The logarithm of the Poisson kernel of a ¹ domain has vanishing mean oscillation

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