The logarithm of the Poisson kernel of a ¹ domain has vanishing mean oscillation
The logarithm of the Poisson kernel of a ¹ domain has vanishing mean oscillation
复制标题
1 域的泊松核的对数具有消失的平均振荡
DOI：
10.1090/s0002-9947-1982-0667174-2
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发表时间：
1982
影响因子：
1.3
通讯作者：
C. Kenig
中科院分区：
文献类型：
--
作者：
D. Jerison;C. Kenig
令 D 为 R" 中的 C1 域，u 为 3D 的调和测度，相对于 D 中的固定极点。然后，du = k da，其中 k 是 D 的泊松核。我们证明 log k 具有 SD 的消失平均振荡。简介。本文的主要目标是研究 R"+ ' 中 C1 域的泊松核的锐正则性性质，n 3= 1。R"+ ' 中的域称为 C1 域，如果由 C1 函数的图局部给出，如果函数只是 Lipschitz，则称为 C1,a，如果函数具有 a 阶 Holder 连续梯度，则称为 C1,a 众所周知（参见 [14]），如果 7) 是 CXa 域 0 < a < 1，且 « 是具有固定极点 X0 E D 的 37) 的调和测度，则 7) 的泊松核，k(Q) — du/do 及其倒数 k~\Q) 是 a 阶的 Holder 连续。这意味着 | log k(Q) — log k(Q') \< C | Q Q' |a for Q, Q' E 37)。在这篇文章中，我们分析了 C1 域的情况，即 a = 0。很容易看出，对于 C1 域，k 和 l/k 不需要有界，因此现在已经众所周知。很多时候，当 L°° 估计值失效时，适当的替代是 BMO 估计值，其中 BMO 表示 John 和 Nirenberg 的有界平均振荡函数空间。在 [3] 中，B. Dahlberg 表明，在 Lipschitz 域上 log k G BMO(dD)，即 Λ f logk--r [logkdo o(A)J¿, »(AJA do<+oo，其中 A 表示表面球。 37)（所有相关定义参见§1），D. Sarason 引入了 BMO 的子空间，他将其称为 VMO（消失平均振荡函数），它与连续函数与 L°° 的关系相同，专门针对 37)，如果 jajTL,^/^-^// do = 0，则 / G VMO(oD)。编辑于 10 月 29 日收到， 1981. 1980 数学学科分类。第一作者是 NSF 博士后研究员，第二作者是 Alfred P. Sloan 研究员。781 ©1982 美国数学学会。 0002-9947/81/0000-0392/S04.50 许可或版权限制可能适用于再分发；请参阅 http://www.ams.org/journal-terms-of-use 782 D. S. JERISON 和 C. E. KENIG 因为人们预计 k 对于 C1 域比 Lipschitz 域具有更多的规律性，B. Dahlberg [5] 提出了是否C1 域 log k E VMO(oD)。事实上，如果 4> 将单位圆共形映射到 D，则 Lindelöf 的经典定理表明 arg $' 是连续的，因为任何连续函数的共轭函数都在 [13] 中。此外，Pommerenke 在 [12] 中推广了这个结果。在平面域 7) 中，当且仅当 37) 是渐近平滑的，即 KQu Qi)/\ Q\ 02 I1 as | Qx Q2|0，其中 l(Qx, Q2) 是 Qx 和 Q2 之间的 37) 的最短弧的长度时，Fabes、Kenig 和 Neri [6] 获得了针对 Dahlberg 问题的部分结果。在[13]中，Sarason 证明了 VMO 是 BMO 范数中连续函数空间的闭包。在[6]中，我们证明了 log k 属于 BMO 范数中 Lx 的闭包。在本文中，我们对任何 n > 1 的问题给出了肯定的回答。我们实际上证明了一个更强的结果，这让人想起 Pommerenke 定理。我们证明了如果 7) 是一个有界域，可以局部地写为以下图。 Lipschitz 函数具有任意小的 Lipschitz 范数，然后 log k E VMO(oD) 。我们的定理表明 logk(x) 在 C1 扰动下在 BMO 范数中是稳定的（参见 3.4）。在本节中，我们设置了本文所需的符号和回忆结果。大写字母 X 和 Y 将表示 R"+' 中域 D 的点，并且 ( X, Y > 将是 R"+1 中的内积。 x 和 y 是 R" 中的内积，| X\= (X, X)l/2 和 | x \= (x ■ x)]/2 分别表示 R"+1 和 R" 中的欧几里德长度。字母 c 将用于表示在不同情况下不一定相同的常数。由于 ti 始终是 D 的边界点，因此不再提及。 37) 表示为 Q，NQ 表示在 Q 处垂直于 D 的外部单位。 B(Q,r)= [XER"+l:\XQ\<r), r > 0。域 7) C R"+1 是 Lipschitz 域，如果存在 8 个，使得对于每个 g G 37) 存在一个球 B(Q, r) 和一个以 Q 为原点的 R"+1 的等距坐标系 (x, t)，其中B(Q, r)PD = B(Q, r) P {(x, t):xER",t> <p(x)} 对于某些 Lipschitz 函数 <p: R" -» R 满足 (1.1) <p(0) = 0 且 II v<p\\x < 8 < oo。为了方便起见，我们总是假设 8 < -r^。这并不影响我们的推理，因为我们关心的是当 8 趋向于零时会发生什么。令 o 为 3D 的表面度量。对于 37) 的表面球 A，我们的意思是 B(Q, r) P aD 对于某些 Q E 37) 且 r > 0。用 ux 表示极点位于 X 的 D 的谐波测量，即 37) 上的测量满足 u(X) = jSDfdux 对于每个许可证或版权限制可能适用于再分发；请参阅 http://www.ams.org/journal-terms-of-use C1 域 783 的泊松核，有界调和函数 u 具有连续边界值 / 在 C0(37) 中。根据 1.3，存在一个密度 kx，使得 dux — kxdo。显然 (1.2) kx>0 且 f kxdo= 1。 ÏM+1 在上半空间 R+ = {(x, t): x 6 R", /> 0) 的众所周知的情况下，密度是通常的泊松核 kY(Q) — P,(x, y) = cnt(\ x — y \2 + r2)-(n+1)/2，其中 Y = (y, t), Q = (x,0), c„ = T((n + l)/2)/w(n+1)/2。固定点 X0 G D。核函数为 K(X, Q) = dux/dux°(Q)。因此，kx(Q) = K(X, Q)kXo(Q)。令 <p 满足 (1.1)。对于 x G R"，令 A(x, r) {(y, <p(y)): | x y |< r, y G R"}。因为 8 < -fo, A(x, r) 和表面球 A = B((x, <p(x)), r) n 37) 在所有后续结果中都是可互换的。我们不会区分它们。定理 1.3 (Dahlberg [3])。令 <p 满足 (1.1)。7) = {(x, t): x E R", t > <p(x)}。那么 (a) ux 和 o 对于所有 X E D 都是绝对连续的。 (b)kx = do3x/do E L2(aD, do), 和 1/2 ¿y/Mô)2Mô)) ^(¿(Mß)^))。对于所有 X E D\ B((0,0), 4) 和所有表面球 A C A(0,2)。此外，常数 c 仅取决于 8。 (c) 对于 dD 的几乎每个 Q(do)，kx(Q) = lim^0(d/dt)G(Q tNQ, X)，其中 G 是 D 的格林函数。(b) 的一个简单结果是
Let D be a C1 domain in R", and u the harmonic measure of 3D, with respect to a fixed pole in D. Then, du = k da, where k is the Poisson kernel of D. We show that log k has vanishing mean oscillation of SD. Introduction. The main goal of this article is to study the sharp regularity properties of the Poisson kernel for C1 domains in R"+ ', n 3= 1. A domain in R"+ ' is called a C1 domain if it is given locally by graphs of C1 functions. It is called a Lipschitz domain if the functions are merely Lipschitz, and is called C1,a if the functions have a gradient which is Holder continuous of order a. It is well known (see [14]) that if 7) is a CXa domain 0 < a < 1, and « is the harmonic measure of 37) with a fixed pole X0 E D, then the Poisson kernel of 7), k(Q) — du/do, and its reciprocal k~\Q) are Holder continuous of order a. This means that | log k(Q) — log k(Q') \< C | Q Q' |a for Q, Q' E 37). In this article we analyze the case of C1 domains, i.e. a = 0. It is very easy to see already in two dimensions that for C1 domains k and l/k need not be bounded, and hence cannot be continuous. It is well known by now that many times when L°° estimates break down, the appropriate replacement are BMO estimates, where BMO denotes the space of functions of bounded mean oscillation of John and Nirenberg. In [3], B. Dahlberg showed that on a Lipschitz domain log k G BMO(dD), i.e., Ä\ f logk-—-r [logkdo a. o(A)J¿, »(AJA do<+oo, where A denotes a surface ball of 37) (see §1 for all the relevant definitions). In [13], D. Sarason introduced a subspace of BMO, which he called VMO (functions of vanishing mean oscillation), which bears the same relationship to BMO that continuous functions bear to L°°. Specialized to the context of 37), / G VMO(oD) if jajTL,^/^-^// do = 0. Received by the editors October 29, 1981. 1980 Mathematics Subject Classification. Primary 31B25; Secondary 42B99. ' The first author is an NSF postdoctoral fellow, and the second was supported in part by the NSF. The second author is an Alfred P. Sloan Fellow. This research was carried out while the first author was visiting the University of Chicago. 781 ©1982 American Mathematical Society 0002-9947/81/0000-0392/S04.50 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 782 D. S. JERISON AND C. E. KENIG Because one expects k to have more regularity for C1 domains than for Lipschitz domains, B. Dahlberg [5] posed the question of whether for C1 domains log k E VMO(oD). When n — 1, this result follows by conformai mapping. In fact, if 4> maps the unit disc conformally onto D, a classical theorem of Lindelöf shows that arg $' is continuous. But then log | $' | is in VMO because the conjugate function of any continuous function is in VMO by results in [13]. Moreover, in [12], Pommerenke generalized this result. He showed that for a simply connected domain 7) in the plane, log | í>' | is in VMO if and only if 37) is asymptotically smooth, i.e. KQu Qi)/\ Q\ 02 I1 as | Qx Q2|0, where l(Qx, Q2) is the length of the shortest arc of 37) between Qx and Q2. In higher dimensions, Fabes, Kenig and Neri [6] obtained some partial results towards Dahlberg's problem. In [13] Sarason showed that VMO is the closure of the space of continuous functions in the BMO norm. In [6] it was shown that log k belongs to the closure of Lx in the BMO norm. In this paper we answer Dahlberg's question in the affirmative for any n > 1. We actually prove a stronger result, reminiscent of Pommerenke's theorem. We prove that if 7) is a bounded domain which can be written locally as the graph of Lipschitz functions with arbitrarily small Lipschitz norms, then log k E VMO(oD) . Our theorem shows that logk(x) is stable in BMO norm under C1 perturbations (see 3.4). 1. In this section we set up notations and recall results needed throughout this paper. Capital letters X and Y will denote points of a domain D in R"+', and ( X, Y > will be the inner product in R"+1. Lower case letters x and y are reserved for points of R", and x ■ y will be the inner product in R". | X\= (X, X)l/2 and | x \= (x ■ x)]/2 denote the Euclidean length in R"+1 and R", respectively. The letter c will be used to denote constants that are not necessarily the same in different occurrences. Their dependence on the dimension will not be mentioned, since ti is fixed throughout. Points of the boundary of D, 37), are denoted Q, and NQ will denote the outer unit normal to D at Q. B(Q,r)= [XER"+l:\XQ\<r), r > 0. A domain 7) C R"+1 is a Lipschitz domain if there exists 8 such that for each g G 37) there exist a ball B(Q, r) and an isometric coordinate system (x, t) of R"+1 with Q as origin for which B(Q, r)PD = B(Q, r) P {(x, t):xER",t> <p(x)} for some Lipschitz function <p: R" -» R satisfying (1.1) <p(0) = 0 and II v<p\\x < 8 < oo. For convenience we will always assume that 8 < -r^. This does not affect our reasoning because we are concerned with what happens as 8 tends to zero. Let o be the surface measure of 3D. By a surface ball A of 37) we mean B(Q, r) P aD for some Q E 37) and r > 0. Denote by ux the harmonic measure of D with pole at X, that is, the measure on 37) satisfying u(X) = jSDfdux for every License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE POISSON KERNEL OF A C1 DOMAIN 783 bounded harmonic function u with continuous boundary values / in C0(37)). According to 1.3 there is a density kx such that dux — kxdo. Evidently (1.2) kx>0 and f kxdo= 1. ÏM+1 In the well-known case of the upper half-space R+ = {(x, t): x 6 R", / > 0) the density is the usual Poisson kernel kY(Q) — P,(x, y) = cnt(\ x — y \2 + r2)-(n+1)/2, where Y = (y, t), Q = (x,0), and c„ = T((n + l)/2)/w(n+1)/2. Fix a point X0 G D. The kernel function is K(X, Q) = dux/dux°(Q). Thus, kx(Q) = K(X, Q)kXo(Q). Let <p satisfy (1.1). For x G R", let A(x, r) {(y, <p(y)): | x y |< r, y G R"}. Because 8 < -fo, A(x, r) and the surface ball A = B((x, <p(x)), r) n 37) are interchangeable in all of the succeeding results. We will not distinguish between them. Theorem 1.3 (Dahlberg [3]). Let <p satisfy (1.1). 7) = {(x, t): x E R", t > <p(x)}. Then (a) ux and o are mutually absolutely continuous for all X E D. (b)kx = do3x/do E L2(aD, do), and 1/2 ¿y/Mô)2Mô)) ^(¿(Mß)^)). for all X E D\ B((0,0), 4), and all surface balls A C A(0,2). Moreover, the constant c depends only on 8. (c) For almost every Q(do) of dD, kx(Q) = lim^0(d/dt)G(Q tNQ, X), where G is the Green function of D. An easy consequence of (b) is that