−1}. Then one defines the square function of f (or of u) by

S^{2}f(x_{o}) = ∫_{Γ(x_{0})} t^{2−n}|∇u(x, t)|^{2}dxdt

where Γ(x_{o}) is the cone {(x, t) : |x − x_{o}| ≤ t} with vertex at x_{o}.

Another important operator associated with an extension u of f is the nontangential maximal function u^{∗}, defined by u^{∗}(x_{o}) = sup{u(x, t) : (x, t) ∈ Γ(x_{o})}. The nontangential maximal function enters this theory in a natural way, when the object of investigation is nontangential limits of harmonic functions (or more generally, the boundary behavior of solutions to some elliptic equation). Thus, if f ∈ L^{p}(R^{n−1}) then we would like to solve the Dirichlet problem for Laplace’s equation Δu = 0 in R_{+}^{n} with data f on the boundary in the sense that u(x, t) converges to f nontangentially, that is, as (x, t) approaches x_{o} through points of Γ(x_{o}) To control such limits, one studies a related maximal operator. This is familiar from, for example, Lebesgue’s differentiation theorem where the Hardy-Littlewood maximal operator appears as the essential operator of study. The important estimate in the solution of the Dirichlet problem above is that ||u^{∗}||_{L^{p}(R^{n−1})} be controlled by ||f||_{L^{p}(R^{n−1})}, which fact gives uniqueness in this problem.

The basic relation between the nontangential maximal function and the square function is the equivalence of ||u^{∗}||_{L^{p}(R^{n−1})} and ||S(u)||_{L^{p}(R^{n−1})} for all 0 < p < ∞, for u harmonic and decaying at ∞. (See [BG],[St].) In addition, when 1 < p ≤ ∞,

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 35J25, 35B65; Secondary 35J05.

the L^{p} Dirichlet problem on R_{+}^{n} described above can be solved uniquely. Hence, in particular we have the estimate ||f||_{L^{2}(R^{n−1})} = c_{n}||S(f)||_{L^{2}(R^{n−1})}, where c_{n} is constant which depends only the dimension and aperture of the cones used to define S. It is worthwhile to sketch the proof of this L^{2} result since it is not hard and it shows how the equation is used. We have

∫_{R^{n−1}} S^{2}(u)(x)dx = ∫∫_{(y,t)∈R_{+}^{n}} |∇u(y, t)|^{2}t^{2−n}{∫_{R^{n−1}} 1_{Γ(x)}(y, t)dx}dydt
= c_{n} ∫∫_{(y,t)} |∇u(y, t)|^{2}tdtdy
= ∫_{R^{n−1}} u^{2}(x, 0)dx = ∫ f^{2}(x)dx.

In the last equality we used the fact that 2|∇u(y, t)|^{2} = Δ(u^{2}) for harmonic u and then applied the divergence theorem.

Both the square function and maximal function can be defined for other extensions as well as on other domains. For instance, for f defined on R^{n−1} one has a Calderón-Torchinsky representation of f by a solid integral in the upper half space R_{+}^{n}:

(0.1)  f(x) = ∫_{0}^{∞} f ∗ Ψ_{t}(y) ∗ Ψ_{t}(y − x)\frac{dt}{t}

where Ψ_{t}(y) = t^{−(n−1)}Ψ(y/t) satisfies ∫_{0}^{∞} |Ψ̂(s)|^{2} \frac{ds}{s} = 1. (To verify (0.1), apply the Fourier transform to both expressions.) Then the natural square function associated to this expression will be ∫∫_{Γ(x)} |f ∗ Ψ_{t}(y)|^{2}t^{−n}dxdt. On suitably regular domains, one may use the solvability of some equation to get an extension of a function defined on the boundary, and then define a square function associated to this extension. The domain in question should satisfy a uniform cone condition so that to each boundary point is associated a family of cones, with varying axes but uniform aperture. Thus if Ω is a Lipschitz domain, it satisfies a uniform interior and exterior cone condition, and if Γ(Q) is the cone with vertex at Q ∈ ∂Ω, one defines

S(u)^{2}(Q) = ∫_{Γ(Q)} |∇u(X)|^{2}δ(X)^{2−n}dX; and u^{∗}(Q) = sup{u(X) : X ∈ Γ(Q)}

where δ(X) is the distance of X to ∂Ω. When u is harmonic and properly normalized, Dahlberg ([D2]) showed that

(0.2)  ||u^{∗}||_{L^{p}(∂Ω,dσ} ˜= ||Su||_{L^{p}(∂Ω,dσ)}, for all 0 < p < ∞

with comparability constants depending only the Lipschitz character of Ω. If p ≥ 2 we also have that ||u^{∗}||_{L^{2}} and ||f||_{L^{2}} are comparable when u is the harmonic extension of f to Ω ([D1]). The “2” here is sharp for the reason that the Dirichlet problem for harmonic functions is uniquely solvable in the class of Lipschitz domains only for p ≥ 2.

Estimates for square functions have been particularly useful in the theory of boundary value problems in roughly two ways. In the first place, the square function is the tool for handling Riesz transforms and, secondly, such estimates give sharp interior regularity of solutions in terms of Sobolev spaces. We can illustrate the first principle with an example from complex function theory. Let u be a harmonic function in the upper half plane with boundary values f in L^{2}(R). Let ũ be its conjugate function - that is, u + iũ is analytic. Then ũ(x, 0) = H(f) where H is the Hilbert transform. Since, by the Cauchy-Riemann equations, |∇u|^{2} = |∇ũ|^{2}, the square functions of u and ũ are equal. Consequently, one sees that ||H(f)||_{L^{2}} ˜= ||f||_{L^{2}} is equivalent to the L^{2} comparability of a harmonic funtion and its square function. The higher dimensional singular integral operators which generalize the Hilbert transform are the Riesz transforms. Alternatively, one can define the Riesz transforms in terms of derivatives of harmonic functions, and this is a point of view which can be extended to solutions of other equations. Thus, if u(x, t) = f ∗ P_{t}(x) is harmonic in R_{+}^{n} and v(x, t) is defined by u(x, t) = ∂v(x, t)/∂t then R_{j}(f), the jth Riesz transform is simply ∂u(x, 0)/∂x^{j} . That is, the L^{2} estimates for the square function of a harmonic function in the upper half space imply that

(0.3)  ∫_{R^{n−1}} ∑_{j} |D_{x_{j}}u|^{2}dx = ∫_{R^{n−1}} |D_{t}u|^{2}dx.

In general, from the fact that S(f) belongs to L^{p} one can conclude that Riesz transforms of f belong to L^{p} as well, and this will be one essential application in some of the following material.

The discussion of sharp interior Sobolev space regularity of solutions is postponed until Section 1, when some further definitions and terminology are introduced. The rest of the paper is organized into short sections, each containing an application of square function inequalities to some elliptic boundary value problem. Except for some notation, each section may be read independently. Section 1 introduces most of this notation and discusses boundary value problems for Laplace’s equation on Lipschitz domains. This includes a brief description of the work of Jerison and Kenig on the inhomogeneous Dirichlet problem with data in trace spaces as well as some extensions of these results to the biharmonic operator in non-smooth domains. In Section 2, we discuss the Agmon-Miranda maximum principle for higher order elliptic operators. This weak version of the maximum principle has been the object of investigation in non-smooth domains. General results asserting its validity have been obtained in [PV1] and [PV2], and for second order systems in [DK], using methods of harmonic analysis which require square function estimates. (There are analogous results in polygonal domains. See for example [MP], [MR1] and [MR2].) In sections 3 and 4 we return to the subject of second order elliptic operators of divergence form. That is, we study operators of form L = div∇A where A is a matrix of bounded measurable coefficients. There is a great deal now known about what sort of additional smoothness conditions one needs to place on the matrix A in order to solve a Dirichlet or Neumann boundary value problem. But in all of these cases, the matrix is assumed to be real valued and symmetric. The exact nature of the role played by both of these assumptions is little understood, except that there are very few techniques and ideas that work in their absence. In Section 3, we consider some problems for complex valued matrices and give an elementary proof of a well-known result in one dimension which is connected to a question of Kato. In Section 4, we show how square function estimates can be used to imply solvability of the Dirichlet problem for L = div∇A even when the (real) matrix is not symmetric.

The results described in Sections 1,2 and 4 have been or will be published elsewhere; therefore, we won’t include too many details. But the ideas in Section 3 appear here only, so the presentation is a bit more detailed. In any case, the main point of this article is to connect several disparate areas of elliptic boundary value problems through a common thread, a tool and a concept which had its origins in harmonic analysis.

Portions of this paper describe work done jointly with other co-authors - we gratefully acknowledge our collaborations with V. Adolfsson, B. Dahlberg, R. Fefferman, C. Kenig, T. Toro, and G. Verchota.

§1 Regularity of solutions

Our first example of this type of regularity result is the solvability of the Dirichlet, Neumann and regularity problems for Laplace’s equation in Lipschitz domains ([D1], [JK1], [V1]).

Given a domain Ω in R^{n}, which is regular for the Dirichlet problem, there is a family of measures on ∂Ω associated with the Laplacian as follows. For each continuous function f defined on ∂Ω, there exists a harmonic function u_{f} having boundary data f which belongs to C(Ω̄). By the maximum principle for harmonic functions, the map f → u_{f}(X) from continuous functions on ∂Ω, whose value at f is the harmonic extension evaluated at a point X in Ω, is a continuous linear functional. Hence, there is a representing measure dω^{X} for each X in Ω such that u_{f}(X) = ∫_{∂Ω} f(Q)dω^{X}(Q). From the Harnack principle, these measures are all mutually absolutely continuous - so we shall fix a point Y in the domain, say the origin, and refer to dω^{Y} = dω as the harmonic measure for Ω. Then the regularity of harmonic functions in Ω is directly connected to the relationship between the harmonic measure of a domain and the n − 1 dimensional Lebesgue measure on the boundary.

When Ω is a domain with Lipschitz boundary in R^{n}, Dahlberg showed that dω and surface measure dσ were mutually absolutely continuous. Indeed he proved a much stronger fact. To state this precisely, we need to define the weight classes B^{p} and A^{p}. For a more complete discussion of these matters, see the book [GC-RdF]. Let B(∂Ω) denote the class of all surface balls on the boundary, i.e. the intersection of spheres in R^{n} centered at a boundary point with ∂Ω. Given a weight ω_{1} and a measure dω_{2}, we say that ω_{1} belongs to A^{p}(dω_{2}) iff

sup_{R∈B(∂Ω)}(\frac{1}{ω_{2}(R)} ∫_{R} ω_{1}dω_{2})(\frac{1}{ω_{2}(R)} ∫_{R} ω_{1}^{−1/(p−1)}dω_{2})^{p−1} < ∞

A weight ω_{1} belongs to the class B^{p}(dω_{2}) if there exists a constant C such that for every R ∈ B(∂Ω), it satisfies the reverse Holder inequality:

(\frac{1}{ω_{2}(R)} ∫_{R}(ω_{1})^{p}dω_{2})^{1/p} ≤ C(\frac{1}{ω_{2}(R)} ∫_{R} ω_{1}dω_{2}).

For example, let’s take (dω_{2}) to be Lebesgue measure dx. Then a weight ω belongs to the class A^{p}(dx) if and only if the Hardy-Littlewood maximal operator is bounded on L^{p}(R^{n}, ωdx) ([Mu]).

The class A^{∞} can be defined to be ⋃_{p>0} A^{p} = ⋃_{p>0} B^{p}. There are also intrinsic definitions which show that an A^{∞} condition is a quantitative version of mutual absolute continuity.

The result of [D1] is that for any Lipschitz domain Ω in R^{n}, harmonic measure and surface measure are mutually absolutely continuous - that is, there exists a weight k such that dω = kdσ. In addition, the weight k belongs B^{2}(dσ); by duality, this is equivalent to asserting that k^{−1} belongs to A^{2}(dω), where kdσ = dω. Since the nontangential maximal function of u_{f} is bounded by a weighted Hardy-Littlewood maximal operator M_{ω}f - just replace Lebesgue measure in the definition of the maximal operator by weighted Lebesgue measure - one obtains solvability of the Dirichlet problem with data in L^{2} by controlling ||M_{ω}f||_{L^{2}(∂Ω)}. That is, the problem

(1.1)  \{ \begin{matrix}Δu = 0 & in Ω \\ lim_{X→Q,X∈Γ(Q)} u(X) = f(Q) & Q ∈ ∂Ω, \end{matrix}

with ||u^{∗}||_{L^{2}(∂Ω,dσ} < C||f||_{L^{2}(∂Ω,dσ} can be solved uniquely with a constant C that depends only the Lipschitz character of Ω. From the equivalence of S(u) and u^{∗} in L^{p}(∂Ω), and for p = 2 in particular, it follows that ∫_{Ω} |∇u|^{2}δ(X)dX is finite. This says that u belongs to the Sobolev space W^{1/2,2}(Ω) of functions with 1/2 derivative in L^{2}(Ω). This interior regularity of solutions with boundary data in L^{2} is sharp for this class of domains. Let’s look at the analogous result for the regularity problem ([JK1]) for the Laplacian. In (1.1), assume now that f(Q) has a tangential derivative in L^{2} (f ∈ W^{1,2}(∂Ω)). Now the object is to obtain L^{2} estimates on all derivatives of u: i.e. ||(∇u)^{∗}||_{L^{2}(∂Ω,dσ)} ≤ C||f||_{W^{1,2}(∂Ω,dσ)}. (Recalling (0.3), observe that this inequality shows that the normal derivative of u, \frac{∂u}{∂N} is bounded above by its tangential derivatve in L^{2} on ∂Ω.) Again by using the square function, one obtains sharp interior regularity of solutions, namely that for W^{1,2} data, u ∈ W^{3/2,2}(Ω).

In [JK2], Jerison and Kenig study the inhomogeneous Dirichlet problem for Δ in Lipschitz domains. They consider the problem

(1.2)  \{ \begin{matrix}Δu = & f in Ω \\ u|_{∂Ω} = & 0 \end{matrix}

where f is in the Sobolev space W^{α,p}(Ω) or a Besov space B^{α,p}(Ω). When the ∂Ω is smooth, (1.2) is uniquely solvable for u satisfying ||u||_{W^{α+2,p}(Ω)} ≤ C||f||_{W^{α,p}(Ω)} when 1 < p < ∞ and α + 2 > 1/p. On Lipschitz, or even C^{1} domains, this problem is not no longer solvable in exactly this range - the strategy of reduction to the homogeneous problem doesn’t work for certain of these function spaces. This reduction fails when the restriction theorem for Sobolev spaces fails, i.e., when the trace space on the boundary is not what one expects from extrapolating from the smooth situation. In [JK2], they find the sharp range of p and α for which the problem above is solvable on these non-smooth domains. For example, if p = 2, then one may take α in the range −3/2 < α < −1/2. There were two important tools in their paper - the investigation of traces of Sobolev spaces on the boundary, and characterizations of Sobolev/Besov spaces of harmonic functions. In particular, it was shown that, if u is harmonic in Ω, p > 1 and 0 ≤ α ≤ 1 then

u ∈ W^{α,p}(Ω) ⇔ u ∈ B^{α,p}(Ω) ⇔ ∫_{Ω} |δ(X)^{1−α}∇u|^{p}dX + ∫_{Ω} |u|^{p}dX < ∞.

When α = 1/2 and p = 2, the solid integral expression which characterizes membership in the corresponding Sobolev spaces is exactly the L^{2}(∂Ω) norm of S(u). In this way, the square function estimates play a key role in their results.

In [AP], the analogous inhomogeneous problem was treated for functions biharmonic in Lipschitz or C^{1} domains - the clamped plate problem. The appropriate square function estimates had already been established - the main new difficulties arose in the interpolation of boundary spaces of arrays of functions. In the next section, we limit the discussion to the homogeneous problem for the biharmonic equation, and describe a completely different use for the square function estimates in this context.

§2 Higher Order Elliptic Operators

Let us begin by focussing on the biharmonic operator, Δ^{2} - later we shall mention some results for more general higher order operators. The biharmonic operator arises in physical problems, such as theory of elastostatics, as well as the clamped plate problem - solving Δ^{2}u = f with zero Dirichlet data. The difficulty in dealing with solutions to these higher order equations is that many of the useful principles one has in the second order situation no longer obtain. For instance, we have no principle of positivity of solutions, there is no natural measure on the boundary associated with solving the Dirichlet problem akin to harmonic measure, and there is no maximum principle for solutions on general domains. There are weak maximum principles - but these so-called maximum principles of Agmon-Miranda type had been established only on smooth domains. Indeed, an example due to Mazya-Nazarov-Plameneskii [MNP] shows that such a maximum principle may fail if the domain is the exterior of a cone in R^{5}. (This is to be contrasted to positive results obtained by Mazya et al for polygonal domains in R^{3} by asymptotic methods.)

In several papers with G. Verchota, we investigated the questions of solvability of the L^{p} Dirichlet problems for Δ^{2} as well as for more general constant coefficient homogeneous operators L = ∑_{|α|=2m} a_{α}D^{α}. Of particular interest to us is the p = ∞ case of the Dirichlet problem, since it is one way of formulating the weak maximum principle of Agmon-Miranda type for gradients of solutions. We showed ([PV1], [PV2]) that this maximum principle was valid on any C^{1} domain in R^{n} for any n, and on any Lipschitz domain in R^{n} as long as n ≤ 3. We also found a counterexample in dimension n = 4 that showed that this three dimensional result is sharp. Let us state precisely what the L^{p} Dirichlet problem is for the biharmonic operator on such domains. As in the harmonic case the boundary data is taken on in the sense of nontangential convergence. Our Dirichlet problem then takes the following form: given f ∈ W^{1,p}(∂Ω) and g ∈ L^{p}(∂Ω), we want to solve

(2.1)  \{ \begin{matrix}Δ^{2}u = 0 & in Ω \\ u = f & a.e. on ∂Ω \\ \frac{∂u}{∂N} = g & a.e. on ∂Ω \end{matrix}

with the estimate ||(∇u)^{∗}||_{L^{p}} ≤ C(||f||_{W^{1,p}(∂Ω)} + ||g||_{L^{p}(∂Ω)}). The L^{2} Dirichlet problem for Δ^{2} was solved in the class of Lipschitz domains in [DKV] in 1984. The issue left open was whether or not this result was optimal. Our work with Verchota showed that the answer was dimension dependent in the sense that only in R^{3} could the L^{p} Dirichlet problem be solved for all 2 ≤ p ≤ ∞. While it is known that there is a p > 2 for which the L^{p} Dirichlet problem is not uniquely solvable in higher dimensions, the optimal result remains open. Finally, the analagous results for the Dirichlet problem for general constant coefficient elliptic operators was only recently carried out in [PV4].

There is another boundary problem - the regularity problem - whose L^{p} solvability near p = 1 turns out to be crucial for the validity of maximum principles. Here the highest order data g in (2.1) above should belong to the boundary Sobolev space W^{1,p}(∂Ω) and the desired estimate is on ||(∇∇u)^{∗}||_{L^{p}} in terms of the data. Obviously there will be some delicacy in formulating what is meant by \frac{(∂)^{k}u}{(∂N)^{k}}, when the normal vector is not smooth nor even continuous. Roughly speaking, one is forced to consider arrays of functions whose components satisfy a compatibility condition and which have tangential derivatives in L^{p}. The first formulation of these boundary conditions was given in Verchota [V2]. (See also [JW].)

For second order elliptic systems on Lipshitz domains, this regularity problem had been solved in the range p < 2 in [DK] in R^{3}. In fact, their result was the first indication that these problems might be solvable in three dimensions even if not uniquely solvable in general.

Itturns out that the Dirichlet problem is dual in a certain strong sense to the regularity problem. This duality is effected by means of theorems which yield the appropriate square function estimates. In the case of the biharmonic operator, the theorem ([PV3]) says that for any Lipschitz domain Ω ⊂ R^{n}, if u is biharmonic in Ω and satisfies ∇u(P) = 0 for some P ∈ Ω, then

(2.2)  for all 0 < p < ∞, ||S(∇u)||_{L^{p}(∂Ω)} ˜= ||(∇u)^{∗}||_{L^{p}(∂Ω)}.

For operators of order 6 and higher, like the polyharmonic operators Δ^{m}, one can compare the L^{p} norms of the nontangential maximal function and square function of (∇^{m−1}u). When in addition the Dirichlet problem (2.1) is solvable, a result like (2.2) gives sharp Sobolev space regularity of solutions, just as we obtained in the harmonic case.

Let us now sketch how the square function estimates are used in the proof of a weak maximum principle for gradients of solutions to the biharmonic equation in Lipschitz domains. (For the biharmonic operator, we only need the square function inequalities (theorem (0.1)) for harmonic functions. To obtain a similar result for operators L of order higher than 6, one need the results of [DKPV] for all such L.) In R^{3}, it turns out that there is an estimate for the Green’s function of the operator Δ^{2} which is related to the solvability of the L^{p} regularity problem in the range 1 < p < 2. If G(X, Y) is the Green’s for Δ^{2} in the domain Ω, i.e., Δ^{2}G(X, .) = δ_{X}(.) with G(X, .) and \frac{∂G(X,.)}{∂N} vanishing on ∂Ω, the key estimate is the following:

(2.3)  ||(Δ_{Q}G(X, Q))^{∗}||_{L^{1}(∂Ω)} ≤ Cdist(X, ∂Ω)

where C depends only on dimension and the Lipschitz character of Ω. In particular, this is a dilation invariant estimate and it suffices to establish this when dist(X, ∂Ω) = 1. It is exactly this L^{1} estimate which fails in dimensions greater than 4 - however it is valid in all dimensions for slightly smoother domains (C^{1}).

Assuming the estimate (2.3), let us suppose that we have an L^{2} solution to the Dirichlet problem (2.1) whose highest order data belongs to L^{∞}(∂Ω) and whose lowest order data has a tangential derivative in L^{∞}(∂Ω). That is, on the boundary of the domain, the full gradient of the biharmonic function u is bounded. We want to prove that ∇u belongs to L^{∞} in the interior of the domain as well. Our solution has a boundary integral representation in terms of the Green’s function for Ω:

\begin{matrix}u(X) = & ∫_{∂Ω)} \frac{∂u(Q)}{∂N}Δ_{Q}G(X, Q)dσ(Q) + ∫_{∂Ω)} u(Q)\frac{∂}{∂N} Δ_{Q}G(X, Q)dσ(Q) \\ = & I + II \end{matrix}

The object is to show that ∇u(X) is bounded by a constant depending only the L^{∞}(∂Ω) norm of the boundary data. Consider first term I above. This is a biharmonic function of X in Ω and by standard interior estimates for solutions, it suffices to bound the quotient u(X)/dist(X, ∂Ω). But by (2.3), this is immediate since the normal derivative of u is bounded.

The theorem (0.1) is required to handle the lower order data in term II above. The difficulty with this term is the presence of three derivatives on the Green’s functions and “not enough” derivatives on u. To remedy this one introduces the Riesz transforms {R_{j}} in the following way. Recall that if h is harmonic, then the normal derivative of h can be expressed in terms of tangential derivatives by the relationship

\frac{∂h}{∂N} = ∑_{j=1}^{n−1} \frac{∂h}{∂T_{j}} R_{j}(h).

This relationship can be verified by writing h as D_{x_{n}}v, using the equation Δv = 0 and the fact that the R_{j}(h) = D_{x_{j}}. The point is that the normal derivative of a harmonic function can be expressed in terms of tangential derivatives. The strategy then is to convert a normal derivative of Δ_{Q}G(X, Q) to tangential derivatives and integrate by parts, putting the tangential derivatives on u. Hence we must control Riesz transforms of Δ_{Q}G(X, Q). But we have an L^{1} estimate for the nontangential maximal function, hence we have an L^{1} estimate for the square function. It is in terms of the square function that control of the Riesz transforms is obtained (recall the discussion about the Hilbert transform in the setting of the complex plane). By this means we estimate ||R_{j}(Δ_{Q}G(X, Q))||_{L^{1}} from above by Cdist(X, ∂Ω).

For operators of higher order, Δ^{3} for example, the solution admits a boundary representation involving lower order data and one uses the strategy of converting normal derivatives to tangentials and then integrating the boundary integrals by parts. Certain higher order Riesz transforms are introduced in this fashion; as illustrated above, the square function estimates for general elliptic L provide the necessary tool to handle these terms. As a consequence of the estimate on Green’s functions and the theorems of ([DKPV]), one obtains the weak maximum principle for all constant coefficient homogeneous elliptic operators in Lipschitz domains in R^{3}.

§3 Second Order Elliptic Operators with Complex Coefficients

In what follows, we are working in the upper half space R_{+}^{n}. Some of the results we will be discussing have analogs in other domains, especially those for real valued operators, but others are particular to this setting. Let us start with some background in the case of real valued equations.

Let φ(x) be a Lipschitz function of x ∈ R^{n−1} − so that there exists a constant M such that for all x and y, |φ(x) − φ(y)| ≤ M|x − y|. The change of variable (x, t) → (x, t − φ(x) maps the domain Ω = {(x, t) : t > φ(x)} to R_{+}^{n} by a “flattening” of the boundary. Solutions u to Δu = 0 in Ω are mapped to solutions of an operator L in R_{+}^{n} which is elliptic and of divergence form. That is L = divA∇ where A is real and symmetric and for which there exists a λ such that for all ζ⃗ ∈ R^{n}, Aζ⃗.ζ⃗ > λ|ζ⃗|^{2}. Because the Jacobian of this change of variable contains the terms \frac{∂φ}{∂x_{j}}, the matrix A has coefficients which are at best bounded measurable functions of x. By the De Giorgi [De] - Nash [Na] - Moser [Mo] theory of such equations, solutions to such equations (being a priori only in the Sobolev space W_{loc}^{1,2}) are in fact Holder continuous of some order. But solutions do not necessarily possess any further regularity which would be implied by, for example, being able to solve the Dirichlet problem with data in L^{2}. On the other hand, for the operators which come from a change of variable as above, one can indeed solve this Dirichlet problem, by Dahlberg’s theorem (1.1). So the question is: what is special about the matrix above?

The key observation is that the coefficients of A, while merely L^{∞} functions of the x ∈ R^{n−1} variable, are in fact independent of the t variable, t ∈ R_{+}. Of course not every matrix of coefficients which are independent of the t variable arises from a bi-Lipschitzian change of variable, but it is interesting to pursue the question of whether or not it is precisely this independence which imparts good properties to solutions of such operators. This insight is correct, and was first remarked by Jerison and Kenig in [JK1]. Having found a different approach to proving Dahlberg’s result (1.1) for harmonic functions in Lipschitz domains, they used their method to generate similar theorems for variable coefficient equations. (A fairly complete picture of how such “smoothness” in the t-direction yields solvability of Dirichlet problems is given in [FKP].) Their approach begins with an identity, rediscovered in this context, and due originally to Rellich. If Lu(x, t) = 0 for (x, t) ∈ R_{+}^{n}, and e⃗ = (0, ...0, 1), then

(3.3)  div(A∇u.∇u e⃗) = 2 div(D_{t}uA∇u)

where D_{t}u = ∇u.e⃗ is the normal derivative of u. There are only two things one needs about the real valued matrix A to prove (3.1) - the symmetry of the matrix and the fact that the coefficients are time independent so that D_{t}(A∇u) = A∇D_{t}u. Integrating (3.1) yields the boundary integral identity: ∫_{R^{n−1}} A∇u.∇udx = 2 ∫_{R^{n−1}} D_{t}uA∇u.e⃗ dx. This in turn implies, using the ellipticity of A, that ∫_{R^{n−1}} |∇_{x}u|^{2}dx ˜= ∫_{R^{n−1}} |D_{t}u|^{2}dx. Hence, the normal and tangential derivatives of solutions have comparable L^{2} norms, exactly as in inequality (0.3) for harmonic functions. This consequence of the Rellich identity will imply solvability of the Dirichlet, regularity and Neumann problems ([JK1],[KP]) for this class of operators. But when the matrix A is allowed to take on complex values, the identities above fail. There is no analog of the Rellich identity, and so this approach to solvability of these boundary value problems fails. However, beyond the failure of the approach, it is not yet understood what the truth of the matter is - and, at least in higher dimensions, no one knows if these boundary value problems are solvable or not. (There is some very new work of G. Verchota and A. Vogel which shows that there are counterexamples to solvability of L^{p} Dirichlet problems for certain constant coefficient complex valued elliptic equations in domains with angles in R^{2}.)

There has been a good deal of interest in the solvability of the regularity and Neumann problems for a certain subclass of second order elliptic complex valued equations. Consider L = D_{t}^{2} + div_{x}B(x)∇_{x}, in which the elliptic complex valued

matrix A associated to L is A = ( \begin{matrix}B(x) & 0 \\ 0 & 1 \end{matrix} ) and define T to be the unbounded

operator T = \sqrt{−div_{x}B(x)∇_{x}}. The solution u of the Dirichlet problem for L, Lu = 0, u(x, 0) = f is given by u(t) = e^{−tT} f. Hence, D_{t}u(x, t) = −e^{−tT} T f. The equivalence, were it true, between the L^{2} norms of tangential and normal derivatives of solutions to L then takes the form

(3.2)  ||T f||_{L^{2}(R^{n−1})} ˜= ||∇_{x}f||_{L^{2}(R^{n−1})}

which says that the domain of T is the Sobolev space W^{1,2}(R^{n−1}). It is this latter question, concerning the domain of such operators, which has attracted a great deal of attention, and the estimate (3.2) has come to be known as the “Kato square root conjecture”. The reformulation of (3.2) in terms of a boundary value problem first appeared in [K], and it is in this form that we discuss the problem.

The validity of (3.2) for solutions to L having this particular structure in R_{+}^{n} is known only in special circumstances: for n = 2 it is true, and in higher dimensions there are a number of partial results (see [K] §2.5 for various references). The n = 2 result is due to Coifman, McIntosh and Meyer [CMcM] in their celebrated paper on the boundedness in L^{2} of the Cauchy integral on Lipschitz curves. In essence, the same proof, or method of proof, gave both results. The approach that was so successful here involves the method of multilinear singular integrals. Unfortunately, in higher dimensions, the method is apparently limited, and has yielded positive results only for operators which are perturbations of constant coefficient operators. We shall describe here an approach to understanding the one dimensional result which is fairly elementary and illuminates the role of the Cauchy integral. That is, we shall sketch a proof of the estimate

(3.3)  ∫_{R^{n−1}} |∇_{x}u|^{2}dx ˜= ∫_{R^{n−1}} |D_{t}u|^{2}dx

for solutions u to D_{t}^{2} + D_{x}a(x)D_{x}, for complex a. (More generally, the proof works for the class of operators D_{t}^{2} + b(x)D_{x}a(x)D_{x}, where Re a(x) > c, Re b(x) > c.) The idea is to establish certain square function estimates. The close connection between the existence of quadratic estimates and this problem of Kato has been formulated in a general abstract framework by McIntosh, and as such is not a new idea. What is new here is the way in which one can exploit the Cauchy operator on Lipschitz curves to prove the quadratic estimates which have (3.3) as a consequence.

Recall that the Cauchy integral operator over a curve Λ in the complex plane is defined by

C_{Λ}f(z) = \frac{1}{2πi} ∫_{Λ} \frac{f(ζ)dζ}{ζ − z} .

When Λ is the graph of a Lipschitz function φ(x), then C_{Λ}f|_{Λ} defines a bounded operator operator on L^{2}. This is the result of Calderón [Ca] for small Lipschitz norm, and [CMcM] in the general case. There is also now an elementary proof of this fact due to Coifman, Jones and Semmes [CJS] which in fact relies on square function estimates itself. That is, the principal value operator C_{φ} is bounded on L^{2}(R) where

C_{φ}f(x, φ(x)) = \frac{1}{πi} ∫_{−∞}^{+∞} \frac{f(y, φ(y))}{(x − y) + i[φ(x) − φ(y)]}(1 + iφ′(y))dy.

In [KM], it was shown that C_{φ} is the square root of a differential operator: if b(x) = 1/(1 + iφ(x)), then

\sqrt{−bD_{x}bD_{x}} = C_{φ}D_{x}.

In particular, if

A = ( \begin{matrix}b(x) & 0 \\ 0 & b^{−1}(x) \end{matrix} ) ,

and u is the solution to divA∇u = 0 with u(0) = g ∈ W^{1,2}(R), then T g = C_{φ} \frac{d}{dx}g is a bounded operator on L^{2}. One can also see the connection between the equivalence in L^{2} of the Neumann and regularity data by writing the solution explicitly using a change of variable. Thus, if a(x) is a complex valued function of x, bounded from below, and \frac{dz}{dx} = 1/a(x) then D_{z} = a(x)D_{x} and the operator D_{t}^{2} + aD_{x}aD_{x} becomes D_{z}^{2} + D_{t}^{2}. A solution u to D_{t}^{2} + aD_{x}aD_{x} can be found by writing u(x, t) = F (z(x)+it)−F (z(x)−it) where F (w) is an analytic function of w above a curve. Representing F (w) as the Cauchy integral F (w) = ∫_{Λ} \frac{f(ζ)dζ}{w−ζ} (with Λ the curve determined by {(x, z(x)) : x ∈ R}) we recover the Cauchy integral when we compute D_{t}u and D_{x}u at t = 0. In [CM], Coifman and Meyer used a similar change of variable, and an analytic “continuation” to obtain the Cauchy integral from the usual Hilbert transform.

The only point we are emphasizing here is that there is one special operator of

the form divA∇ where A = ( \begin{matrix}a(x) & 0 \\ 0 & a^{−1}(x) \end{matrix} ) for which the Dirichlet, Neumann

and regularity problems are solvable in L^{2}. And one reason that this operator is so well behaved is that the differential operator a(x)D_{x} is always obtained from a simpler one by means of a change of variable - indeed, it seems that part of the difficulty in higher dimensions is that this fact has no analog. On R it emerges that a(x)D_{x}a(x)D_{x} is the unigue operator which controls the others, in a sense we shall now make precise.

Let L = divA∇ where A = ( \begin{matrix}a(x) & 0 \\ 0 & 1 \end{matrix} ). The first observation is that solutions

to L (decaying at infinity) satisfy the square function estimate

(3.4)  ∫_{R} |u(x, 0)|^{2}dx ˜= ∫∫_{R_{+}^{2}} t|∇u(x, t)|^{2}dxdt.

This is easily enough established by an integration by parts:

∫_{R} |u(x, 0)|^{2}dx = ∫∫_{R_{+}^{2}} D_{t}(|u(x, t)|^{2})D_{t}(t)dxdt

= ∫∫_{R_{+}^{2}} 2t[|D_{t}u(x, t)|^{2} − u(x, t)D_{x}\overline{a(x)}D_{x}\overline{u(x, t)} − \overline{u(x, t)}D_{x}a(x)D_{x}u(x, t)]dxdt

= ∫∫_{R_{+}^{2}} 2t[|D_{t}u(x, t)|^{2} + 2Re(a(x))|D_{x}u(x, t)|^{2}dxdt.

(Notice that u and its conjugate ū satisfy different equations. This does not present an obstacle in proving this simple identity; however, in establishing the more subtle properties of solutions to L, this is, in fact, the main obstacle.)

We observe that when u is a solution to L then so is D_{t}u, since the coefficients of A are independent of t. Hence one can use (3.4) to relate the L^{2} norm of D_{t}u on R^{n−1} to a solid integral in the upper half space:

∫_{R} |D_{t}u(x, 0)|^{2}dx ˜= ∫∫_{R_{+}^{2}} |∇u(x, t)|^{2}tdxdt

Clearly (3.3) would follow if we had also a similar identity for the term ∫_{R} |D_{x}u|^{2}dx. The difficulty in establishing such an identity is that D_{x}u is not a solution to L. Hence we contrive to represent a(x)D_{x}u as a solution of some elliptic equation by introducing ũ, which satisfies the first order system of equations

\begin{aligned}D_{t}ũ & = a(x)D_{x}u \\ D_{x}ũ & = −D_{t}u. \end{aligned} (3.5)

(The divergence-free vector (a(x)D_{x}u, D_{t}u) is a curl.) The ũ thus constructed will

also satisfy a divergence form equation, namely L̃u = 0 where L̃ = div ( \begin{matrix}1 & 0 \\ 0 & a^{−1}(x) \end{matrix} ) ∇.

This latter fact is special to two dimensions.

Suppose now that we could prove square function estimates for solutions to L̃, i.e., that when L̃w = 0, we have ∫_{R} S^{2}(w)dx = ∫_{R} w^{2}dx. Applying this to w = D_{t}ũ we find that

∫_{R} |D_{t}ũ|^{2}dx ˜= ∫∫_{R_{+}^{2}} |∇D_{t}ũ(x, t)|^{2}tdxdt.

In virtue of (3.5), we have |D_{t}ũ|^{2} = |a(x)D_{x}u|^{2} ˜= |D_{x}u|^{2}, and |∇D_{t}ũ(x, t)|^{2} ˜= |∇D_{t}u(x, t)|^{2}. Inotherwords, the square function estimates for solutions to L̃, when combined with the known estimates (3.4) for solutions to L yield (3.2) in the twodimensional case.

It remains to establish the estimate above for solutions to L̃, and this is a nontrivial task. It seems at first that L̃ is an operator which differs nonessentially from L. However, the simple integration by parts argument which worked for L can not be made to work for L̃. For this operator, the fact that ũ and its conjugate satisfy different equations presents a fundamental obstacle to an “easy” proof.

The aim is to prove (3.2) for solutions v to L̃. Instead of working with |v|^{2} = v\overline{v}, the idea is to replace v̄ by w where w(x, 0) = \overline{v(x, 0)} and w satisfies the equation

div ( \begin{matrix}a(x) & 0 \\ 0 & a^{−1}(x) \end{matrix} ) ∇w = 0. This is the equation whose solution has the special

representation in terms of the Cauchy integral, and for which the desired square function estimates are known. We now illustrate how to use this to bound the term ∫ |v|^{2}dx from above by ∫∫ |∇D_{t}v(x, t)|^{2}tdxdt.

∫_{R} |v|^{2}a^{−1}(x)dx = ∫_{R} vwa^{−1}(x)dx
= ∫∫_{R_{+}^{2}} D_{t}(vw)D_{t}(t)a^{−1}(x)dxdt
= ∫∫_{R_{+}^{2}}[D_{t}^{2}(v)w + D_{t}^{2}(w)v + D_{t}(v)D_{t}(w)]ta^{−1}(x)dxdt

= ∫∫_{R_{+}^{2}}[wD_{x}aD_{x}(v) + vD_{x}āD_{x}(w) + D_{t}(v)D_{t}(w)a^{−1}(x)]tdxdt

= ∫∫_{R_{+}^{2}}[D_{x}(w)aD_{x}(v) + D_{x}(v)āD_{x}(w) + D_{t}(v)D_{t}(w)a^{−1}(x)]tdxdt

Taking the real part gives

\begin{aligned}∫_{R} |v|^{2}dx & ≤ C\sqrt{∫∫_{R_{+}^{2}} |∇v|^{2}tdxdt}\sqrt{∫∫_{R_{+}^{2}} |∇w|^{2}tdxdt} \\ & ≤ C\sqrt{∫∫_{R_{+}^{2}} |∇v|^{2}tdxdt}\sqrt{∫_{R} |w|^{2}dx}, \end{aligned}

where C depends on ||a||_{L^{∞}}. Because |w(x, 0)| = |v(x, 0)|, this gives the inequality, and half of the the equivalence (3.3). The other half can be proven similarly.

The idea above can also be used to prove square function estimate for solutions to operators such as D_{t}^{2} + a(x)D_{x}b(x)D_{x} and hence provide an elementary proof of the known ([KM]) boundedness properties of \sqrt{a(x)D_{x}b(x)D_{x}}. See the work [AT] for more details on the properties of the square root of this operator.

To make this idea work in higher dimensions, even for a some interesting subclass of divergence form operators, two things are needed. The first is a nontrivial example of an operator that plays the role of a(x)D_{x}a(x)D_{x}. Is this somehow related to changes of variable of the Laplacian, or is there a more sophisticated point of view which needs to be developed? Secondly, one needs to replace the role of the first order systems (3.4) by some n-dimensional counterpart. The most straightforward analog yields a rather intractable system of equations instead of the simpler single equation L̃ in two dimensions.

§4 Second Order Divergence Form Elliptic Operators: Real but not Symmetric

The results described in this section are joint with C. Kenig and T. Toro. We are interested in understanding the role played by the hypothesis of symmetry of the matrix A in the solvability of the Dirichlet problem for operators L = divA∇ when the coefficients of A are merely bounded and measurable. Let’s take a very simple

example. Suppose A is the 2 × 2 matrix in R^{2}: A = ( \begin{matrix}1 & γ(x) \\ −γ(x) & 1 \end{matrix} ). Clearly

A is elliptic - the effect of the skew-symmetry is that the bilinear form associated to A is the same as that for the identity matrix. In the symmetric case, ellipticity places a strong restriction on the size of the off diagonal terms, but here there is no constraint on γ(x), except that we assume it is an L^{∞} function. Still, this example is so simple that it was a surprise to find the analysis of the operator associated to it, just on the upper half plane, so complicated. In any case, this is one example of the type of matrix that can be treated by the results of [KPT].

Let Ω be a Lipschitz domain in R^{n} and u a function in Ω with ||u||_{L^{∞}} ≤ 1. If Δ is a surface ball on the boundary of Ω, i.e., Δ is the intersection of a sphere B(Q) in R^{n} centered at a point Q on ∂Ω with ∂Ω, then the “tent region” or Carleson region over Δ is defined to be T (Δ) = B(Q) ∩ Ω. A measure μ is called a Carleson measure on Ω if there exists a constant C such that μ(T (Δ)) ≤ Cσ(Δ) where dσ is the surface measure on ∂Ω and Δ is any surface ball on the boundary. There are a couple of basic properties of harmonic functions which motivate Definition (4.1) below. In the first place, if u is a bounded harmonic function in, say, the upper half space R_{+}^{n}, then the measure t|∇u(x, t)|^{2}dxdt is Carleson. This is the main property of harmonic functions exploited in [FSt] to prove that the dual of the Hardy space H^{1} is the John-Nirenberg space BMO ([JN]), since it turns out that harmonic extensions of BMO functions have this same Carleson property. Indeed, this property of harmonic functions arises in many other situations - the proof of the Corona theorem, for example - and all these proofs would have been made easier if only |∇u(x, t)|dxdt were Carleson instead. This is not in general true, but there is an interesting substitute, which we shall describe after introducing the concept of ϵ - approximability in a domain.

Definition (4.1). Let u be defined in a domain Ω in R^{n}. Then u is ϵ - approximable if there exists a C^{∞} function φ in Ω such that |u − φ| < ϵ in Ω and, there exists a constant C_{ϵ} > 0 such that |∇φ| is Carleson, i.e.,

sup_{Δ∈∂Ω} ∫∫_{T (Δ)} |∇φ(X)|dX ≤ C_{ϵ}σ(Δ).

In [G], Garnett proves that any bounded harmonic function on the upper half plane is ϵ - approximable for any ϵ with a constant C_{ϵ} on the order of ϵ^{−6}. As he remarks in the notes section of the book, this theorem was motivated by the work of Varopoulos [Va]. Following this result, Garnett derives a corollary which he terms a “quantitative version of Fatou’s theorem”. This is then the first connection between ϵ - approximability and quantitative properties of boundary limits of harmonic functions. This approximability result was extended by Dahlberg in [D3] to the case of bounded harmonic functions in Lipschitz domains. He used in fact the equivalence in L^{2} between the square function and the nontangential maximal function of harmonic functions to prove this. This equivalence, and its implications for approximating solutions of second order elliptic operators, can be carried much further, as the following theorems show. From now on, L will denote an arbitrary real second order divergence form elliptic operator with bounded measurable coefficients, not assumed to be symmetric. Associated to L is the elliptic boundary representing measure dω_{L} whose existence is guaranteed by the Riesz representation theorem, since there is a strong maximum principle valid for all such L.

Theorem 4.2 [KPT]. Let L = divA∇ and let Ω be a Lipschitz domain in R^{n}. Suppose that every solution u to Lu = 0 is ϵ - approximable in Ω for every ϵ. Then dω_{L} belongs to A^{∞}(∂Ω, dσ).

The conclusion of the theorem says that dω_{L} belongs to A^{∞}(∂Ω, dσ),which is equivalent to asserting that dω_{L} belongs to A^{p}(∂Ω, dσ) for some p, and this is in turn equivalent to the statement that the L^{p} Dirichlet problem is solvable for some p. The result becomes more interesting when combined with the following criterion for understanding when solutions to L are ϵ - approximable.

Theorem 4.3 [KPT]. Let L be as above. Suppose that ||u^{∗}||_{L^{2}(∂Ω,dσ)} ˜= ||S(u)||_{L^{2}(∂Ω,dσ)} on every Lipschitz domain Ω for all suitable normalized solutions u to L in Ω. (Assume u vanishes at some point X_{o} in Ω whose distance to the boundary comparable to the diameter of Ω.) In addition, the comparability constants in this L^{2} equivalence should depend only the Lipschitz character of Ω. Then, in any such domain Ω, every bounded solution u to Lu = 0 is ϵ - approximable.

The conclusion of Theorem 4.3 is also true if instead of assuming comparability of u^{∗} and its square function in L^{2}, their norms are comparable in L^{p} for some 0 < p < ∞. The theorem can be further weakened by assuming that the Lipschitz domains on which u^{∗} and its square function are comparable are domains with small Lipschitz constant. This is a technical but important fact in the actual applications of this result.

To prove Theorem 4.3, we first observe that if one can compare the nontangential maximal function and square function of solutions, then Dahlberg’s proof of approximability can be carried out. That is, in addition to the maximum principle and the Harnack principle for solutions, the important ingredient in his proof for harmonic functions is this square function estimate. Then, a purely real variable argument proves that ϵ - approximability implies the A^{∞} property of dω_{L}.

This investigation was motivated by the following question. Suppose L = divA∇ is elliptic and real valued, and A has bounded measurable coefficients. Consider L acting on functions defined in R_{+}^{n} and further suppose that the coefficients of A are independent of the t (transverse) variable. This was the case, for example, with our divergence form operators in Section 3 which arose from a bi - Lipschitzian change of variable from the Laplacian. However, let us drop the assumption that the matrix A is symmetric. (Hence it cannot be an operator that arises via a change of variable from a symmetric one.) Then, are the measures dω_{L} and dσ mutually absolutely continuous, and is the Dirichlet problem for L solvable with L^{2} data as in the symmetric case?

In two dimensions we can prove that the nontangential maximal function and square functions of solutions to such L are comparable in L^{2}. Thus, the theorems above provide a partial resolution of this question, showing that the L^{p} Dirichlet problem is solvable for such L for some p larger than 2. Whether or not the L^{2} result is true remains open. There are some higher dimensional generalizations of this fact, but even in two dimensions the result has interesting consequences for non-divergence form operators. As L. Escauriaza has pointed out to us, every nondivergence second order symmetric elliptic operator in two variables can be written as a divergence form (non-symmetric) operator. Then Theorems 4.2 and 4.3 above imply the first such solvability results for non-divergence second order operators with merely bounded measurable coefficients.

Concluding remarks. Many authors could write a paper with this theme - the application of square function inequalities to problems in PDE - and the choices of examples might be very different. Naturally, the examples chosen here reflect our current research interests, but there are certainly many other possible choices which are both significant and interesting. Furthermore, one is not limited to the consideration of elliptic equations nor of boundary value problems in the sense discussed above. And finally, these estimates of Littlewood-Paley type are merely one way in which the ideas and techniques of harmonic analysis play an increasingly important role in PDE.

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