Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 355–384

EXISTENCE OF BIG PIECES OF GRAPHS FOR PARABOLIC PROBLEMS

Steve Hofmann, John L. Lewis, and Kaj Nyström

University of Missouri, Department of Mathematics Columbia, MO 65211-0001, U.S.A.; hofmann@math.missouri.edu University of Kentucky, Department of Mathematics Lexington, KY 40506-0027, U.S.A.; john@ms.uky.edu Umeå University, Department of Mathematics SE-90187 Umeå, Sweden; kaj.nystrom@math.umu.se Current address: Föreningssparbanken, Group Financial Risk Control SE-10534 Stockholm, Sweden; kaj.nystrom@foreningssparbanken.se

Abstract. In this paper we define parabolic chord arc domains and generalize, to a parabolic setting, a theorem of Semmes concerning quantitative approximation by graphs with small Lipschitz constants.

1. Background and notation

In this paper, motivated by applications to the analysis of parabolic partial differential equations (see [HLN]) and parabolic singular integrals on domains not locally given by graphs, we define parabolic chord arc domains and generalize, to a parabolic setting, a theorem of Semmes concerning quantitative approximation by graphs with small Lipschitz constants.

If X = (x_{0}, x), we note for n > 1 that [LM], [HL] considered graph domains of the form Ω̃ = {(X, t) ∈ R^{n+1} : x_{0} > ψ(x, t)}, where ψ = ψ(x, t) : R^{n} → R had compact support and satisfied

(1.1) |ψ(x, t) − ψ(y, s)| ≤ b_{1}(|x − y| + |s − t|^{1/2}) < ∞ for all (x, t), (y, s) ∈ R^{n},

and

(1.2)  ∥D_{1/2}^{t}ψ∥_{∗} ≤ b_{2} < ∞.

2000 Mathematics Subject Classification: Primary 42B20, 35K05.

The first and second author were supported by grants from NSF.—This paper was partially conceived at the Mittag-Leffler Institute during a special year in potential theory and partial differential equations. J.L. Lewis and K. Nyström would like to thank the Institute for its gracious hospitality.

Here D_{1/2}^{t}ψ(x, t) denotes the \frac{1}{2} derivative in t of ψ(x, ·), x fixed. This half derivative in time can be defined by way of the Fourier transform or by

D_{1/2}^{t}ψ(x, t) ≡ ĉ∫_{R} \frac{ψ(x, s) − ψ(x, t)}{|s − t|^{3/2}} ds

for properly chosen ĉ. Also ∥ · ∥_{∗} denotes the bounded mean oscillation norm on R^{n} taken with respect to rectangles of side length r in the space direction and r^{2} in the time direction (see the display following (2.37) for a definition). If n = 1 the above authors considered graph domains Ω for which (1.1), (1.2) held with ψ = ψ(t). Through the work of [H], [LM], [HL] it has become clear that this is the right smoothness condition to impose on the domain, from the perspective of mutual absolute continuity of caloric measure with respect to a certain projective surface measure as well as from the point of view of parabolic singular integrals. In this paper we consider sets not necessarily locally given by graphs but impose conditions on the set in such a manner that the conditions allow us to extract (in a scale invariant fashion) big and very big pieces of time-varying graphs from the set, i.e., graphs given by a functions ψ satisfying the type of smoothness conditions stated above. The results of this paper are applied in [HLN] where we obtain parabolic analogues of results by Kenig–Toro in [KT], [KT1] concerning regularity and free boundary regularity on Reifenberg flat chord arc domains.

To formulate our results we need some definitions and notation. Let (X, t), X = (x_{0}, …, x_{n−1}), t ∈ R denote a point in R^{n+1} and for given r > 0 set

C_{r}(X, t) = {(Y, s) : |Y − X| < r, |t − s| < r^{2}}.

Given a Borel set F ⊂ R^{n+1} let \overline{F}, ∂F denote the closure, boundary of F, respectively, and put σ(F) = ∫_{F} dσ_{t} dt where dσ_{t} is (n−1)-dimensional Hausdorff measure on the time slice F ∩(R^{n}×{t}). Let Ω be a connected open set in R^{n+1} . We say that ∂ Ω separates R^{n+1} and is δ_{0} Reifenberg flat, 0 < δ_{0} ≤ 1/10, if given any (X, t) ∈ ∂Ω, R > 0, there exists an n-dimensional plane P̂ = P̂(X, t, R) containing (X, t) and a line parallel to the t axis with unit normal n̂ = n̂(X, t, R) such that

(1.3)  \begin{aligned}\{(Y, s) + rn̂ ∈ C_{R}(X, t) : (Y, s) ∈ P̂, r > δ_{0}R\} & ⊂ Ω, \\ \{(Y, s) − rn̂ ∈ C_{R}(X, t) : (Y, s) ∈ P̂, r > δ_{0}R\} & ⊂ R^{n+1} \ Ω. \end{aligned}

For short we shall just say ∂ Ω separates R^{n+1} when (1.3) holds for some δ_{0} . Note that if ∂ Ω separates R^{n+1} , then a line segment drawn parallel to n̂ and with endpoints in each of the sets in (1.3), also intersects ∂ Ω. Below we will, for (X, t) ∈ ∂ Ω and r > 0, by Δ(X, t, r) denote a surface cube defined as Δ(X, t, r) = ∂ Ω ∩ C_{r}(X, t).

We say that ∂ Ω satisfies an (M, R) Ahlfors condition, M ≥ 4, if for all (X, t) ∈ ∂ Ω and 0 < r ≤ R,

(1.4)  σ(∂ Ω ∩ C_{r}(X, t)) ≤ Mr^{n+1}.

Using the above note, (1.3), (1.4), the fact that Hausdorff measure does not increase under a projection we deduce for 0 < r ≤ R, (X, t) ∈ ∂ Ω,

(1.5)  (\frac{1}{2}r)^{n+1} ≤ σ(∂ Ω ∩ C_{r}(X, t)) ≤ Mr^{n+1},

whenever ∂ Ω separates R^{n+1} and satisfies an (M, R) Ahlfors condition. Next let

(1.6)  d(F_{1}, F_{2}) = inf{|X − Y | + |s − t|^{1/2} : (X, t) ∈ F_{1}, (Y, s) ∈ F_{2}},

denote the parabolic distance between the sets F_{1} , F_{2} and for Ω as in (1.5) set

(1.7)  γ(Z, τ, r) = inf_{P}[r^{−n−3} ∫_{∂ Ω∩C_{r}(Z,τ)} d({(Y, s)}, P )^{2} dσ(Y, s)]

where the infimum is taken over all n-dimensional planes P containing a line parallel to the t axis. Let

(1.8)  dν(Z, τ, r) = γ(Z, τ, r) dσ(Z, τ)r^{−1} dr.

We say that ν is a Carleson measure on [∂ Ω ∩ C_{R}(Y, s)] × (0, R) if there exists M_{1} < ∞ such that whenever (X, t) ∈ ∂ Ω and C_{ϱ}(X, t) ⊂ C_{R}(Y, s), we have

(1.9)  ν([C_{ϱ}(X, t) ∩ ∂ Ω] × (0, ϱ)) ≤ M_{1}ϱ^{n+1}.

The least such M_{1} in (1.9) is called the Carleson norm of [∂ Ω∩C_{R}(Y, s)]×(0, R). We write ∥ν∥_{+} for the Carleson norm of ν when (1.9) holds for all ϱ > 0. As in [DS], [DS1] we say that ∂ Ω is uniformly rectifiable (in the parabolic sense) if ∥ν∥_{+} < ∞ and (1.5) holds all R > 0. We remark that the work in [DS], [DS1] was in turn motivated by earlier work of [Jo], [Jo1], where L^{∞} versions of (1.7) were introduced. If ∂ Ω separates R^{n+1} and is uniformly rectifiable, we shall call Ω a parabolic regular domain. Finally in analogy with [KT] we say that Ω is a chord arc domain with vanishing constant if Ω is a parabolic regular domain and

(1.10)  sup_{{(X,t)∈∂ Ω \atop 0<ϱ≤r}}[ϱ^{−(n+1)}ν([C_{ϱ}(X, t) ∩ ∂ Ω] × (0, ϱ))] → 0 as r → 0.

In the following we are interested in showing the existence of big pieces of graphs, in particular we prove for parabolic regular domains that if δ_{0} in (1.3) is sufficiently small, then ∂ Ω contains big pieces of time varying graphs in the following sense. Given (X, t) ∈ ∂ Ω and R > 0, there exists after a possible rotation in the space variable, a function ψ as in (1.1), (1.2) and Ω̃ as above such that for some M_{2} = M_{2}(M, ∥ν∥_{+}, δ_{0})

(1.11)  M_{2}σ(∂ Ω̃ ∩ Δ(X, t, R)) ≥ R^{n+1}.

We will also obtain a parabolic analogue of Semmes decomposition, [S], concerning very big pieces of time varying graph domains when ∥ν∥_{+} is sufficiently small.

Given Ω as in (1.3) let (X, t) ∈ ∂ Ω and let ∥ν∥ be the Carleson norm of ν on Δ(X, t, 800R) × (0, 800R) (see (1.9)). We first prove

Theorem 1. If Ω satisfies (1.3), (1.5), and ∥ν∥ < ∞, then there exists c(M) such that if 0 < δ_{0} ≤ c(M)^{−1} in (1.3), then ∂ Ω contains big pieces of time-varying graphs in the sense of (1.11) with b_{1} + b_{2} ≤ c(M)(1 + ∥ν∥)^{1/2} in (1.1), (1.2). Moreover for (X, t), R as in (1.11), we can make the construction so that either Ω̃ ∩ C_{2R}(X, t) ⊂ Ω or Ω ∩ C_{2R}(X, t) ⊂ Ω̃.

Concerning very big pieces we obtain the following parabolic version of the theorem of Semmes for chord arc domains with small constants (see [S], [S1], [KT]).

Theorem 2. Let Ω, δ_{0}, (X, t) be as in Theorem 1 and suppose for some δ̂, 0 < δ̂ ≤ δ_{0} , that ∥ν∥ = δ^{κ} with δ ≤ δ̂ and κ = 24(n + 3) max{n − 1, 1}. If δ̂ = δ̂(M) is small enough, then Δ(X, t, R) contains very big pieces of time-varying graphs with small constants in the following sense. There exists after a possible rotation in the space variable, ψ, Ω̃, c_{2} = c_{2}(M) satisfying (1.1), (1.2) and

(a) b_{1} + b_{2} ≤ c_{2}δ.

(b) σ(Δ(X, t, R) \ ∂ Ω̃) + σ(∂ Ω̃ ∩ C_{R}(X, t) \ ∂ Ω) ≤ e^{−1/(c_{2}δ)}R^{n+1} .

(c) Δ(X, t, 2R) = G ∪ B, where G ⊂ ∂ Ω̃ and σ(B) ≤ e^{−1/(c_{2}δ)}R^{n+1} .

(d) If p denotes the orthogonal projection of R^{n+1} on {0} × R^{n}, d( · , ·) is as in (1.6), and (Y, s) ∈ B, then |y_{0} − ψ(y, s)| ≤ c_{2}δd({p(Y, s)}, p(G)),

(e) We can make the construction so that

either Ω̃ ∩ C_{R}(X, t) ⊂ Ω or Ω ∩ C_{R}(X, t) ⊂ Ω̃.

The rest of the paper is devoted to the proof of Theorem 1 and 2. The two sections of the paper are organized as follows. In Section 2 we start by jointly treating Theorem 1 and 2. Our proof scheme is motivated by the work in [DS]. Because of the refined analysis needed in the proof of Theorem 2 just the proof of Theorem 1 is completed in that section. In Section 3 we finish the proof of Theorem 2.

Finally in a remark following the proof of Theorem 2, we point out the differences between chord arc domains with small constants in the sense of [KT] and our definition (using smallness of the Carleson norm in (1.9)). In the first case it turns out that there are several equivalent definitions, including one which is an elliptic analogue of our definition. However, in the parabolic situation our definition is the only correct one, as we show by examples.

2. Proof of the theorems

In the sequel c denotes a positive constant ≥ 1, depending only on n. In general c(a_{1}, …, a_{m}) denotes a positive constant ≥ 1, depending only on n, a_{1}, …, a_{m}, not necessarily the same at each occurrence. Given Ω as in (1.3) let (X, t) ∈ ∂ Ω and let ∥ν∥ be the Carleson norm of ν on Δ(X, t, 800R) × (0, 800R) (see (1.9)).

Proof. If n = 1 then (d) of Theorem 2, ψ(y, s) should be replaced by ψ(s). To simplify matters we shall assume that n > 1 in the proof of Theorems 1 and 2. At the end of the proofs, we briefly treat the case n = 1. We shall also just make the construction so that Ω̃ ∩ C_{R}(X, t, R) ⊂ Ω.

We begin the proof of both theorems together since their proofs involve many of the same steps. We introduce for (Z, τ) ∈ ∂ Ω, r > 0,

(2.1)  γ_{∞}(Z, τ, r) = inf_{P} sup_{(Y,s)∈Δ(Z,τ,r)} \frac{d(Y, s, P)}{r},

where the infimum is taken over all n planes P containing a line parallel to the t axis. Given (Z, τ), r as above we claim that

(2.2)  γ_{∞}(Z, τ, r)^{n+3} ≤ 16^{n+3}γ(Z, τ, 2r).

To prove (2.2) we note that the infimum in the definition of γ (see (1.7)) is actually obtained for some plane say P̃. Clearly there exists (Z_{1}, τ_{1}) ∈ Δ̄(Z, τ, r) with

ϱ = rγ_{∞}(Z, τ, r) ≤ d({(Z_{1}, τ_{1})}, P̃ ).

From (1.5) we have σ(Δ(Z_{1}, τ_{1}, ϱ/8)) ≥ (ϱ/16)^{n+1} . Moreover every point in Δ(Z_{1}, τ_{1}, ϱ/8) is contained in Δ(Z, τ, 2r) and lies at least \frac{1}{2}ϱ from P̃. Thus, (2.2) is valid. Integrating (2.2) with respect to dσ(z, τ)r^{−1} dr over Δ(Y, s, 4ϱ) × (0, 4ϱ) and making simple estimates we conclude that

(2.3)  γ_{∞}(Y, s, ϱ)^{n+3} ≤ c∥ν∥

whenever (Y, s) ∈ Δ(X, t, 100R) and 0 < ϱ ≤ 100R. From (2.3) we see in the situation of Theorem 2 that there exists c_{3} = c_{3}(n) with

(2.4) γ_{∞}(Y, s, ϱ) ≤ c_{3}δ^{κ/(n+3)} for all (Y, s) ∈ Δ(X, t, 100R), 0 < ϱ ≤ 100R.

Let δ_{1} = δ_{0} in the proof of Theorem 1 and δ_{1} = 2c_{3}δ^{κ/(n+3)} in the proof of Theorem 2. From the definition of γ_{∞} we see when δ_{1} = 2c_{3}δ^{κ/(n+3)} < 1/10 that (1.3) still holds with (X, t), R, δ_{0} replaced by (Y, s), ϱ, δ_{1} provided P̂(Y, s, ϱ) is replaced by a plane through (Y, s) which is parallel to the plane that minimizes γ_{∞}(Y, s, ϱ). From now on we denote this minimizing plane (for ease of notation) by P̂(Y, s, ϱ) whenever (Y, s) ∈ Δ(X, t, 100R) and 0 < ϱ ≤ 100R. In the proof of Theorems 1 and 2 we assume, as we may, thanks to (1.3) that P̂(X, t, R) = {(Y, s) ∈ R^{n+1} : y_{0} = −10δ_{1}R} and

\begin{aligned}\{(−10δ_{1}R + r, y, s) & ∈ C_{R}(X, t), r > δ_{1}R\} ⊂ Ω, \\ n̂(X, t, R) = e_{0} & = (1, 0, …, 0). \end{aligned}

As in Theorem 2 let p(Y, s) = (0, y, s) be the orthogonal projection of R^{n+1} onto {0} × R^{n} and observe from (1.3) as in (1.5) that

(2.5)  (\frac{1}{2}R)^{n+1} ≤ H^{n}(p[Δ(X, t, R)]),

where H^{n} denotes Hausdorff n measure on {0} × R^{n}.

Let E be the set of all points (Y, s) ∈ Δ(X, t, R) for which there exists r, 0 < r ≤ R, with

(2.6)  H^{n}(p[Δ(Y, s, r)]) ≤ θr^{n+1},

where θ = 1000^{−(n+1)}M^{−1} . Thus E ⊂ Δ(X, t, R) consists of those points (Y, s) such that Δ(Y, s, r) has a small projection on {0} × R^{n} for some r > 0. Using a well-known covering argument we see there exists a covering of E by sets \{Δ(Y_{i}, s_{i}, r_{i})\} with \{Δ(Y_{i}, s_{i}, \frac{1}{5}r_{i})\} pairwise disjoint and (Y_{i}, s_{i}) ∈ E. Moreover (2.6) holds with Y, s, r replaced by Y_{i}, s_{i}, r_{i} for each i and 0 < r_{i} ≤ R. Using (1.5), (2.5) we get

H^{n}(p(E)) ≤ ∑_{i} H^{n}(p[Δ(Y_{i}, s_{i}, r_{i})])

≤ θ ∑_{i} r_{i}^{n+1} ≤ 10^{n+1}θ ∑_{i} σ(Δ(Y_{i}, s_{i}, \frac{1}{5}r_{i}))

≤ 10^{n+1}θσ(Δ(X, t, 2R)) ≤ 2^{−(n+2)}R^{n+1}

≤ \frac{1}{2}H^{n}(p[Δ(X, t, R)]).

Thus if F = Δ(X, t, R) \ E, then

(2.7)  H^{n}(p(F)) ≥ \frac{1}{2}H^{n}(p[Δ(X, t, R)]) ≥ 2^{−(n+2)}R^{n+1}.

Next suppose (Y, s) ∈ F, 0 < r ≤ \frac{1}{2}R, n̂ is the normal in (1.3) defined relative to P̂(Y, s, r) and η = |⟨n̂, e_{0}⟩|. We consider two cases. First if η ≤ 1/10 observe that P̂(Y, s, r) and p(R^{n+1}) intersect in an (n − 1)-dimensional plane containing a line parallel to the t axis on which p is the identity mapping. Also e′ = e_{0} − ⟨n̂, e_{0}⟩n̂ lies in a plane through 0 that is parallel to P̂(Y, s, r), |e′| ≥ \frac{1}{2} , and |p(e′)| ≤ η. Hence p maps K = C_{r}(Y, s) ∩ P̂(Y, s, r) onto a set with H^{n} measure kηr^{n+1} , where c^{−1} ≤ k ≤ c. Since each point of Δ(Y, s, r) lies at most δ_{1}r from P̂ and p(K) is convex we conclude from the above analysis that

c^{−1}(η + δ_{1})r^{n+1} ≤ H^{n}(p[Δ(Y, s, r)]) ≤ c(η + δ_{1})r^{n+1}.

From this inequality we deduce for 1/10 ≥ η, that there exists c_{+} = c_{+}(n) ≥ 1, such that if (2.6) is false, then

(2.8)  (c_{+}M)^{−1} ≤ η

On the other hand, if η ≥ 1/10, then (2.8) clearly holds so (2.8) is true in both cases. We note that if (Y, s) ∈ F, (the set with big projections), then (2.8) is true for 0 < r ≤ R. Next we use (2.8) to show that if (Y, s), (Z, τ) ∈ F, then

(2.9)  |Z − Y | ≤ cM(|z − y| + |s − t|^{1/2}).

In fact if r = 2(|Z − Y | + |s − τ|^{1/2}) and r ≥ R, then (2.9) follows from (1.3) for δ_{1}, P̂(X, t, R). Otherwise we write Z = Z′ + Z′′, Y = Y′ + Y′′ where (Y′, s), (Z′, τ) are the orthogonal projections of (Y, s), (Z, τ) onto P̂(Y, s, r) and

(2.10)  |Y′′ − Z′′| ≤ δ_{1}r.

Moreover using the same argument and notation as in the proof of (2.8) we have

(2.11)  |z′ − y′| ≥ η|Z′ − Y′|.

Using (2.8), (2.10), (2.11), the triangle inequality and δ_{1} ≤ η/1000, we get

|Z − Y | ≤ δ_{1}r + (1/η)|z′ − y′| ≤ r/100 + (1/η)|z − y| ≤ r/100 + c_{+}M|z − y|.

Clearly this inequality implies (2.9). From (2.9) we see that p is invertible on F ∩ (R^{n−1} × {s}) for each s ∈ R. Thus there exists ψ^{∗}: p(F) → R with

(2.12)  |ψ^{∗}(y, s) − ψ^{∗}(z, τ)| ≤ cM(|y − z| + |s − τ|^{1/2})

and (ψ^{∗}(y, s), y, s) ∈ F for all (y, s) ∈ p(F).

We now use our Carleson measure assumption and argue as in [DS, Chapter 13]. Let

f(Z, τ) = ∫_{0}^{100R} γ(Z, τ, r)r^{−1} dr, (Z, τ) ∈ Δ(X, t, R),

and observe from (1.8), (1.9) that

∫_{Δ(X,t,100R)} f(Z, τ) dσ(Z, τ) ≤ ∥ν∥(100R)^{n+1}.

From this inequality and ‘Tchebychev’s inequality’ we find for A ≥ 1000,

σ({(Z, τ) ∈ Δ(X, t, 100R) : f(Z, τ) ≥ A^{n+1}∥ν∥}) ≤ (100R/A)^{n+1} ≤ (R/10)^{n+1}.

Using the above inequality, (2.7), the fact that Hausdorff measure does not increase under a projection we deduce the existence of a closed set F_{1} = F_{1}(A) with F_{1} ⊂ F,

(2.13)  f(Z, τ) ≤ A^{n+1}∥ν∥, (Z, τ) ∈ F_{1},

and

(2.14)  H^{n}(p(F_{1})) ≥ 2^{−(n+3)}R^{n+1}.

We remark that we shall later obtain better estimates for the σ measure of a set F_{1} as above, when A is small (depending on δ). To extend ψ^{∗} off of p(F_{1}) we identify R^{n} with {0} × R^{n} and put

Q_{r}(z, τ) = {(y, s) ∈ R^{n} : |y_{i} − z_{i}| < r, i = 1, …, n − 1, |s − τ| < r^{2}}

for (z, τ) ∈ R^{n}, r > 0. Let \{\overline{Q}_{i} = \overline{Q}_{r_{i}}(x̂_{i}, t̂_{i})\} be a Whitney decomposition of R^{n} \ p(F_{1}) into rectangles. That is, Q_{i} ∩ Q_{j} = ∅, i ≠ j, and

(2.15)  10^{−10n}d(Q_{i}, p(F_{1})) ≤ r_{i} ≤ 10^{−8n}d(Q_{i}, p(F_{1})).

Let {v_{i}} be a partition of unity adapted to {Q_{i}}. Thus,

(2.16)

(a) ∑_{i} v_{i} ≡ 1 on R^{n} \ p(F_{1}),

(b) v_{i} ≡ 1 on Q_{i} and v_{i} ≡ 0 in R^{n} \ \overline{Q}_{2r_{i}}(x̂_{i}, t̂_{i}) for all i,

(c) v_{i} is infinitely differentiable on R^{n} with

r_{i}^{−l}|\frac{∂^{l}}{∂x^{l}} v_{i}| + r_{i}^{−2l}|\frac{∂^{l}}{∂t^{l}} v_{i}| ≤ c(l, n) for l = 1, 2, ….

In (c), ∂^{l}/∂x^{l} denotes an arbitrary lth partial in the space variable x. Set

C_{ϱ}(z, τ) = {(y, s) ∈ R^{n} : |z − y| < ϱ, |τ − s| < ϱ^{2}} whenever (y, s) ∈ R^{n}, ϱ > 0.

Next for each i choose (x′_{i}, t′_{i}) ∈ p(F_{1}) with
ϱ_{i} = d({(x′_{i}, t′_{i})}, Q_{i}) = d(p(F_{1}), Q_{i})

and set Λ = \{i : \overline{Q}_{i} ∩ \overline{C}_{2R}(x, t) ≠ ∅\}. We consider two cases. Under the assumptions of Theorem 1 we put

(2.17)  ψ(y, s) = \{ \begin{matrix}ψ^{∗}(y, s), (y, s) ∈ p(F_{1}), \\ ∑_{i∈Λ}(ψ^{∗}(x′_{i}, t′_{i}) + c^{∗}Mϱ_{i})v_{i}(y, s), (y, s) ∈ R^{n} \ p(F_{1}). \end{matrix}

With ψ now defined on R^{n} we can use (2.12), (2.15)–(2.17), and the usual Whitney type argument to get that (1.1) holds with b_{1} replaced by cM. One needs to be slightly careful when (y, s) is in the closure of two cubes say Q_{i}, Q_{j} with i ∈ Λ, j ∉ Λ. However this case follows easily from the fact that |ψ^{∗}| ≤ cMR and |∂v_{k}/∂y_{l}|(y, s) ≤ c/R for 1 ≤ l ≤ n − 1, k = i, j. Also ψ ≡ 0 in R^{n} \ C_{4R}(X, t) and Ω̃ ∩ C_{2R}(X, t) ⊂ Ω for c^{∗} large enough since otherwise (2.8) would be false. The proof of Theorem 1 is now complete except for (1.2). We continue under the scenario of Theorem 2. Let

h(K_{1}, K_{2}) = sup_{(Y,s)∈K_{1}} d({(Y, s)}, K_{2}) + sup_{(Y,s)∈K_{2}} d({(Y, s)}, K_{1})

be the parabolic Hausdorff distance between K_{1} and K_{2} . From (1.3) it is easily deduced that for (Z, τ), (Y, s) ∈ Δ(X, t, 100R) and 0 < \frac{1}{4}r ≤ ϱ ≤ 4r ≤ 100R, we have

(2.18)

(α) |n̂(Y, s, r) − n̂(Z, τ, ϱ)| ≤ 10^{5}δ_{1} whenever |Y − Z| + |s − τ|^{1/2} ≤ 4r,
(β) h(P̂(Y, s, r) ∩ C_{r}(Y, s), P̂(Z, τ, ϱ) ∩ C_{r}(Y, s)) ≤ 10^{5}δ_{1}r, (Z, τ),
(Y, s), r, ϱ as in (α).

We now extend the definition of ψ^{∗} from p(F_{1}) to R^{n}. Let

\begin{aligned}(X′_{i}, t′_{i}) & = (ψ^{∗}(x′_{i}, t′_{i}), x′_{i}, t′_{i}), \\ P̂(X′_{i}, t′_{i}, ϱ_{i}) & = \{(φ_{i}(y, s), y, s) : (y, s) ∈ R^{n}\} for i ∈ Λ \end{aligned}

which defines φ_{i}: R^{n} → R and as in [DS] put

(2.19)  ψ̂(y, s) = \{ \begin{matrix}ψ^{∗}(y, s), & for (y, s) ∈ p(F_{1}), \\ ∑_{i∈Λ} φ_{i}(y, s)v_{i}(y, s) & when (y, s) ∈ R^{n} \ p(F_{1}). \end{matrix}

We note from (2.15) that

(2.20)  r_{i}/r_{j} ≤ ϱ_{i}/r_{j} ≤ c whenever \overline{Q}_{i} ∩ \overline{Q}_{j} ≠ ∅.

We claim for (y, s), (z, τ) ∈ R^{n} that

(2.21)

(+) |φ_{i}(y, s) − φ_{i}(z, τ)| ≤ cM(|y − z| + |s − τ|^{1/2}) for all i,
(++) |φ_{i}(y, s) − φ_{j}(y, s)| ≤ cM^{2}δ_{1} min(ϱ_{i}, ϱ_{j}) when (y, s) ∈ \overline{Q}_{i}
and \overline{Q}_{i} ∩ \overline{Q}_{j} ≠ ∅.

(2.21)(+) is a direct consequence of (2.8). Also from (2.18), (2.8) and elementary geometry, we have for (y, s) ∈ \overline{Q}_{i} and \overline{Q}_{i} ∩ \overline{Q}_{j} ≠ ∅,

|φ_{i}(y, s) − φ_{j}(y, s)| ≤ cMd({(φ_{i}(y, s), y, s)}, P̂(X′_{j}, t′_{j}, ϱ_{j})) ≤ cM^{2}δ_{1} min(ϱ_{i}, ϱ_{j})

which is (2.21)(++).

From (2.18)–(2.21) and (2.16) it is easily deduced for δ_{1} small enough that

(2.22)  \begin{aligned}(∗) & |ψ̂(y, s) − ψ̂(z, τ)| ≤ cM(|y − z| + |s − τ|^{1/2}), \\ (∗∗) & |\frac{∂^{2}}{∂y_{l}∂y_{k}} ψ̂|(y, s) + |\frac{∂}{∂s}ψ̂|(y, s) ≤ cM^{2}δ_{1}ϱ_{i}^{−1} for (y, s) ∈ \overline{Q}_{i}, \\ & 1 ≤ l, k ≤ n − 1. \end{aligned}

For example, if (y, s) ∈ \overline{Q}_{i} and λ = \{j : \overline{Q}_{j} ∩\overline{Q}_{i} ≠ ∅\} ⊂ Λ, then ∂^{2}φ_{j}/∂y_{l}∂y_{k} ≡ 0 since ϕ_{j} is linear so

\frac{∂^{2}ψ̂}{∂y_{l}∂y_{k}} = ∑_{j∈λ}[\frac{∂φ_{j}}{∂y_{l}} \frac{∂v_{j}}{∂y_{k}} + \frac{∂φ_{j}}{∂y_{k}} \frac{∂v_{j}}{∂y_{l}} + φ_{j} \frac{∂^{2}}{∂y_{l}∂y_{k}} v_{j}] = T_{1} + T_{2} + T_{3}.

From (2.21)(++) we deduce first that |∇φ_{i} − ∇φ_{j}| ≤ cM^{2}δ_{1} when \overline{Q}_{i} ∩ \overline{Q}_{j} ≠ ∅ and second at (y, s) ∈ \overline{Q}_{i} that

|T_{1}| + |T_{2}| ≤ ∑_{j∈λ}[|\frac{∂φ_{i}}{∂y_{l}} − \frac{∂φ_{j}}{∂y_{l}} | |\frac{∂v_{j}}{∂y_{k}} | + |\frac{∂φ_{i}}{∂y_{k}} − \frac{∂φ_{j}}{∂y_{k}} | |\frac{∂v_{j}}{∂y_{l}} |] ≤ cM^{2}δ_{1}ϱ_{i}^{−1}.

Also at (y, s),

|T_{3}| ≤ ∑_{j∈λ} |φ_{i} − φ_{j}||\frac{∂^{2}}{∂y_{l}∂y_{k}} v_{j}| ≤ cM^{2}δ_{1}ϱ_{i}^{−1}.

Combining these inequalities we get (2.22)(∗∗) in this case.

Let Γ = {(ψ̂(y, s), y, s) : (y, s) ∈ R^{n}} and suppose that γ′, ν′ are the measures in (1.7)–(1.9) corresponding to Γ.

We shall prove that ∥ν′∥_{+} ≤ c(M)(δ_{1}^{2} + ∥ν∥). That is, we shall show that we can control the relevant Carleson norm with respect to Γ using δ_{1} and the local Carleson norm, ∥ν∥, defined relative to Δ(X, t, 800R). To begin we claim that if (Y, s) ∈ Γ ∩ \overline{C}_{100R}(X, t), then

(2.23)  d({(Y, s)}, ∂ Ω) ≤ c̃M^{2}δ_{1}d({(y, s)}, p(F_{1})).

To prove this inequality note that if (Y, s) ∈ F_{1} , then this inequality is trivally true so assume (Y, s) = (ψ̂(y, s), y, s) with (y, s) ∈ \overline{Q}_{i}. If \overline{Q}_{i} ∩ \overline{Q}_{j} ≠ ∅ for some j ∉ Λ, then from (2.15) we have cϱ_{i} ≥ R which implies (2.23) in view of (1.3). Otherwise, from (2.21) (++), (1.3) we have

\begin{aligned}d(\{(Y, s)\}, ∂ Ω) & ≤ d(\{(Y, s)\}, P̂(X′_{i}, t′_{i}, ϱ_{i})) + cδ_{1}ϱ_{i} \\ & ≤ |ψ̂(y, s) − ϕ_{i}(y, s)| + cδ_{1}ϱ_{i} ≤ cM^{2}δ_{1}ϱ_{i}. \end{aligned}

Thus (2.23) is true. We continue under the assumption that δ_{1} is so small that c̃M^{2}δ_{1} ≤ 10^{−6} .

Let Γ_{i} = {(Y, s) ∈ Γ : (y, s) ∈ Q_{i}} and put ĉ = 2c̃M^{2} . Then from (2.23) we see there exists a covering of \overline{Γ}_{i} ⊂ C_{100R}(X, t) by cylinders \{C_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j})\} with centers, (Z_{j}, τ_{j}) ∈ ∂ Ω and {C_{ĉδ_{1}ϱ_{i}/5}(Z_{j}, τ_{j})} pairwise disjoint. We note that each point in ∪C_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j}) lies in at most c(n) cylinders in the covering as follows from disjointness of the smaller cylinders and the fact that if \overline{C}_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j}) ∩ \overline{C}_{ĉδ_{1}ϱ_{i}}(Z_{l}, τ_{l}) ≠ ∅ then \overline{C}_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j}) ⊂ C_{4ĉδ_{1}ϱ_{i}}(Z_{l}, τ_{l}). Let P be a plane containing a line parallel to the t axis. For (Z, τ) ∈ \overline{Γ}_{i} ∩ \overline{C}_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j}) we have

d({(Z, τ)}, P ) ≤ cM^{2}δ_{1}ϱ_{i} + min_{(Y,s)∈Δ̄(Z_{j},τ_{j},ĉδ_{1}ϱ_{i})} d({(Y, s)}, P ).

Let (X̂, t̂) ∈ F_{1}, \overline{C}_{r}(X̂, t̂) ⊂ C_{80R}(X, t), Γ_{i,j} = Γ_{i} ∩ C_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j}) and put

∂ Ω_{i} = {(Y, s) ∈ ∂ Ω : d({(Y, s)}, Γ_{i}) ≤ 2ĉδ_{1}ϱ_{i}}.

We note that if \overline{Γ}_{i,j} ∩ C_{r}(X̂, t̂) ≠ ∅, then C_{ĉδ_{1}ϱ_{i}}(Z_{j}, τ_{j}) ⊂ C_{r}(X̂, t̂, \frac{5}{4}r) as follows from (2.15) and the fact that ϱ_{i} ≤ 2r. Also, σ( \overline{Γ}_{i}) ≤ cMϱ_{i}^{n+1} since from (2.22)(∗)

(2.24) dσ(Y, s) = \sqrt{1 + |∇ψ̂(y, s)|^{2}} dy ds ≤ 2MdH^{n}(y, s)  for (Y, s) ∈ Γ.

Here ∇ψ̂ denotes the gradient of ψ̂ in the space variable only. Using these notes, the above inequality, (1.5), we get

∫_{Γ_{i}∩C_{r}(X̂,t̂)} d({(Y, s)}, P )^{2} dσ(Y, s) ≤ ∑_{j} ∫_{Γ_{i,j}∩C_{r}(X̂,t̂)} d({(Y, s)}, P )^{2} dσ(Y, s)

≤ c(M^{2}δ_{1}ϱ_{i})^{2}σ(Γ_{i} ∩ C_{r}(X̂, t̂))

+ cM ∑_{j} ∫_{C_{5r/4}(X̂,t̂)∩Δ(Z_{j},τ_{j},ĉδ_{1}ϱ_{i})} d({(Y, s)}, P )^{2} dσ(Y, s) (2.25)

≤ c(M)[δ_{1}^{2}ϱ_{i}^{n+3} + ∫_{C_{5r/4}(X̂,t̂)∩∂ Ω_{i}} d({(Y, s)}, P )^{2} dσ(Y, s)].

Finally we observe for fixed l that ∂ Ω_{l} ∩ ∂ Ω_{k} ≠ ∅ for at most c(n) integers k as we find from (2.15). We conclude from this observation upon summing (2.25) that if

ξ = ξ(X̂, t̂, r) = {i : Γ_{i} ∩ C_{r}(X̂, t̂) ≠ ∅},

then

∫_{Γ∩C_{r}(X̂,t̂)} d({(Y, s)}, P )^{2} dσ(Y, s)

≤ c(M)[δ_{1}^{2} ∑_{i∈ξ} ϱ_{i}^{n+3} + ∫_{∂ Ω∩C_{5r/4}(X̂,t̂)} d({(Y, s)}, P )^{2} dσ(Y, s)]. (2.26)

Let γ′, ν′ be the measures in (1.7)–(1.9) corresponding to Γ. Using (2.22), (2.26) we shall show that

(2.27)  ∥ν′∥_{+} ≤ c(M)(δ_{1}^{2} + ∥ν∥),

where ∥ν′∥_{+} denotes the Carleson norm of ν′ relative to Γ × (0, ∞) as defined after (1.9). To do this first suppose (X̂, t̂) ∈ F_{1} and C_{r}(X̂, t̂) ⊂ C_{80R}(X, t). Then from (2.26) and the fact that F_{1} ⊂ ∂ Ω ∩ Γ we get

(2.28)  γ′(X̂, t̂, r) ≤ c(M)[δ_{1}^{2} ∑_{i∈ξ}(ϱ_{i}/r)^{n+3} + γ(X̂, t̂, \frac{5}{4}r)].

For given ϱ̂ > 0, (Ẑ, τ̂) ∈ Γ with Cϱ̂(Ẑ, τ̂) ⊂ C_{20R}(X, t), we integrate (2.28) over F_{1} ∩ Cϱ̂(Ẑ, τ̂). If F_{1} ∩ Cϱ̂(Ẑ, τ̂) = ∅ the following inequality is trivially true. Otherwise, summing and interchanging the order of integration we obtain with r′_{i}(X̂, t̂) = d({(X̂, t̂)}, Γ_{i}) + ϱ_{i},

ν′[F_{1} ∩ Cϱ̂(Ẑ, τ̂) × (0, ϱ̂)] = ∫ ϱ̂ ; 0 ∫_{F_{1}∩Cϱ̂(Ẑ,τ̂)} γ′(X̂, t̂, r) dσ(X̂, t̂)r^{−1} dr

≤ c(M)δ_{1}^{2} ∫_{F_{1}∩Cϱ̂(Ẑ,τ̂)}( ∑_{i∈ξ(X̂,t̂,ϱ̂)} ∫ ϱ̂ ; r′_{i}(X̂,t̂)(ϱ_{i}/r)^{n+3}r^{−1} dr) dσ(X̂, t̂)

+ c(M)ν[F_{1} ∩ C_{5ϱ̂/4}(Ẑ, τ̂) × (0, 2ϱ̂)] (2.29)

≤ c(M)δ_{1}^{2} ∑_{i∈ξ(Ẑ,τ̂,ϱ̂)} ∫_{F_{1}∩Cϱ̂(Ẑ,τ̂)}(\frac{ϱ_{i}}{r′_{i}(X̂, t̂)})^{n+3} dσ(X̂, t̂) + c(M)∥ν∥ϱ̂^{n+1}

≤ c(M)[δ_{1}^{2} ∑_{i∈ξ(Ẑ,τ̂,ϱ̂)} ϱ_{i}^{n+1} + ∥ν∥ϱ̂^{n+1}]

≤ c(M)(δ_{1}^{2} + ∥ν∥)ϱ̂^{n+1},

where we have used (1.5) to estimate the last integral. Next for r > 0, (x̂, t̂) ∈ R^{n} let

κ(x̂, t̂, r) = r^{−(n+3)} inf_{L} ∫_{C_{r}(x̂,t̂)} |ψ̂(y, s) − L(y)|^{2} dy ds,

where the infimum is over all linear functions of y (only). We note that

(2.30)  (c(M))^{−1}κ(x̂, t̂, r) ≤ γ′(X̂, t̂, r) ≤ c(M)κ(x̂, t̂, r)

as we see from (2.24), (2.22)(∗), and an argument similar to the one used in (2.8). Using (2.30), (2.22)(∗∗) and Taylor’s theorem we deduce for 0 < r ≤ \frac{2}{3}ϱ_{i} and (x̂, t̂) ∈ \overline{Q}_{i} that

γ′(X̂, t̂, r) ≤ c(M)κ(x̂, t̂, r) ≤ c(M)δ_{1}^{2}r^{2}ϱ_{i}^{−2}.

If \frac{2}{3}ϱ_{i} < r ≤ 8ϱ_{i}, then this inequality is also valid as we see from (1.3) with (X, t), R replaced by (X′_{i}, t′_{i}), 8ϱ_{i}. Thus

(2.31)  ν′[ \overline{Γ}_{i} × (0, 8ϱ_{i})] ≤ c(M)δ_{1}^{2}ϱ_{i}^{n+1}.

If 8ϱ_{i} < r, then from (2.15), (2.30) we see for (x̂, t̂) ∈ \overline{Q}_{i} that

γ′(X̂, t̂, r) ≤ c(M)γ′(X′_{i}, t′_{i}, \frac{3}{2}r).

Integrating this inequality, using (2.28) as well as (2.13) we find for (X̂, t̂) ∈ \overline{Γ}_{i}, \overline{Γ}_{i} ⊂ C_{40R}(X, t) and Cϱ̂(X̂, t̂) ⊂ C_{40R}(X, t) that

\begin{aligned}∫ ϱ̂ ; 8ϱ_{i} γ′(X̂, t̂, r)r^{−1} dr & ≤ c(M)f(X′_{i}, t′_{i}) + c(M)δ_{1}^{2} ∫ ϱ̂ ; 2ϱ_{i} ∑_{j∈ξ(X′_{i},t′_{i},2r)}(ϱ_{j}/r)^{n+3}r^{−1} dr \\ & ≤ c(M)∥ν∥ + c(M)δ_{1}^{2} ∑_{j∈ξ(X′_{i},t′_{i},2ϱ̂)}(\frac{ϱ_{j}}{ϱ_{i} + ϱ_{j} + d(Q_{i}, Q_{j})})^{n+3}. \end{aligned}

Integrating the above inequality over \overline{Γ}_{i} with respect to σ and summing for i ∈ ξ(Ẑ, τ̂, 2ϱ̂) we obtain when Cϱ̂(Ẑ, τ̂) ⊂ C_{20R}(X, t) is as in (2.29)

∑_{i∈ξ(Ẑ,τ̂,2ϱ̂)} ∫_{\overline{Γ}_{i}} ∫ ϱ̂ ; 8ϱ_{i} γ′(X̂, t̂, r)r^{−1} dr dσ(X̂, t̂) ≤ c(M)∥ν∥ϱ̂^{n+1}

+ c(M)δ_{1}^{2} ∑_{j∈ξ(Ẑ,τ̂,2ϱ̂)}(ϱ_{j})^{n+3} ∫_{R^{n}}(ϱ_{j} + d({(y, s)}, {(x′_{j}, t′_{j})}))^{−(n+3)} dy ds

≤ c(M)[∥ν∥ϱ̂^{n+1} + δ_{1}^{2} ∑_{j∈ξ(Ẑ,τ̂,2ϱ̂)} ϱ_{j}^{n+1}]

≤ c(M)(δ_{1}^{2} + ∥ν∥)ϱ̂^{n+1}.

Combining this inequality with (2.31) we deduce first that

ν′[(Γ \ F_{1}) ∩ Cϱ̂(X̂, t̂) × (0, ϱ̂)] ≤ c(M)(δ_{1}^{2} + ∥ν∥)ϱ̂^{n+1}

and thereupon in view of (2.29) that

(2.32)  ν′[Γ ∩ Cϱ̂(Ẑ, τ̂) × (0, ϱ̂)] ≤ c(M)(δ_{1}^{2} + ∥ν∥)ϱ̂^{n+1}

which is the desired inequality when Cϱ̂(Ẑ, τ̂) ⊂ C_{20R}(X, t). To handle the other cases we make the following observations. (2.33)(a) If (X̂, t̂) ∈ Γ, and 0 < r < d({(X̂, t̂)}, C_{4R}(X, t)), then γ′(X̂, t̂, r) = 0.

(2.33)(b) If r ≥ R, (X̂, t̂) ∈ Γ, then γ′(X̂, t̂, r) ≤ c(M)δ_{1}^{2}(R/r)^{n+3} . To prove (2.33)(a) recall that ψ̂ ≡ 0 on R^{n}\C_{4R}(x, t). I.e., d({(Y, s)}, {0}×R^{n}) = 0 for (Y, s) ∈ C_{r}(X̂, t̂) which clearly implies (a). (b) follows from the same recollection as in (a) and the fact that h(Γ, {0} × R^{n}) ≤ cδ_{1}R. From (2.33) we see that if ϱ̂ ≥ R, then

ν′(Cϱ̂(Ẑ, τ̂) × (0, ϱ̂)) ≤ ν′(Cϱ̂(Ẑ, τ̂) × (0, R)) + c(M)δ_{1}^{2}ϱ̂^{n+1}
≤ ν′(C_{10R}(X, t) × (0, R)) + c(M)δ_{1}^{2}ϱ̂^{n+1}
≤ c(M)(δ_{1}^{2} + ∥ν∥)ϱ̂^{n+1}.

If ϱ̂ ≤ R and Cϱ̂(Ẑ, τ̂))∩(R^{n+1} \∂C_{20R}(X, t)) ≠ ∅, then ν(Cϱ̂(Ẑ, τ̂)×(0, ϱ̂)) = 0 thanks to (2.33). We conclude that (2.32) holds whenever (Ẑ, τ̂) ∈ Γ and ϱ̂ > 0. Hence (2.27) is valid.

In order to study the implications of (2.27) we shall need some more notation. Let φ be an infinitely differentiable real-valued function on R^{n} with compact support in C_{1}(0, 0) (i.e, φ ∈ C_{0}^{∞}(C_{1}(0, 0))) and set

\begin{aligned}φ_{λ}(z, τ) & ≡ λ^{−(n+1)}φ(z/λ, τ/λ^{2}), \\ ∥φ∥_{∞} & = max\{|φ(z, τ)| : (z, τ) ∈ R^{n}\}. \end{aligned}

Next for θ: R^{n} → R, define the convolution of φ with θ by

φ ∗ θ(z, τ) = ∫_{R^{n}} φ(z − y, τ − s)θ(y, s) dy ds

whenever this convolution makes sense. For short we write φ_{λ}θ for φ_{λ}∗θ. Suppose now that φ, as above, annihilates constants and the coordinate functions y_{i}, 0 ≤ i ≤ n − 1. That is

∫_{Rn} φ(y, s) dy ds = 0 and ∫_{R^{n}} y_{i}φ(y, s) dy ds = 0.

Observe for each linear function L of y only that

(φ_{λ} ∗ ψ̂)^{2}(ẑ, τ̂) = [φ_{λ} ∗ (ψ̂ − L)]^{2}(ẑ, τ̂) ≤ c∥φ∥_{∞}λ^{−(n+1)} ∫_{C_{λ}(ẑ,τ̂)} |ψ̂ − L|^{2} dy ds.

From this observation, the definition of κ, (2.30), and (2.27) we see that

∫_{0}^{ϱ}∫_{C_{ϱ}(z,τ)} λ^{−3}(φ_{λ} ∗ ψ̂)^{2}(y, s) dy ds dλ ≤ c∥φ∥_{∞} ∫_{0}^{ϱ}∫_{C_{ϱ}(z,τ)} λ^{−1}κ(y, s, λ) dy ds dλ

≤ c(M)∥φ∥_{∞}(δ_{1}^{2} + ∥ν∥)ϱ^{n+1}. (2.34)

Let P ∈ C_{0}^{∞}(C_{1}(0, 0)) be an even nonnegative function on R^{n} with ∫_{R^{n}} P (w) dw ≡ 1. We note that if φ = φ(z, τ) is any of the functions,

(2.35)  λ\frac{∂P_{λ}}{∂λ} , λ^{2} \frac{∂P_{λ}}{∂τ} , λ^{2} \frac{∂^{2}P_{λ}}{∂z_{i}^{2}} , 1 ≤ i ≤ n − 1,

then φ annihilates linear functions of y and ∥φ∥_{∞} ≤ c as is easily seen. From (2.34), (2.35), we conclude

(2.36) \begin{aligned}∫_{0}^{ϱ}∫_{C_{ϱ}(z,τ)}(λ^{−1} & [\frac{∂(P_{λ}ψ̂)}{∂λ}]^{2} + λ[\frac{∂(P_{λ}ψ̂)}{∂τ}]^{2} + λ[\frac{∂^{2}(P_{λ}ψ̂)}{∂z_{i}^{2}}]^{2}) dy ds dλ \\ & ≤ c(δ_{1}^{2} + ∥ν∥)ϱ^{n+1} for 1 ≤ i ≤ n − 1. \end{aligned}

We now return to the proof of Theorem 1. In this case we claim that (2.36) holds with ψ̂ replaced by ψ (see (2.17)) and δ_{1} replaced by 1. Indeed, in view of (1.1) for ψ we can use the plan of the previous proof to conclude first that (2.27) holds with δ_{1} replaced by 1 and second that (2.36) is valid. In fact the argument is much simpler as we do not have to make precise estimates on the constants.

Proof of Theorem 1. We use (2.36) with δ_{1} = 1 to show that

(2.37)  ∥D_{1/2}^{t}ψ∥_{∗} ≤ c(M)(1 + ∥ν∥)^{1/2}

thereby completing the proof of Theorem 1. Here ∥ · ∥_{∗} denotes the norm in the space BMO(R^{n}) defined in the following way. Given g: R^{n} → R, locally integrable with respect to Lebesgue n measure, let C = C_{r}(z, τ) and let |C| be the Lebesgue n measure of C. Set

g_{C} = |C|^{−1} ∫_{C} g dy ds.

Then g ∈ BMO(R^{n}) with norm ∥g∥_{∗} if and only if

∥g∥_{∗} = sup_{C}{|C|^{−1} ∫_{C} |g − g_{C}| dy ds} < ∞.

To prove (2.37), let β ∈ C_{0}^{∞}(C_{2ϱ}(z, τ)), 0 ≤ β ≤ 1, with β ≡ 1 on C_{3ϱ/2}(z, τ) and

ϱ^{−l}|\frac{∂^{l}}{∂y^{l}} β| + ϱ^{−2l}|\frac{∂^{l}}{∂s^{l}} β| ≤ c(l, n)

at each point of R^{n}. If C = C_{ϱ}(z, τ), we put

ψ = [(ψ − ψ_{C})β] + [βψ_{C} + (1 − β)ψ] = ψ_{1} + ψ_{2}.

We first consider ψ_{1} . Given ε, 0 < ε < ϱ/100, we note that

(2.38) \begin{aligned}∫_{R^{n}}(D_{1/2}^{t}P_{ε}ψ_{1})^{2} dy ds & = ∫_{R^{n}}(D_{1/2}^{t}P_{ϱ}ψ_{1})^{2} dy ds \\ & − ∫_{ε}^{ϱ}∫_{R^{n}} 2D_{1/2}^{t}(\frac{∂}{∂λ}P_{λ}ψ_{1})D_{1/2}^{t}(P_{λ}ψ_{1}) dy ds dλ \\ & = ∫_{R^{n}}(D_{1/2}^{t}P_{ϱ}ψ_{1})^{2} dy ds \\ & + a ∫_{ε}^{ϱ}∫_{R^{n}} ℋ [\frac{∂}{∂λ}(P_{λ}ψ_{1})]\frac{∂}{∂t}(P_{λ}ψ_{1}) dy ds dλ \\ & = V_{1} + V_{2}. \end{aligned}

In (2.38), a is a constant and ℋ is the Hilbert transform on R defined by

ℋ k(s) = PV ∫_{R} k(τ̂)(s − τ̂)^{−1} ds

for k a real valued function. In the above display we have k = (∂/∂λ)(P_{λ}ψ_{1})(y, ·). From (1.1) we get |∂(P_{ϱ}ψ_{1})/∂s| ≤ c(M)ϱ^{−1} at each point in C_{8ϱ}(z, τ) and this function vanishes elsewhere in R^{n}. From this inequality we find for |y − z| ≤ 4ϱ that

|D_{1/2}^{t}P_{ϱ}ψ_{1}|(y, s) = |c ∫_{R} \frac{P_{ϱ}ψ_{1}(y, s) − P_{ϱ}ψ_{1}(y, t̂)}{|s − t̂|^{3/2}} dt̂| ≤ c(M) min\{1, \frac{ϱ^{3}}{|s − τ|^{3/2}} \}.

If |y − z| > 4ϱ, then D_{1/2}^{t}P_{ϱ}ψ_{1}(y, s) ≡ 0. Using these inequalities and integrating over R^{n}, we conclude that

(2.39)  |V_{1}| ≤ c(M)ϱ^{n+1}.

To handle V_{2} we note that ℋ is a bounded operator with norm ≤ c from L^{2}(R) to L^{2}(R)—the usual space of Lebesgue square integrable functions on R. Using this note and Hölder’s inequality we get from (2.37),

(2.40)  \begin{aligned}|V_{2}|^{2} & ≤ c(∫_{0}^{ϱ}∫_{R^{n}} λ^{−1}[\frac{∂}{∂λ}(P_{λ}ψ_{1})]^{2} dy ds) \\ & · (∫_{0}^{ϱ} ∫_{R^{n}} λ[\frac{∂}{∂t}(P_{λ}ψ_{1}))]^{2} dy ds) \\ & = V_{21} · V_{22}. \end{aligned}

To estimate V_{22} we observe for (y, s) ∈ C_{4ϱ}(z, τ), 0 < λ ≤ ϱ, and

L(x̂) = β(y, s) + ∑_{i=1}^{n−1} \frac{∂}{∂y_{i}} β(y, s)(y_{i} − x̂_{i})

that

|\frac{∂}{∂s}(P_{λ}ψ_{1})|(y, s)

≤ |_{∫R^{n}} \frac{∂}{∂t̂}(P_{λ})(y − x̂, s − t̂)(ψ(x̂, t̂) − ψ_{C})(β(x̂, t̂) − L(x̂)) dx̂ dt̂|

+ ∑_{i=1}^{n−1}|\frac{∂}{∂y_{i}} β(y, s)_{∫R}(y_{i} − x̂_{i})\frac{∂}{∂t̂}(P_{λ})(y − x̂, s − t̂)(ψ(x̂, t̂) − ψ(y, s)) dx̂ dt̂|

+ |β(y, s)\frac{∂}{∂s}P_{λ}ψ(y, s)|

≤ c(M)/ϱ + c|\frac{∂}{∂s}P_{λ}ψ|(y, s),

where we have used (1.1), (2.35), and Taylor’s theorem to estimate the integrals. If (y, s) ∈ R^{n} \ C_{4ϱ}(z, τ), then (∂/∂s)(P_{λ}ψ_{1})(y, s) = 0. Similarly, for (y, s) ∈ C_{4ϱ}(z, τ), 0 < λ ≤ ϱ,

|\frac{∂}{∂λ}(P_{λ}ψ_{1})|(y, s) ≤ c(M)λ/ϱ + c|\frac{∂}{∂λ}P_{λ}ψ|(y, s)

and this function vanishes elsewhere in R^{n}. Putting these inequalities into (2.40) and using (2.36) we obtain first that

|V_{2}|^{2} ≤ c(M)(1 + ∥ν∥)^{2}ϱ^{2n+2}

and second in view of (2.38), (2.39) that

∫_{R^{n}}(D_{1/2}^{t}P_{ε}ψ_{1})^{2} dy ds ≤ c(M)(1 + ∥ν∥)ϱ^{n+1}.

Using this inequality, the inequalities directly above (2.39), and letting ε → 0, we deduce that D_{1/2}^{t}P_{ε}ψ_{1} converges weakly in L^{2}(R^{n}) to a function ζ satisfying the same inequalities as D_{1/2}^{t}P_{ε}ψ_{1} . Using weak convergence, we see that ψ_{1}(y, ·) = cI_{1/2} ∗ ζ(y, ·), where I_{1/2}(s) = |s|^{−1/2} . This equality implies that ζ = D_{1/2}^{t}ψ_{1} exists and satisfies

(2.41)  ∫_{R^{n}}(D_{1/2}^{t}ψ_{1})^{2} dy ds ≤ c(M)(1 + ∥ν∥)ϱ^{n+1}.

Consider now integrals involving ψ_{2} . Observe that ψ_{2} is constant on C_{3ϱ/2}(z, τ). Thus for (y, s) ∈ C,

D_{1/2}^{t}ψ_{2}(y, s) = c ∫_{\{t̂:|t̂−τ|≥9ϱ^{2}/4\}} \frac{ψ_{2}(y, t̂) − ψ_{2}(y, s)}{|s − t̂|^{3/2}} dt̂.

Using this equality, ψ_{2} = (ψ_{C} − ψ)β + ψ, and (1.1) we deduce from estimates similar to the above that if

α = c ∫_{\{t̂:|t̂−τ|≥9ϱ^{2}/4\}} \frac{ψ(z, t̂) − ψ(z, τ)}{|τ − t̂|^{3/2}} dt̂,

then |D_{1/2}^{t}ψ_{2} − α| ≤ c(M) on C. Thus,

(2.42)  ∫_{C} |D_{1/2}^{t}ψ_{2} − α| dy ds ≤ c(M)ϱ^{n+1}.

From (2.42) and (2.41) we see that D_{1/2}^{t}ψ exists and is integrable on C. Moreover, since

|α − (D_{1/2}^{t}ψ)_{C}| ≤ |C|^{−1} ∫ |α − D_{1/2}^{t}ψ(y, s)| dy ds

it follows from (2.41), (2.42) and Hölder’s inequality that

∫_{C} |D_{1/2}^{t}ψ − (D_{1/2}^{t}ψ)_{C}| dy ds ≤ 2 ∫_{C} |D_{1/2}^{t}ψ_{2} − α| dy ds + 2 ∫_{C} |D_{1/2}^{t}ψ_{1}| dy ds

≤ c(M)ϱ^{n+1} + cϱ^{(n+1)/2}(∫_{R^{n}} |D_{1/2}^{t}ψ_{1}|^{2} dy ds)^{1/2} (2.43)

≤ c(M)(1 + ∥ν∥)^{1/2}ϱ^{n+1}.

Hence D_{1/2}^{t}ψ ∈ BMO (R^{n}) with ∥D_{1/2}^{t}ψ∥_{∗} ≤ c(M)(1 + ∥ν∥)^{1/2} . The proof of Theorem 1 is now complete.

3. Proof of Theorem 2

We return to the proof of Theorem 2. We shall use (2.36) and the fact that ∥ν∥ ≤ δ_{1} to show as in the proof of Theorem 1 that

(3.1)  \begin{matrix}(a)  ∫_{R^{n}}(D_{1/2}^{t}ψ̂)^{2} dy ds ≤ c(M)δ_{1}^{2}R^{n+1}, \\ (b)  ∑_{i=1}^{n−1} ∫_{R^{n}}(\frac{∂ψ̂}{∂y_{i}} )^{2} dy ds ≤ c(M)δ_{1}^{2}R^{n+1}. \end{matrix}

The proof of (3.1)(a) is essentially the same as (2.41) with ψ_{1} , ϱ replaced by ψ̂, R, only now we use the fact that |ψ̂| ≤ c(M)δ_{1}R to estimate V_{1} while the estimate for V_{2} follows directly from (2.36). We omit the details. As for (b) we have for 0 ≤ i ≤ n − 1,

∫_{R^{n}}(\frac{∂}{∂y_{i}} ψ̂)^{2} dy ds = ∫_{R^{n}}(\frac{∂}{∂y_{i}} P_{R}ψ̂)^{2} dy ds

− 2 ∫_{0}^{R}∫_{R^{n}}(\frac{∂^{2}}{∂y_{i}∂λ}P_{λ}ψ̂)(\frac{∂}{∂y_{i}} P_{λ}ψ̂) dy ds dλ

≤ c(M)δ_{1}^{2}R^{n+1} + |∫_{0}^{R}∫_{R^{n}}(\frac{∂}{∂λ}P̂_{λ}ψ)(\frac{∂^{2}}{∂y_{i}^{2}} P̂_{λ}ψ) dy ds dλ|

≤ c(M)δ_{1}^{2}R^{n+1} + (∫_{0}^{R}∫_{R^{n}} λ^{−1}(\frac{∂}{∂λ}P_{λ}ψ)^{2} dy ds dλ)^{1/2}

· (∫_{0}^{R} ∫_{R^{n}}λ(\frac{∂^{2}}{∂y_{i}^{2}} P_{λ}ψ)^{2} dy ds dλ)^{1/2}

≤ c(M)δ_{1}^{2}R^{n+1}.

Thus (3.1) is true. From (3.1) we shall deduce the existence of a closed set F_{2} ⊂ F_{1} ⊂ Δ(X, t, R) with

(3.2)

(α) σ(F_{2}) ≥ 2^{−(n+4)}R^{n+1},
(β) |n̂(Y, s, r) − e_{0}| ≤ c(M)δ_{1}^{1/[8(n−1)]} for all (Y, s) ∈ F_{2}
and 0 < r ≤ R.

We remark that the introduction of ψ̂ and most of our effort so far was made to get us in a position to prove (3.2). To this end if g: R^{n} → R and g is locally integrable on R^{n} we let ℳ g be the n-dimensional Hardy–Littlewood maximal function defined by

ℳ g(z, τ) = sup_{r>0} |C_{r}(z, τ)|^{−1} ∫_{C_{r}(z,τ)} |g|(y, s) dy ds.

Similarly let ℳ^{(1)}g( · , s) denote the (n − 1)-dimensional Hardy–Littlewood maximal function of g( · , s) taken with respect to balls and let

∇ψ̂(z, τ) = (\frac{∂}{∂z_{1}} ψ̂, …, \frac{∂}{∂z_{n−1}} ψ̂)(z, τ), (z, τ) ∈ R^{n},

be the spatial gradient of ψ̂ whenever these partial derivatives exist. Recall that ψ̂ has compact support in C_{4R}(X, t). From (3.1)(b), the Hardy–Littlewood maximal theorem, and weak type estimates we first deduce the existence of a closed set J ⊂ (t − 16R^{2}, t + 16R^{2}) with

(3.3)

(i) H^{1}[(t − 16R^{2}, t + 16R^{2}) \ J] ≤ δ_{1}R^{2},
(ii) _{∫C_{8R}(x,t)}ℳ^{(1)}(|∇ψ̂|)(y, s) dy ≤ c(M)δ_{1}^{1/2}R^{n−1} for all s ∈ J.

If s ∈ J we can again use weak type estimates and the Hardy–Littlewood maximal theorem to get a closed set G_{1}(s) ⊂ C_{4R}(x, t) ∩ (R^{n−1} × {s}) with

(3.4)  \begin{matrix}(+)  H^{n−1}[C_{4R}(x, t) ∩ (R^{n−1} × \{s\}) \ G_{1}(s)] ≤ δ_{1}^{1/4}R^{n−1}, \\ (++)  ℳ^{(1)}(|∇ψ̂|)(y, s) ≤ c(M)δ_{1}^{1/4} for all (y, s) ∈ G_{1}(s). \end{matrix}

In (3.3), (3.4), H^{1} , H^{n−1} denote Hausdorff one and (n − 1)-dimensional measure on R^{n}. For fixed s ∈ J let χ_{1}( · , s) denote the characteristic function of C_{4R}(x, t) ∩ (R^{n−1} × {s}) \ G_{1}(s). Using (3.3), (3.4) and the above argument once again we get for each s ∈ J a closed set G_{2}(s) ⊂ G_{1}(s) with

(3.5)  \begin{matrix}(∗)  H^{n−1}[C_{4R}(x, t) ∩ (R^{n−1} × \{s\}) \ G_{2}(s)] ≤ δ_{1}^{1/8}R^{n−1}, \\ (∗∗)  ℳ^{(1)}(χ_{1})(y, s) ≤ c(M)δ_{1}^{1/8} for all (y, s) ∈ G_{2}(s). \end{matrix}

We shall use (3.3)–(3.5) and (2.22)(∗) to show that if s ∈ J and (y, s) ∈ G_{2}(s) ∩ C_{2R}(x, t), then

(3.6) |ψ̂(y, s) − ψ̂(z, s)| ≤ c(M)δ_{1}^{1/[8(n−1)]}|y − z|, whenever |y − z| ≤ R.

In fact if (z, s) ∈ G_{1}(s) and r = |y − z|, then from basic Sobolev inequalities and (3.4)(++) we find

|ψ̂(y, s) − ψ̂(z, s)| ≤ c|y − z|[ℳ^{(1)}(|∇ψ̂|)(y, s) + ℳ^{(1)}(|∇ψ̂|)(z, s)] ≤ c(M)δ_{1}^{1/4}r.

Otherwise (i.e. (z, s) ∉ G_{1}(s)) from (3.5)(∗∗) we see there exists (z̃, s) ∈ G_{1}(s) with |z − z̃| < c(M)δ_{1}^{1/[8(n−1)]}r. Using (2.22)(∗) and the above inequalities we conclude that

\begin{aligned}|ψ̂(y, s) − ψ̂(z, s)| & ≤ |ψ̂(y, s) − ψ̂(z̃, s)| + |ψ̂(z̃, s) − ψ̂(z, s)| \\ & ≤ c(M)δ_{1}^{1/4}r + c(M)δ_{1}^{1/[8(n−1)]}r ≤ c(M)δ_{1}^{1/[8(n−1)]}r. \end{aligned}

Thus (3.6) is true. Next we note from (3.1)(a) that

∫_{R}∫_{R^{n}} \frac{(ψ̂(y, s) − ψ̂(y, τ))^{2}}{(s − τ)^{2}} dy ds dτ = c ∫_{R^{n}}(D_{1/2}^{t}ψ̂)^{2} dy ds ≤ c(M)δ_{1}^{2}R^{n+1},

where the first equality can be proved by way of the Fourier transform. Using this inequality and weak type estimates we see that if

f_{1}(z, τ) = ∫_{−8R}^{8R} \frac{(ψ̂(z, τ + h) − ψ̂(z, τ))^{2}}{h^{2}} dh,

then there exists G′ closed with G′ ⊂ C_{4R}(x, t) and

(3.7) (i) |C_{4R}(x, t) \ G′| ≤ δ_{1}R^{n+1}, (ii) f_{1}(z, τ) ≤ c(M)δ_{1} for all (z, τ) ∈ G′.

We claim that if (y, s) ∈ G′ ∩ C_{2R}(x, t), then

(3.8)  |ψ̂(y, s) − ψ̂(y, τ)| ≤ cδ_{1}^{1/4}|s − τ|^{1/2} whenever |s − τ| ≤ R^{2}.

In fact if τ − s = r^{2} > 0, then

r^{−4} ∫_{r^{2}}^{2r^{2}}(ψ̂(y, s + h) − ψ̂(y, s))^{2} dh ≤ cf_{1}(y, s) ≤ c(M)δ_{1},

thanks to (3.7)(ii). Using weak type estimates we see that there exists a closed set J′ ⊂ (r^{2}, 2r^{2}) with

(3.9) (A) H^{1}[(r^{2}, 2r^{2}) \ J′] ≤ δ_{1}^{1/2}r^{2},

(B) |ψ̂(y, s) − ψ̂(y, ŝ)| ≤ c(M)δ_{1}^{1/4}r for all ŝ ∈ J′.

Using (3.9)(A) we conclude first the existence of ŝ ∈ J with 0 ≤ ŝ − τ ≤ 2δ_{1}^{1/2}r^{2} and second from (2.22)(∗), (3.9)(B) that

\begin{aligned}|ψ̂(y, s) − ψ̂(y, τ)| & ≤ |ψ̂(y, s) − ψ̂(y, ŝ)| + |ψ̂(y, ŝ) − ψ̂(y, τ)| \\ & ≤ c(M)δ_{1}^{1/4}r + c(M)δ_{1}^{1/4}r ≤ c(M)δ_{1}^{1/4}r. \end{aligned}

Thus (3.8) is valid. Let χ be the characteristic function of C_{4R}(x, t) \ G′ and note from (3.7) as well as our usual argument that there exists a closed set G′′ ⊂ G′ with

(3.10) (a) |C_{4R}(x, t) \ G′′| ≤ δ_{1}^{1/2}R^{n+1},

(b) ℳ (χ)(z, τ) ≤ c(M)δ_{1}^{1/2} for all (z, τ) ∈ G′′.

We put G_{2} = ⋃_{t∈J} G_{2}(t) and G = G′′ ∩ G_{2} . Observe from (3.3)(i), (3.5) that

|C_{4R}(x, t) \ G_{2}| ≤ c(M)(δ_{1} + δ_{1}^{1/8})R^{n+1}

which in view of (3.10) implies

(3.11)  |C_{4R}(x, t) \ G| ≤ c(M)(δ_{1}^{1/8} + δ_{1}^{1/2})R^{n+1} = c(M)δ_{1}^{1/8}R^{n+1}.

Let (y, s) ∈ G ∩ C_{2R}(x, t) and (z, τ) ∈ C_{R}(y, s). If r = |y − z| + |s − τ|^{1/2} , then from (3.10)(b) we see there exists (z̃, τ̃) ∈ G′∩C_{r}(y, s) with d({(z, τ)}, {(z̃, τ̃)}) ≤ c(M)δ_{1}^{1/[2(n+1)]}r. From this inequality, (2.22)(∗), (3.6) and (3.8) we deduce

|ψ̂(y, s) − ψ̂(z, τ)| ≤ |ψ̂(y, s) − ψ̂(z̃, τ̃)| + |ψ̂(z̃, τ̃) − ψ̂(z, τ)|

≤ |ψ̂(y, s) − ψ̂(z̃, s)| + |ψ̂(z̃, s) − ψ̂(z̃, τ̃)| + c(M)δ_{1}^{1/[2(n+1)]}r (3.12)

≤ c(M)(δ_{1}^{1/[8(n−1)]} + δ_{1}^{1/4} + δ_{1}^{1/[2(n+1)]})r ≤ c(M)δ_{1}^{1/[8(n−1)]}r.

We now can prove (3.2). We choose F_{2} ⊂ F_{1} closed with p(F_{2}) ⊂ G and |p(F_{2})| ≥ 2^{−(n+4)}R^{n+1} . This choice is possible for δ_{1} > 0 sufficiently small, as we see from (2.14) and (3.11). From this choice, the fact that Hausdorff measure does not increase under a projection we get (3.2)(α). To prove (3.2)(β) let (Y, s) ∈ F_{2} and 0 < r ≤ R. Then (y, s) ∈ G so from (3.12) we find that each point of Γ ∩ C_{r}(Y, s) lies within parabolic distance c(M)δ_{1}^{1/[8(n−1)]}r of a point of the plane through (Y, s) with normal e_{0} . Also from (2.23) every point in Γ ∩ C_{r}(Y, s) lies within parabolic distance c(M)δ_{1}r of a point of ∂ Ω. Finally from (1.3), (2.18) we see that each point of Δ(Y, s, r) lies within parabolic distance cδ_{1}r of the plane through (Y, s) with normal n̂(Y, s, r). Since ∂ Ω separates R^{n+1} and n̂(X, t, R) = e_{0} , we conclude from basic geometry that all of the above can only hold if (3.2)(β) is valid.

Next we use (3.2), a John–Nirenberg type argument and the fact that (X, t) ∈ ∂ Ω, R > 0 are arbitrary to prove

Lemma 3.13. Let δ_{2} = δ_{1}^{1/[24(n−1)]} and for given (X, t) ∈ ∂ Ω, R > 0, let

K = {(Y, s) ∈ Δ(X, t, R) : |n̂(Y, s, r) − n̂(X, t, R)| ≤ δ_{2} for 0 < r ≤ R}.

If 0 < δ_{2} ≤ δ̃ and δ̃ = δ̃(M) is small enough, then there exists c̄ = c̄(M) ≥ 1 with

σ(Δ(X, t, R) \ K) ≤ e^{−1/(c̄δ_{2})}R^{n+1}.

Proof. Again we assume that e_{0} = n̂(X, t, R). Given R̃, R ≤R̃ ≤ 2R, set

Ê(λ) = Ê(λ,R̃)

= {(Y, s) ∈ Δ(X, t,R̃) : |n̂(Y, s, r) − ê_{0}| > λδ_{2}^{2} for some r, 0 < r ≤R̃}.

We show there exists θ = θ(M), 0 < θ < 1, and c(M) ≥ 1 such that for some R̃ ∈ [R, 2R],

(3.14)  σ(Ê(k + 1)) ≤ θσ(Ê(k)) + c(M)e^{−1/[c(M)δ_{2}]}R^{n+1}

for k = 1, …, k_{0} − 1. Here k_{0} is the largest positive integer for which k_{0} ≤ δ_{2}^{−1} . Once (3.14) is proved we can iterate this inequality k_{0} − 1 times starting from k = 1 to get Lemma 3.13 by iteration. In fact if (Y, s) ∈ Ê(k + \frac{1}{2}) we let

ϱ = ϱ(Y, s, k) = inf\{r : 0 < r ≤R̃ and |n̂(Y, s, r) − e_{0}| ≤ (k + \frac{1}{2})δ_{2}^{2}\}.

Then from (2.18) we see for 0 < δ_{2} ≤ δ̃, (Y, s) ∈ Ê(k + \frac{1}{2}), and δ̃ > 0 sufficiently small that

(3.15) kδ_{2}^{2} ≤ (k + \frac{1}{2})δ_{2}^{2} − c(M)δ_{1} ≤ |n̂(Z, τ, 5ϱ) − e_{0}| ≤ (k + \frac{1}{2})δ_{2}^{2} + c(M)δ_{1}

for all (Z, τ) ∈ Δ(Y, s, 5ϱ). As above, using a well-known covering lemma, we find {Δ(Y_{i}, s_{i}, ϱ_{i})} with

(3.16)  \begin{matrix}(a)  Ê(k + \frac{1}{2}) ⊂ ⋃ Δ(Y_{i}, s_{i}, 5ϱ_{i}), \\ (b)  Δ(Y_{j}, s_{j}, ϱ_{j}) ∩ Δ(Y_{i}, s_{i}, ϱ_{i}) = ∅, i ≠ j. \end{matrix}

Let ξ′ be the set of positive integers i for which Δ(Y_{i}, s_{i}, ϱ_{i})∩(R^{n+1}\Δ(X, t,R̃)) ≠ ∅ and let ξ be the rest of the positive integers that are indices in the above union. From (2.18) and the triangle inequality it is easily deduced that

|n̂(Y_{i}, s_{i}, ϱ_{i}) − e_{0}| ≤ cδ_{1} log(R/ϱ_{i}).

Also from (3.15) we have |n̂(Y_{i}, s_{i}, ϱ_{i}) − e_{0}| ≥ δ_{2}^{2} . Combining these inequalities we see that

ϱ_{i} ≤ e^{−δ_{2}^{2}/(cδ_{1})}R ≤ e^{−1/(cδ_{2})}R.

Hence if c_{∗} = c_{∗}(M) is large enough and r′ = e^{−1/(c_{∗}δ_{2})}R, then

⋃_{i∈ξ′} Δ(Y_{i}, s_{i}, ϱ_{i}) ⊂ Δ(X, t,R̃ + r′) \ Δ(X, t,R̃ − r′).

We now choose R̃ ∈ [R, 2R] so that

σ[Δ(X, t,R̃ + r′) \ Δ(X, t,R̃ − r′)] ≤ Mr′(1000R)^{n}.

The existence of R̃ follows from Ahlfors regularity of ∂ Ω (see (1.5)) and an easy counting argument using the fact that there are at least 1/(8r′) disjoint ‘rings’ of the above type contained in Δ(X, t, 2R) \ Δ(X, t, R). From the above inequalities it follows that

(3.17)  σ( ⋃_{i∈ξ′} Δ(Y_{i}, s_{i}, ϱ_{i})) ≤ Mr′(1000R)^{n}.

If i ∈ ξ, then from (3.2) with R, X, t replaced by ϱ_{i}, Y_{i}, s_{i}, there exists L_{i} ⊂ Δ(Y_{i}, s_{i}, ϱ_{i}) such that for 0 < r ≤ ϱ_{i} and (Z, τ) ∈ Δ(Y_{i}, s_{i}, ϱ_{i}),

(3.18)  \begin{matrix}(i)  |n̂(Z, τ, r) − n̂(Y_{i}, s_{i}, ϱ_{i})| ≤ c(M)δ_{2}^{3}, \\ (ii)  σ(L_{i}) ≥ 2^{−(n+4)}ϱ_{i}^{n+1}. \end{matrix}

Let L = ⋃_{i∈ξ} L_{i}. Then from (3.18)(ii), (3.16)(a), and (1.5) we get

(3.19)  \begin{aligned}σ(Ê(k + 1)) & ≤ σ(⋃ Δ(Y_{i}, s_{i}, 5ϱ_{i})) ≤ c(M)σ(⋃ Δ(Y_{i}, s_{i}, ϱ_{i})) \\ & ≤ c(M)σ( ⋃_{i∈ξ} Δ(Y_{i}, s_{i}, ϱ_{i})) + c(M)r′R^{n} \\ & ≤ c(M)σ(L) + c(M)r′R^{n}. \end{aligned}

From the definition of ϱ_{i} and (3.18)(i) we deduce for i ∈ ξ and (Z, τ) ∈ L_{i} that

|n̂(Z, τ, r) − e_{0}| < (k + 1)δ_{2}^{2} for 0 < r ≤R̃

provided 0 < δ_{2} ≤ δ̃ and δ̃ = δ̃(M) > 0 is small enough. Consequently,

L_{i} ∩ Ê(k + 1) = ∅

and from (3.15) we have L_{i} ⊂ Ê(k). Thus L ⊂ Ê(k) \ Ê(k + 1) and so by (3.19)

σ(Ê(k)) ≥ σ(L) + σ(Ê(k + 1)) ≥ (1 + c(M)^{−1})σ(Ê(k + 1)) − c(M)r′R^{n}.

Clearly this inequality is equivalent to (3.14). The proof of Lemma 3.13 is now complete.

We shall also need

Lemma 3.20. Let f be as defined below (2.12) and δ_{2} as in Lemma 3.13. If 0 < δ_{2} ≤ δ′ and δ′ = δ′(M) > 0 is sufficiently small, then there exists c^{∗} = c^{∗}(M) ≥ 1 such that

σ({(Z, τ) ∈ Δ(X, t, R) : f(Z, τ) ≥ δ_{2}}) ≤ e^{−1/(c^{∗}δ_{2})}R^{n+1}.

Proof. Let

a(Y, s, ϱ) = σ(Δ(Y, s, ϱ))^{−1} ∫_{Δ(Y,s,ϱ)} f(Z, τ) dσ(Z, τ).

We shall show that f ∈ BMO[Δ(X, t, R)] defined with respect to σ. In fact we prove

(3.21) sup_{Δ(Y,s,r)⊂Δ(X,t,R)}(σ(Δ(Y, s, r))^{−1} ∫_{Δ(Y,s,r)} |f − a(Y, s, r)| dσ) ≤ c(M)δ_{1}^{2}

It is well known that (3.21) implies the conclusion of Lemma 3.20 for Ahlfors regular sets and in fact one can prove this by an argument similar to the one used in proving Lemma 3.13. Thus we prove only (3.21). To do this we first claim that if ϱ_{1} = |Y − Z| + |s − τ|^{1/2} ≤ \frac{1}{2}ϱ ≤ \frac{1}{2}R, then

(3.22)  I = |∫_{ϱ}^{100R}[γ(Z, τ, r) − γ(Y, s, r)]r^{−1} dr| ≤ c(M)δ_{1}^{2}(ϱ_{1}/ϱ).

To prove (3.22) note for r ≥ ϱ that

γ(Y, s, r − ϱ_{1}) ≤ γ(Z, τ, r) ≤ γ(Y, s, r + ϱ_{1}).

Thus,

(3.23)  \begin{aligned}I & ≤ |∫_{ϱ}^{100R}[γ(Y, s, r − ϱ_{1}) − γ(Y, s, r)]r^{−1} dr| \\ & + |∫_{ϱ}^{100R}[γ(Y, s, r + ϱ_{1}) − γ(Y, s, r)]r^{−1} dr| \\ & = I_{1} + I_{2}. \end{aligned}

From the definition of γ in (1.7), (2.18), and (1.3) we note that

γ(Y, s, r′) ≤ (r′)^{−n−3} ∫_{Δ(Y,s,r′)} d({(X̂, t̂)}, P̂(Y, s, r′))^{2} dσ(X̂, t̂) ≤ c(M)δ_{1}^{2}.

Using this note and changing variables in the integral involving r − ϱ_{1} of (3.23), we get

|I_{1}| ≤ ∫_{ϱ−ϱ_{1}}^{ϱ} γ(Y, s, r′)(r′)^{−1} dr′ + c(M)δ_{1}^{2}ϱ_{1} ∫_{ϱ}^{100R}(r′)^{−2} dr′

+ ∫_{100R−ϱ_{1}}^{100R} γ(Y, s, r′)(r′)^{−1} dr′
≤ c(M)δ_{1}^{2}(ϱ_{1}/ϱ).

A similar estimate holds for |I_{2}|. Putting these estimates in (3.23) we conclude (3.22). Next suppose Δ(Y, s, r) ⊂ Δ(X, t, R) and put f = h+k on Δ(Y, s, r) where

h(Z, τ) = ∫_{0}^{100r} γ(Z, τ, r′)(r′)^{−1} dr′, (Z, τ) ∈ Δ(Y, s, r).

From the hypotheses in Theorem 2

∫_{Δ(Y,s,r)} h dσ ≤ c(M)∥ν∥r^{n+1} ≤ c(M)δ_{1}^{2}r^{n+1}.

Also if â = k(Y, s), then from (3.22) with ϱ = 100r, we obtain

∫_{Δ(Y,s,r)} |k − â| dσ ≤ c(M)δ_{1}^{2}r^{n+1}.

Putting these two estimates together and using

|â − a(Y, s, r)| ≤ σ[Δ(Y, s, r)]^{−1} ∫_{Δ(Y,s,r)} |f − a(Y, s, r)| dσ

we get

\begin{aligned}∫_{Δ(Y,s,r)} |f − a(Y, s, r)| dσ & ≤ 2 ∫_{Δ(Y,s,r)} h dσ + 2 ∫_{Δ(Y,s,r)} |k − â| dσ \\ & ≤ c(M)δ_{1}^{2}r^{n+1}. \end{aligned}

Thus (3.21) is valid and the proof of Lemma 3.20 is complete.

Armed with Lemmas 3.13 and 3.20 we can now retrace our earlier work and finally get Theorem 2. Indeed, from these lemmas we can choose F′ closed,

F′ ⊂ {(Z, τ) ∈ Δ(X, t, 2R) : f(Z, τ) ≤ δ_{2}} ∩ K,

where K is as in Lemma 3.13 with R replaced by 2R, so that for some c ≥ 1 and 0 ≤ δ_{2} ≤ min{δ̃, δ′}, we have

(3.24)  σ(Δ(X, t, 2R) \ F′) ≤ e^{−1/(cδ_{2})}R^{n+1}.

Again we assume that e_{0} = n̂(X, t, R) and that P̂(X, t, R) = {(Y, s) ∈ R^{n+1} : y_{0} = −10δ_{1}R}. Arguing as in (2.9)–(2.11) we conclude the existence of ψ̃: R^{n} → R with

(3.25)  |ψ̃(y, s) − ψ̃(z, τ)| ≤ cδ_{2}[|y − z| + |s − τ|^{1/2}]

whenever (y, s), (z, τ) ∈ p(F′). Let Q_{i}, {v_{i}} be as (2.15), (2.17) with F_{1} replaced by F′. We can extend ψ̃ to R^{n} as in either (2.17) or (2.19). Since the extension in (2.17) is slightly simpler we use it and thus put

(3.26) ψ(y, s) = \{ \begin{matrix}ψ̃(y, s), (y, s) ∈ p(F′), \\ ∑_{i∈Λ}(ψ̃(x′_{i}, t′_{i}) + c^{+}δ_{2}ϱ_{i})v_{i}(y, s) when (y, s) ∈ R^{n} \ p(F′). \end{matrix}

Observe for each (ψ(x′_{i}, t′_{i}), x′_{i}, t′_{i}) ∈ F′ that there is a truncated parabolic cone with vertex at this point whose interior is ⊂ Ω and which has height 4R, axis parallel to e_{0} , and angle opening \frac{1}{2}π − cδ_{2} . Using this fact it is easily seen for c^{+} large enough that if

(3.27)  Ω̃ = {(y_{0}, y, s) ∈ R^{n+1} : y_{0} > ψ(y, s)} then Ω̃ ∩ C_{100R}(X, t) ⊂ Ω.

From the definition of F′ above (3.24) it follows as earlier that

(3.28)  |ψ(y, s) − ψ(z, τ)| ≤ cδ_{2}[|z − y| + |τ − s|^{1/2}].

Also if Γ = {(ψ(z, τ), z, τ) : (z, τ) ∈ R^{n}}, (y, s) ∈ Q_{i}, and

\begin{aligned}(Y, s) & = (ψ(y, s), y, s) ∈ Γ ∩ C_{100R}(X, t), \\ (Y′, s) & = (y_{0}, y, s) ∈ Δ(X, t, 100R) \end{aligned}

then from (1.3), (3.28), and the definition of F′ we deduce

(3.29)  |y_{0} − ψ(y, s)| ≤ |y_{0} − ψ(x′_{i}, t′_{i})| + |ψ(y, s) − ψ(x′_{i}, t′_{i})| ≤ cδ_{2}ϱ_{i}.

Clearly (3.29) yields

(3.30)  d({(Y, s)}, ∂ Ω) + d({(Y′, s)}, Γ) ≤ cδ_{2}ϱ_{i}.

Using (3.28), (3.30), we can now repeat the argument following (2.23) with ∥ν∥, δ_{1} replaced by δ_{2} and ψ̂ by ψ to conclude first as in (2.27) that

(3.31)  ∥ν′∥ ≤ c(M)δ_{2}^{2},

where ν′ is defined relative to ψ and thereupon as in (2.36),

(3.32) \begin{aligned}∫{ϱ \atop 0}∫C_{ϱ}(z,τ) & (λ^{−1}[\frac{∂(P_{λ}ψ)}{∂λ}]^{2} + λ[\frac{∂(P_{λ}ψ)}{∂τ}]^{2} + λ[\frac{∂^{2}(P_{λ}ψ)}{∂z_{i}^{2}}]^{2}) dy ds dλ \\ & ≤ c(M)δ_{2}^{2}ϱ^{n+1}. \end{aligned}

From (3.32) we obtain by the same argument as in (2.37),

(3.33)  ∥D_{1/2}^{t}ψ∥_{∗} ≤ c(M)δ_{2}.

From (3.28) and (3.33) we see that (a) of Theorem 2 is valid with δ_{2} = δ. (c), (d) of Theorem 2 follow from (3.24), (3.30) with G = F′. (e) of this theorem is just (3.27). To prove (b) of Theorem 2, observe that

p(∂ Ω̃ ∩ C_{R}(X, t) \ ∂ Ω) ⊂ C_{R}(x, t) \ p(F′)

and from (1.3), (3.24) that

(3.34)  |C_{R}(x, t) \ p(F′)| ≤ σ(Δ(X, t, 2R) \ F′) ≤ e^{−1/(cδ_{2})}R^{n+1}.

To get (3.34) we have also used the fact (once again) that Hausdorff measure does not increase under a projection. From (2.24) we conclude that

σ(∂ Ω̃ ∩ C_{R}(X, t) \ ∂ Ω) ≤ ∫_{C_{R}(x,t)\p(F′)} \sqrt{1 + |∇ψ|^{2}} dy ds ≤ e^{−1/(cδ_{2})}R^{n+1}.

The above inequality and (3.24) give (b) of Theorem 2. Thus Theorem 2 is true when n > 1.

If n = 1 the only ‘planes’ allowed in (1.3) are lines parallel to the t axis. Using this fact it is easily shown for δ_{0} small enough that (1.3) implies ∂ Ω = {(ψ(t), t) : t ∈ R} for some ψ satisfying (1.1) with (x, t), (y, s) replaced by s, t and b_{1} ≤ cδ_{0} . From (2.30), (2.34) we see that (2.36) holds without the term in z_{i}. Using (2.36) and arguing as in the proof of (2.37) we get both Theorems 1 and 2 with Ω̃ = Ω. The proof of Theorems 1 and 2 are now complete.

Remark. In [KT] the ‘elliptic version’ of Theorem 2 is proved for Ahlfors regular domains Ω which are Reifenberg flat and with locally small chord arc constant, where now chord arc has a different meaning. To make comparisions and simplify matters we give all definitions globally. More specifically suppose Ω ⊂ R^{n}. Then ∂ Ω is said to be Ahlfors regular if (1.5) holds with σ = surface area or H^{n−1} measure on ∂ Ω. ∂ Ω separates R^{n} and is (δ, ∞) Reifenberg flat if (1.3) holds with δ = δ_{0} , only now all (n − 1)-planes are allowed in the definition. Also if in addition to these assumptions,

(3.35)  σ[Δ(X, r)]/(ω_{n}r^{n−1}) ≤ 1 + δ for 0 < r < ∞, X ∈ ∂ Ω,

then Ω is said to be a (δ, ∞) chord arc domain. Here ω_{n} is the volume of the unit ball in R^{n−1} , X ∈ ∂ Ω, and Δ(X, r) = {Y ∈ ∂ Ω : |Y − X| < r}. In [KT] it is shown that a theorem analogous to Theorem 2 which we call “Semmes theorem with small constants” holds for (δ, ∞) chord arc domains provided δ > 0 is sufficiently small. One can also define as in (1.7), γ(Z, r), only now the infimum is taken over all (n − 1)-planes and integration is with respect to surface area on ∂ Ω. ν is defined relative to γ as in (1.8) and ∥ν∥_{+} is as in (1.9) only with balls replacing rectangles. Using the same argument as in Theorem 2 it follows that this version of (1.9) with small constant implies “Semmes theorem with small constants”. On the other hand it is obvious that “Semmes theorem with small constants” (on all scales) implies that Ω is a (δ, ∞) chord arc domain for some small δ. Moreover it can be shown that “Semmes theorem with small constants” implies (1.9) with ∥ν∥_{+} small. For a proof of this, see the argument at the end of Section 7 in [HLN]. A proof in a more general situation is given in [DS1, Part IV, Theorem 1.3]. Thus the ‘elliptic version’ of (1.9) with small constant, the chord arc conditions in [KT], and “Semmes theorem with small constants” are all equivalent in the sense that small constants in one implies small constants in the other. As a consequence of this equivalence it is easily seen that the elliptic analogue of a chord arc domain with vanishing constant as defined in Section 1 is equivalent to the definition in [KT].

On the other hand the parabolic analogue of the global chord arc conditions in [KT] is weaker than (1.9). In fact in [LS] it was shown for n = 1 that if ω is any non increasing function on (0, ∞) with ω(0) = 0, ω(2τ) ≤ 2ω(τ), τ ≥ 0, and ω ≡ 1 for τ ≥ 1, then there exists ψ: R → R, satisfying

(3.36)  |ψ(t) − ψ(s)| ≤ ω(|t − s|), s, t ∈ R.

Moreover, if ∫_{0}^{1} τ^{−2}ω^{2}(τ) dτ = ∞ and D = {(x_{0}, t) : x_{0} > ψ(t)}, then it follows first that ∥D_{1/2}^{t}ψ∥_{∗} = ∞ and second that the Carleson norm in (1.9) calculated relative to ∂D is infinite. Let Ω = {(x_{0}, t) : x_{0} > δψ(t)} and suppose also that ω(τ) ≤ cτ^{1/2} , for τ > 0. Then (1.3) holds with δ_{0} replaced by cδ as follows from (3.36) and

σ[Δ(x_{0}, t, r)]/(2r^{2}) = 1

384 S. Hofmann, J.L. Lewis, and K. Nyström

which is (3.35) with δ = 0 for our σ so that Ω is a parabolic analogue of a (δ, ∞) chord arc domain in the sense of [KT]. Furthermore (1.9) is false for ∂ Ω since it is false for ∂D. Finally from the above equality and (2.4) it follows for n = 1 that (1.9) with small constant implies the conditions in [KT] for a chord arc domain. Thus (1.9) is stronger for n = 1 than the parabolic analogue of the chord arc conditions in [KT]. To get n > 1 examples simply put Ω = {(X, t) ∈ R^{n+1} : x_{0} > ψ(t)θ(x)} where θ ∈ C_{0}^{∞}(R^{n−1}) with θ ≡ 1 on {x : |x| ≤ N}, N large and ψ is as in any of the above examples. Finally

ω(τ) = \{ \begin{matrix}τ^{1/2}[log(1/τ)]^{−1/2} for 0 < τ ≤ e^{−4}, \\ min\{(e^{2}/2)τ, 1\} for τ > e^{−4} \end{matrix}

is an example of an ω which satisfies all of the above conditions.

References

[DS] David, G., and S. Semmes: Singular integrals and rectifiable sets in R^{n}: au-dèa des graphes lipschitziens. - Astérisque 193, 1991.

[DS1] David, G., and S. Semmes: Analysis of and on Uniformly Rectifiable sets. - Amer. Math. Soc. Math. Surveys Monogr. 38, 1993.

[H] Hofmann, S.: Parabolic singular integrals of Calderón type, rough operators, and caloric layer potentials. - Duke Math. J. 90, 1997, 209–259.

[HL] Hofmann, S., and J. Lewis: Solvability and representation by caloric layer potentials in time-varying domains. - Ann. of Math. 144, 1996, 349–420.

[HLN] Hofmann, S., J. Lewis, and K. Nyström: Caloric measure in parabolic flat domains. - Duke Math. J. (to appear).

[Jo] Jones, P.: Square functions, Cauchy integrals, analytic capacity, and harmonic measure. - In: Harmonic Analysis and Partial Differential Equations, edited by J. Garcia-Cuerva. Lecture Notes in Math. 1384, Springer-Verlag, 1989.

[Jo1] Jones, P.: Rectifiable sets and the traveling salesman problem. - Invent. Math. 102, 1990, 129–163.

[KT] Kenig, C., and T. Toro: Harmonic measure on locally flat domains. - Duke Math J. 87, 1997, 501–551.

[KT1] Kenig, C., and T. Toro: Free boundary regularity for harmonic measure and Poisson kernels. - Ann. of Math. 150, 1999, 369–454.

[LM] Lewis, J., and M. Murray: The method of layer potentials for the heat equation in time-varying domains. - Mem. Amer. Math. Soc. 114, 1995.

[LS] Lewis, J., and J. Silver: Parabolic measure and the Dirichlet problem for the heat equation in two dimensions. - Indiana Univ. Math. J. 37, 1988, 801–839.

[S] Semmes, S.: Chord-arc surfaces with small constant, I. - Adv. Math. 85, 1991, 198–223.

[S1] Semmes, S.: Chord-arc surfaces with small constant, II: Good parametrizations. - Adv. Math. 85, 1991, 170–199.

[St] Strichartz, R.S.: Bounded mean oscillation and Sobolev spaces. - Indiana Univ. Math. J. 29, 1980, 539–558.

Received 13 December 2002