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Carleson Measures and Elliptic Boundary Value Problems 3 1 Carleson measures and elliptic boundary value 2 problems 3 4 Jill Pipher 5 Abstract. In this article, we highlight the role of Carleson measures in elliptic boundary value prob- 6 lems, and discuss some recent results in this theory. The focus here is on the Dirichlet problem, with 7 measurable data, for second order elliptic operators in divergence form. We illustrate, through selected 8 examples, the various ways Carleson measures arise in characterizing those classes of operators for 9 which Dirichlet problems are solvable with classical non-tangential maximal function estimates. 10 Mathematics Subject Classification (2010). Primary 42B99, 42B25, 35J25, 42B20. 11 Keywords. Carleson measures, elliptic divergence form equations, A1, boundary value problems. 12 1. Introduction 13 Measures of Carleson type were introduced by L. Carleson in [9] and [10] to solve a problem 14 in analytic interpolation, via a formulation that exploited the duality between Carleson mea- 15 sures and non-tangential maximal functions (defined below). Carleson measures have since 16 become one of the most important tools in harmonic analysis, playing a fundamental role in 17 the study of singular integral operators in particular, through their connection with BMO, 18 the John-Nirenberg space of functions of bounded mean oscillation. We aim to describe, 19 through some specific examples, the ubiquitous role of measures of this type in the theory 20 of boundary value problems, especially with regard to sharp regularity of “elliptic” measure, 21 the probability measure arising in the Dirichlet problem for second order divergence form 22 elliptic operators. Perhaps the first connection between Carleson measures and boundary 23 value problems was observed by C. Feffeman in [18], namely that every BMO function on Rn Rn+1 24 has a harmonic extension to the upper half space + which satisfies a certain Carleson 25 measure condition. This established an important link between solutions to boundary value 26 problems for the Laplacian and the function space BMO. It may be surprising to see the 27 extent to which this link exists for operators other than the Laplacian, and in the context of 28 more general domains. 29 In order to define Carleson measures, we introduce the geometric notion of a Carleson ⊂ Rn f 2 Rn+1 30 region above a cube. If Q is a cube with side length l(Q) set TQ = (x; t) + : 31 x 2 I; 0 < t < l(Q)g, a cube sitting above its boundary face Q. (The notation TQ comes 32 from an equivalent formulation involving “tents” over cubes.) Rn+1 33 Definition 1.1. The measure dµ is a Carleson measure in the upper half space + if there n 34 exists a constant C such for all cubes Q ⊂ R , µ(T (Q)) < CjQj, where jQj denotes the Proceedings of International Congress of Mathematicians, 2014, Seoul 2 Jill Pipher 35 Lebesgue measure of the cube Q. 36 The classical theory of harmonic functions in the upper half space, or the ball, considers p 37 solutions to the Dirichlet problem with measurable, specifically L , data. Given a function p n 38 f 2 L (R ), the convolution of f and the Poisson kernel is an absolutely convergent integral p 39 when 1 < p < 1, giving meaning to the harmonic extension u(x; t) of an L function. 40 And the sense in which this extension u converges to its boundary values is “non-tangential”. n 41 That is, for every x0 2 R , one can define a non-tangential approach region to x0, Γa(x0) = p n 42 f(x; t): jx − x0j < atg. Then if u(x; t) is the Poisson extension of f 2 L (R ), for 43 almost every x0, u(x; t) ! f(x0) as (x; t) 2 Γa(x0) approaches x0. Moreover, one has 44 a non-tangential maximal function estimate, specified below, which yields solvability and p 45 uniqueness of this L Dirichlet problem. 46 The result of C. Fefferman about harmonic functions, which proved to be a powerful tool 47 in harmonic function theory, is this: if u(x; t) is the Poisson extension of f 2 BMO, then jr j2 Rn+1 48 dµ = t u dxdt is a Carleson measure in the upper half space + . The converse also 49 holds for functions that are not too large at 1. 50 In the last several decades, there have been many significant developments in the theory of p 51 boundary value problems with data in L spaces, for harmonic (or poly-harmonic) functions 52 defined in very general domains, and for solutions to second order divergence form (and 53 higher order) elliptic operators with non-smooth coefficients. We will highlight a selection 54 of these developments in which the role of Carleson measures has been decisive. Rn+1 55 For simplicity of notation, we will formulate the results in the upper half plane, + , 56 but in fact these results are more naturally formulated on Lipschitz domains - see the cited 57 references for this generality. In some cases, the perturbation results hold in more general 58 (chord-arc) domains: [41–43]. 59 2. Definitions and background A divergence form elliptic operator L := − div A(x)r; n+1 60 defined in R , where A is a (possibly non-symmetric) (n+1)×(n+1) matrix of bounded 61 real coefficients, satisfies the uniform ellipticity condition nX+1 2 −1 λjξj ≤ hA(x)ξ; ξi := Aij(x)ξjξi; kAkL1(Rn) ≤ λ ; (2.1) i;j=1 for some λ > 0, and for all ξ 2 Rn+1, x 2 Rn. As usual, the divergence form equation is 2 1;2 interpreted in the weak sense, i.e., we say that Lu = 0 in a domain Ω if u Wloc (Ω) and Z Aru · rΨ = 0 ; 2 1 62 for all Ψ C0 (Ω). Rn+1 f 2 Rn × 63 For notational simplicity, Ω will henceforth be the half-space + := (x; t) 64 (0; 1)g even though the results are more naturally formulated on Lipschitz domains. See the 65 cited references for this generality. Carleson measures and elliptic boundary value problems 3 p 66 The solvability of the Dirichlet problem for L with data in L (dx) is a function of a 67 precise relationship between the elliptic measure ! associated to L and Lebesgue measure. 68 The elliptic measure associated to L is analogous to the harmonic measure: it is the 69 representing measure for solutions to L with continuous data on the boundary. n 70 Definition 2.2. A non-negative Borel measure ! defined on R is said to belong to the class 71 A1 if there are positive constants C and θ such that for every cube Q, and every Borel set ⊂ 72 F Q, we have ( ) jF j θ !(F ) ≤ C !(Q): (2.3) jQj 73 A real variable argument shows that a measure, !, belongs to A1(dx) if and only if it is 74 absolutely continuous with respect to Lebesgue measure and there is an exponent q > 1 such 75 that the Radon-Nikodym derivative k := d!/dx satisfies (Z )1/q Z − k(x)qdx ≤ C− k(x) dx ; (2.4) Q Q 76 uniformly for every cube Q. This property is called a reverse-Hölder estimate of order q. 77 If ! is the elliptic measure associated to an operator L, then the existence of such a q > 1 78 is, in turn, equivalent to the solvability of the Dirichlet problem for L with boundary data p 79 f 2 L (for p dual to q), in the sense of non-tangential convergence and non-tangential p 80 estimates on the boundary. These non-tangential estimates are expressed in terms of L 81 bounds on two classical operators associated to solutions: the square function ZZ !1/2 α jr j2 dydt S (u)(x) := u(y; t) n−1 ; (2.5) jx−yj<αt t 82 and the non-tangential maximal function α N∗ (u)(x) := sup ju(y; t)j (2.6) (y;t):jx−yj<αt 83 Precisely, the elliptic measure satisfies a reverse Hölder estimate of order q if and only if the p 84 following L Dirichlet problem is solvable, for p dual to the exponent q: 8 > Rn+1 <Lu = 0 in + p n lim ! u(·; t) = f in L (R ) and n:t: (Dp) :> t 0 kN∗(u)kLp(Rn) < Ckfkp: n 85 Here, the notation “u ! f n:t:” means that lim(y;t)!(x;0) u(y; t) = f(x); for a:e: x 2 R , 2 f 2 Rn+1 j − j g 86 where the limit runs over (y; t) Γ(x) := (y; t) + : y x < t . The constant C 87 depends only on ellipticity and dimension. 88 We will usually suppress the dependence on the aperture α, since the choice of aperture p 89 does not affect the range of available L estimates. 90 Solutions to L are said to satisfy De Giorgi-Nash-Moser bounds when the following local Rn+1 91 Hölder continuity estimates hold. Assume that Lu = 0 in + in the weak sense and 4 Jill Pipher ⊂ Rn+1 2 Rn+1 92 B2R(X) + , X + , R > 0. Then 0 1 1 ( ) Z 2 j − j µ Y Z B 2 dx C ju(Y ) − u(Z)j ≤ C @ juj A ; for all Y; Z 2 BR(X); R jB2R(X) B2R(X) (2.7) 93 for some constants µ > 0 and C > 0. In particular, one can show that for any p > 0 0 1 1 Z p B p dx C ju(Y )j ≤ C @ juj A ; for all Y; Z 2 BR(X): (2.8) jB2R(X) B2R(X) 94 The De Giorgi-Nash-Moser bounds always hold when the coefficients of the underlying 95 equation are real [14, 40, 44], and the constants depend quantitatively only upon ellipticity and 96 dimension.
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