Contribution from:
Mathematical Physics Studies Vol. 27
Perspectives in Analysis
Essays in Honor of Lennart Carleson’s 75th Birthday
Michael Benedicks, Peter W. Jones, Stanislav Smirnov (Eds.)
© Springer Verlag Berlin Heidelberg 2005
ISBN 3-540-30432-0
On Scaling Properties of Harmonic Measure
Peter W. Jones
Department of Mathematics, Yale University, New Haven, CT 06520, USA
jones-peter@yale.edu
1 Introduction
This short article gives an exposition of some open problems concerning harmonic measure for domains in Euclidean space. This is an area where Lennart
Carleson has been a leading figure for about fifty years. Like many other analysts, this author had his career shaped to a large extent by Lennart, and this
provides me an opportunity to both wish him a happy birthday and thank
him for all the assistance he has given me for nigh thirty years.
Recall that the harmonic measure for a base point z0 ∈ Ω is the (unique)
probability measure ω on ∂Ω so that
Z
F (z0 ) = F (z)dω(z)
for all harmonic functions F on Ω that are (modulo small technicalities) continuous up to ∂Ω. Despite decades of intensive research, we are far from understanding the “fine structure” of harmonic measure. Certainly the greatest
achievement here is N. Makarov’s theorem [11] that on any simply connected
planar domain, ω (for any base point) is supported on a set of Hausdorff
dimension one, and ω puts no mass on any set of dimension less than one.
In modern parlance this is written as dim(ω) = 1. A related, surprising theorem due to Carleson [5] is that for “square” Cantor sets (as boundary) in
the plane, dim(ω) < 1, no matter how large the dimension of the Cantor set.
In Rd , d ≥ 3, our knowledge is much less complete, though notable results
due to Bourgain [4], Wolff [12], and others provide some knowledge of basic
scaling laws.
It was understood in the 1990’s (see [6]) that dynamical systems and Conformal Field Theory were intimately tied to these problems, at least in dimension two. It is a curious state of affairs that CFT as it is understood today,
makes no notable predictions for these problems. This is perhaps even more
surprising in light of the great success CFT has had with SLE. Exactly how
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Peter W. Jones
the full story will unfold remains mysterious. My guess is that we are missing
fundamental concepts, and that the future will bring some unsuspected, closer
connections between analysis and statistical physics. If the situation is murky
for d = 2, it is much worse for d ≥ 3. All results known there are the end
products of arguments of real variable type, and therefore cannot yield sharp
constants. Whether there can be a unified approach to all dimensions seems
now to be an untouchable problem. If we are very lucky, the new generation
may give us the solution due to the rise of a young star of Lennart Carleson’s
caliber and vision.
2 Harmonic Measure in the Plane
We discuss here various conjectures concerning the fine structure of harmonic
measure. In the plane one has some extra measure of confidence in these
conjectures due to experience from conformal mappings as well as Conformal
Field Theory. A guiding principle taken from CFT is that all expressions
should be simple and analytic (in the parameter values). We start with some
examples from dimension two and then turn to d ≥ 3 in the following section.
The first conjecture we discuss concerns conformal mappings h : D → Ω,
where Ω is a bounded domain. (All conjectures are the same for mappings h
of {|z| > 1} to Ω, where h(∞) = ∞.) Let t ∈ C be a complex number and
consider the growth of integral means for |(h0 )t |. Define βh (t) ≥ 0 to be the
smallest real number such that
Z 2π
|h0 (reiθ )t |dθ = O (1 − r)−βh (t) .
0
Similarly we define the pressure spectrum,
βsc (t) = sup βh (t) ,
where the supremum is taken over all conformal maps h to bounded domains.
(The “sc” stands for simply connected.) It is a nontrivial fact (the “snowflake
construction”) that for every ε > 0 and every t there is an h such that
Z 2π
|h0 (reiθ )t |dθ ≥ c(1 − r)ε−βsc (t)
0
holds for all r < 1. (See e.g. [6] for the case t = 1.)
The first conjecture we list is
βsc (t) = |t|2 /4 ,
|t| ≤ 2
(1a)
βsc (t) = |t| − 1 ,
|t| ≥ 2 .
(1b)
and
On Scaling Properties of Harmonic Measure
75
The phase transition at |t| = 2 can be (partially) understood by looking at
t = 2 and noticing the bound |h0 (r exp{iθ})| ≤ c(1 − r)−1 , which is due to the
finite area of the image domain. (Thus (1b) holds for t ≥ 2.) The conjecture
in (1) was introduced by Kraetzer [9], but now carries the slightly ungainly
name of the Brennan–Carleson–Jones–Kraetzer Conjecture. (This is because
Brennan’s Conjecture was for the special value t = −2, while Carleson and
Jones had made the conjecture for t = 1.) Actually, I. Binder’s name should be
added to the list of names mentioned because he had the idea of introducing
complex values of t.
If we restrict to real values of t ∈ [−2, 2], the conjecture has two equivalent
forms. The first of these is in terms of the multifractal formalism, which has
the additional feature of adding Beurling’s estimate on harmonic measure in
a more transparent form. Recall that if ω is a (locally finite) Borel measure,
the f (α) spectrum is defined by
f (α) = Dimension(Eα ) ,
where we understand the dimension in question to be the Hausdorff dimension
and where (without getting too technical)
Eα = z | ω D(z, r) ∼ rα .
Here D(z, r) is the disk centered at z and of radius r, and the ∼ symbol means
that this holds as r → 0. The reformulation of (1) (for t ≥ −2) is that for ω
the harmonic measure on a bounded, simply connected domain,
f (α) ≤ 2 − 1/α .
(2)
Again, the conjecture implicitly states that the estimate in (2) is optimal. We
define as before
fsc (α) = sup f (α) ,
where the supremum is now taken over all simply connected domains. (Unlike
βsc , the estimates for f (α) do not depend on whether the domain is bounded
or not.) Thus the conjecture is
fsc (α) = 2 − 1/α ,
α ≥ 1/2 .
(3)
The proof that conjectures (1) and (3) are equivalent involves a stopping
time argument followed by a renormalization procedure. The algebra involved
yields the relationship
t = 2 − 2/α
for α ≥ 1/2. One builds by a snowflake construction a conformal map h where
|h0 (reiθ )| ≤ c(1 − r)ε+1−1/α
and where only this maximal growth is used to estimate βh (t). Conversely,
given h one estimates f (α) by a similar construction.
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Peter W. Jones
This reasoning can be extended to give another (equivalent) formulation
for Hölder domains. Recall that a simply connected domain Ω is called a
γ Hölder domain, 0 < γ < 1, if (any) Riemann map h : D → Ω is in the
Hölder space Λγ , |f (z) − f (w)| ≤ c|z − w|γ , z, w ∈ D (equivalently D). This is
the same as |f 0 (z)| ≤ c0 (1 − |z|)γ−1 . The conjectures (1) and (3) are equivalent
(for the appropriate range of t) to the conjecture
Ω is a γ Hölder domain
=⇒
dim(∂Ω) ≤ 2 − γ .
(4)
The algebra used here is the relation α = 1/γ, and the understanding of (4)
is that the estimate 2 − γ is best possible.
There is yet another formulation that is believed to be equivalent (for
the range t ∈ [−2, 2]) in terms of quasiconformal mappings. For a globally
quasiconformal map F let k be the bound for the lower dilatation. (So 0 ≤ k <
1 and k = 0 if and only if F is conformal.) Let βk (t) be the pressure function
where the supremum is taken over all h having a globally quasiconformal
extension F with bound k. The conjecture for t ∈ [−2, 2] is that
βk (t) = β(kt) .
(5)
Some support for this is given by a remarkable theorem of Smirnov (unpublished), which is one of the few conjectured upper bounds that has been
proven. Smirnov’s result is that if h has an extension F with bound k, then
dim F (S 1 ) ≤ 1 + k 2 .
Recall that Beurling’s theorem states that for a simply connected domain,
ω D(z, r) ≤ cr1/2
where c depends only on the inradius of the base point for harmonic measure.
Here D(z, r) is the disk centered at z with radius r. A trivial consequence of
Beurling’s theorem is
fsc (α) = 0 ,
α < 1/2 ,
because the sets in question are empty.
What is the evidence for conjectures (1), (3) and (4)? First we note that (1)
is true for t ≥ 2. For t = −2 this is Brennan’s conjecture, which is supposed
to correspond to the (Beurling-type) conjecture fsc (1/2) = 0. For t = 1 the
conjecture is due to Carleson–Jones [6], who also provided some computer
evidence. A small surprise there is the tie to coefficient problems for the class
Σ of functions univalent on {|z| > 1} having Laurent expansion
z+
∞
X
n=1
bn z −n .
On Scaling Properties of Harmonic Measure
77
Unlike the case for de Brange’s theorem (|an | ≤ n) for the class S, there seems
to be no natural guess for the maximal size of bn . Instead it is shown that if
Bn = sup|bn |, the supremum being taken over Σ, then
lim
n→+∞
log Bn
= γΣ
log n
exists and (surprise) γΣ = β(1) − 1. (Thus the conjecture that γΣ = −3/4.)
Returning to other evidence, the value α = 1, which corresponds to t =
0, should be understood as the theorem of Makarov [11]: For any simply
connected domain the harmonic measure is supported on a set of (exactly)
Hausdorff dimension one. (More precisely, Makarov’s theorem tells one that
the |t|2 /4 conjecture is correct for t = 0 in the sense that βsc (t) should be
quadratic near t = 0.) Another way of rephrasing these results and conjectures
is the following list:
α = 1/2 ↔ t = −2 ↔ Brennan’s Conjecture and Beurling’s Theorem
α=1
↔ t = 0 ↔ Makarov’s Theorem
α=2
↔ t = 1 ↔ Carleson–Jones Conjecture and `(Γε ) ∼ ε−1/4
By the last entry we mean that if Ω is a bounded domain then Γε is the curve
{z | G(z, z0 ) = ε}, where G is Green’s function for Ω with base point z0 . If we
integrate |h0 | over {G(z, 0) = ε} on the disk (essentially the circle of radius
1 − ε about the origin). The |t|2 /4 conjecture says that the extremal case is
the length of Γε (which is exactly the integral of |h0 |) should be O(ε−1/4 ).
The upshot is that for these three special values there is a corresponding
“physical” quantity that is either known (Beurling’s or Makarov’s Theorem)
or reasonably conjectured (t = 1, α = 2). It would be interesting to see if
there is a longer list of natural, “physical” values of t, α. (Of course t = 2,
α = +∞ corresponds to finite area.)
For the fsc (α) conjecture it should be noted that the asymptotics are
correct. By the results of [7], there are constants c1 and c2 such that for
α ≥ 1,
2 − c1 /α ≤ fsc (α) ≤ 2 − c2 /α .
(6)
Now the study of f (α) is also well defined for general (i.e. non-simply
connected) domains in R2 (as well as in Rd ). The results of [7] are actually for
general domains, and can be expressed as bounds for the “universal” spectrum:
2 − c1 /α ≤ fu (α) ≤ 2 − c2 /α ,
α≥1.
(7)
Here fu (α) = sup f (α), the supremum being taken over all planar domains.
For general domains in R2 one cannot have conjecture (3) for α < 1, as the
correct result there is fu (α) = α. (It pays to use Cantor sets as boundaries.)
One might think that by using Cantor sets one could obtain fu (α) > fsc (α) at
least in the range α > 1. (By the theorem of Jones–Wolff [8], fu (1) = 1.) However, this turns out not to be the case. Binder and Jones have announced [2]:
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Peter W. Jones
fu (α) = fsc (α) ,
α≥1.
(8)
As a consequence, the correct conjecture is
fu (α) = 2 − 1/α ,
α>1,
(9)
with the value α = 1 being a theorem. Note that by (7) the asymptotic
behavior is correct. Before discussing (8) let us remark on a general (d = 2)
philosophy: All such expressions are analytic functions of low complexity. This
philosophy is the simple consequence of adopting a CFT point of view. It seems
curious that conjectures (1) and (3) do not seem to exactly fit into our present
understanding of CFT, even if all arguments (e.g. renormalization) are from
the philosophy of CFT. The point here is that the correct universality class
of domains does not come from the canonical lists of CFT, which include
SLE. Those domains are very far from extremal (for (1) and (3)) because
(e.g. 0 < κ ≤ 4) the two sides of the SLE trace are statistically the same. One
expects the extremal domains for (1) and (3) to be “one sided”, i.e. the exterior
domain does not have extremal behavior. It is, however, not impossible that
CFT could be directly brought to bear on these problems.
The result (8) has a somewhat curious history, and it is unclear if the
methodology used is optimal. In any case, the proof is via a two step argument.
The first step is to show that if a choice of domain Ω and α > 0 are given,
then (for ε > 0) there is a polynomial P , with Julia set J, such that
f (α, J c ) ≥ f (α, Ω) − ε .
The polynomial P has attached to it both good news and bad news. The
good news is that P can be assumed to act hyperbolically on its Julia set
J. In addition, the Julia set can be chosen to be of pure Cantor type, i.e. all
critical points are outside of J and in the basin of infinity, P n (c) → ∞ for
all critical points c. (If Ω is simply connected, one can alternatively demand
that J is a Jordan curve, and P is hyperbolic on J.) The bad news is that as
ε → 0, the polynomial constructed by [2] has degree tending to infinity. This
“bad” behavior is indeed necessary if J is to approximate ∂Ω in the correct
technical sense. Step two in the proof of (8) is to invoke a remarkable result
due to Binder–Makarov–Smirnov: If P and J are of pure Cantor type, there is
a polynomial Q, degree(Q) = degree(P ), such that the Julia set is connected
and
f (α, A∞ ) ≥ f (α, J c ) ,
where A∞ is the basin of infinity for Q. It is worth remarking that while this
theorem of Binder, Makarov and Smirnov appears to be simply the maximum
principle, no elementary proof is known at present. To sum up, to prove (6)
one first “approximates” the boundary of Ω by a polynomial Julia set of pure
Cantor type, so that f (α, J c ) ≥ f (α, Ω) − ε. Then one applies the result
of Binder, Makarov and Smirnov to “homotopy” J c to the compliment of a
˜ This means that for α ≥ 1,
connected Julia set J.
On Scaling Properties of Harmonic Measure
fu (α) = fdsc (α) ,
79
(10)
where the subscript d stands for (conformal) dynamic. It is indeed remarkable
that (10) holds, even if there were early hints in [6] that this would be the
case. (There one built a non-conformal dynamical system.)
The boldest form of the conjecture is that for (10) one need not consider
high degree polynomials, but may restrict to Julia sets for z 2 + c, where c is
in the Mandelbrot set. Though the evidence for this is slight, it is positive. If
this were indeed true, it would show an unexpected form of universality for
quadratic Julia sets. Only slightly stronger is the conjecture that, for all α,
the complement of the Mandelbrot set (M ) is extremal. This is because if c
is a boundary point of M and J is the Julia set for the parameter c, “near c,
M looks like J”.
It should also be remarked that numerical evidence for conjecture (1) was
given by Kraetzer [9]. It seems there is more work to be done here, especially
for complex values of t. A very nice idea has been introduced by D. Beliaev [1].
He defines a class of “conformal snowflakes” by using a particular randomization process. He is then able to get rigorous, computer assisted proofs of
lower bounds for the pressure function β(t).
An amusing aspect of these problems is that it seems likely that the proofs
for upper bounds (e.g. β(t) ≤ |t|2 /4, |t| ≤ 2) will be quite different from those
for the corresponding lower bounds. On this point, the only thing that seems
evident is that one should take unbounded domains with compact boundaries
(of log-capacity one). This is of course also the natural setting for polynomial
dynamics.
3 Harmonic Measure in Rd
Our knowledge of harmonic measure and the f (α) spectrum is much less
complete in Rd , d ≥ 3, than for d = 2. What we do know corresponds to
experience from CFT: Dimension two is special. As an example of this consider
the support of Harmonic measure for a domain Ω. In d = 2 the results of
Makarov and Jones–Wolff show that there is always a set of dimension one
that supports harmonic measure and (Makarov) if Ω is simply connected
one cannot use a smaller dimension. In other words, f (α) < α for α > 1
and for simply connected domains f (1) = 1. This means f (1) is given by
∂Ω = S 1 . Now for d ≥ 3 this might lead one to expect that f (α) < α for
α > d − 1, i.e. there is a set of dimension at most d − 1 that supports harmonic
measure (the Øksendal Conjecture). However, Tom Wolff [12] has provided
counterexamples (of “snowball-type”) showing this is false. On the positive
side we know (Bourgain [4]) that harmonic measure is always supported on a
set of dimension ≤ d−ε(d). The weakest conjecture that fits known properties
for the dimension of the support of harmonic measure is
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Peter W. Jones
dim(ω) ≤ d − ε(d) ,
(11)
where ε(d) → 0 as d → +∞. It is likely that this is relatively easy. As
justification for this conjecture one could cite Jones–Makarov [7], where it is
shown that in Rd ,
d − c1 /αd−1 ≤ fu (α) ≤ d − c2 /αd−1 ,
(12)
where c1 and c2 are dimension dependent.
One of the reasons that we are presently lacking precise knowledge of
dim(ω) in d ≥ 3 is the lack of arguments of complex variables type. A (very)
formal attempt to generalize the proof of Jones–Wolff to higher dimensions
leads to the conjecture
1
(13)
dim(ω) ≤ d −
d−1
because (!) if ∆u = 0 in Rd , then |gradient(u)|p is subharmonic for p ≥
(d − 2)/(d − 1). Corresponding to (13) is the statement
fu d −
1
1 = d−
d−1
d−1
as well as
f (α) < α ,
α>d−
1
.
d−1
If one now makes the supposition that fu (α) = d − cd α−d+1 for large α,
one comes to a somewhat reasonable looking formula. We first denote by α(ω)
the (universal) dimension of harmonic measure, so that by conjecture (13),
α(ω) = d − (d − 1)−1 . Algebra then leads one to the possibility
1
fu (α) = d −
d−1
α(ω)
α
d−1
(14)
for values α ≥ α(ω). Notice that this has the correct asymptotics as α → +∞,
and has fu (α) < α for α > α(ω). If we set d = 2 we also recover the conjecture
fu (α) = 2 − 1/α, α ≥ 1. On the other hand we have f 0 (α(ω)) < 1, which
seems unreasonable. If we wish also to make f 0 (α(ω)) = 1, thus providing for
a smoother phase transition, the next and perhaps more reasonable guess is
f (α) = d −
1−d
1
α + 1 − α(ω)
,
d−1
(15)
for α ≥ α(ω). Of course, whether one chooses (14) or (15), one should have
f (α) = α for 0 ≤ α ≤ α(ω). Conjecture (15) has the advantage that f 0 exists
for all α (and f 0 (α(ω)) = 1). I am unsure whether one can provide any other
justification for (14) and (15) beyond the following: They fit all known facts;
give us the “correct” conjecture in d = 2; are based on “simple” (here rational)
On Scaling Properties of Harmonic Measure
81
functions; and interpolate analytically through dimensions. The interpretation
of (14) or (15) for non-integer dimensions is unclear to me.
As is the case for d = 2, it seems clear that, whatever the correct answer
is for fu (α), the proofs of upper versus lower bounds will be quite different.
An examination of the arguments of [6] and [12] makes it clear that near
extremals for fu (α) can be built from non-conformal dynamical systems. But
in d ≥ 3 there is no natural analogue of Julia sets and/or conformal dynamics,
or rather it seems unlikely that this is how one should proceed. Perhaps there
are natural dynamical systems in d ≥ 3 that give rise to the appropriate
domain, but it would seem that if this were the case, those dynamical systems
are not among the ones we now are aware of. It also seems unlikely that
simply connected domains could be useful when d ≥ 3. This is due to the
fact that the potential theory in d ≥ 3 puts zero capacity on line segments,
and consequently fu (α) = fsc (α) actually holds for all α > 0. (As far as I am
aware, no one has written down the proof of this rather elementary statement.)
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
D. Beliaev: Ph.D. Thesis, KTH, Stockholm (2005)
I. Binder, P.W. Jones: to appear
I. Binder, N.G. Makarov, S. Smirnov: Duke Math. J. 117, 343 (2003)
J. Bourgain: Inventiones Math. 87, 477 (1987)
L. Carleson: Ann. Acad. Sci. Fenn. 10, 113 (1985)
L. Carleson, P.W. Jones: Duke Math. J. 66, 169 (1992)
P.W. Jones, N.G. Makarov: Annals of Maths. 142, 427 (1995)
P.W. Jones, T. Wolff: Acta Math. 161, 131 (1988)
P. Kraetzer: Complex Variable Theory Appl. 31, 305 (1996)
S. Smirnov: unpublished manuscript
N.G. Makarov: Proc. London Math. Soc. 51 (1985)
T. Wolff: Counterexamples with harmonic gradients in R3 . In: Essays on
Fourier Analysis in Honor of E.M. Stein, no 42 (Princeton Univ. Press, Princeton, NJ 1995) pp 321–384