Hardy Spaces on Strongly Pseudoconvex Domains in C^{n} and Domains of Finite Type in C^{2}

Galia Dafni

November 1993

ABSTRACT

We consider Hardy spaces H^{p}, p ≤ 1, of holomorphic functions in the interior of Ω, where Ω is either a strongly pseudoconvex domain in C^{n} or a weakly pseudoconvex domain of finite type in C^{2}. We exhibit an atomic decomposition for the boundary values of these functions (in the sense of distributions), thus showing they belong to the local real Hardy spaces h^{p}(∂Ω).

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Nonisotropic Balls . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Admissible Approach Regions, Maximal Function, and

H^{p}(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Atoms and h^{p}(∂Ω) . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Reproducing Formula . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Atomic Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Size Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 Cancellation Condition . . . . . . . . . . . . . . . . . . . . . . . . 32

6.1 p near 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 p small . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A Existence of Boundary Values 43

B Moment Conditions for h^{p} 47 1 h^{p}(R^{n}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2 h^{p} on Spaces of Homogeneous Type . . . . . . . . . . . . . . . . 50 3 h^{p}(∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1 Introduction

Given a bounded domain Ω ⊂ C^{n} with smooth boundary, there are two notions of Hardy spaces one can associate to it. On the one hand, one can define the holomorphic Hardy space H^{p}(Ω), 0 < p < ∞, to be the space of all functions F holomorphic in Ω and satisfying

sup_{δ>0} ∫_{S_{δ}} |F |^{p}dσ_{δ} < ∞,

where the integrals are taken over shells S_{δ} approximating the boundary. On the other hand, for 0 < p ≤ 1, one can consider real Hardy spaces on ∂Ω as spaces of distributions f on ∂Ω which admit an atomic decomposition

f = ∑ λ_{k}a_{k},

where the numbers λ_{k} satisfy

∑ |λ_{k}|^{p} < ∞,

and the a_{k} are atoms, i.e. functions supported in balls B_{k} satisfying a size condition and a cancellation condition. For ∂Ω, since it is compact, the correct notion is that of the local Hardy spaces h^{p}(∂Ω) defined by Goldberg ([Gol]), where one requires cancellation only for atoms supported in small balls. This definition depends on an appropriate choice of a family of balls at each point on the boundary. Such families of nonisotropic balls have been studied extensively when Ω is strongly pseudoconvex (see [S]) or Ω is pseudoconvex of finite type in C^{2} (see [NRSW]).

One can ask what is the relation between the boundary values of holomorphic H^{p} functions, considered as distributions on ∂Ω, and the real Hardy spaces h^{p}(∂Ω). For the case of the unit ball in C^{n}, Garnett and Latter [GL], using the properties of the Poisson-Szëgo kernel, showed that the holomorphic Hardy spaces are the Szëgo projection of atomic Hardy spaces. For p = 1, this question is related to issues of duality with BMO spaces, as discussed in [KL1]. The main theorem of this paper answers this question for the kinds of domains specified above.

Theorem 1.1 Let Ω be a strongly pseudoconvex domain in C^{n} or a pseudoconvex domain of finite type in C^{2}. If F belongs to H^{p}(Ω), 0 < p ≤ 1, then the boundary values f of F, considered as a distribution on ∂Ω, belongs to h^{p}(∂Ω), and ∥f∥_{h^{p}} ≤ C∥F ∥_{H^{p}}.

This theorem is stated more precisely in Section 4. The background and definitions required in the statement of the theorem and its proof can be found in Section 2. The proof itself, following in general outline the method used by Wilson ([W]) to prove atomic decomposition for H^{p}(R_{+}^{n+1}), is divided over several sections, starting with the definition of the atoms in Section 4, which is based on the reproducing formula shown in Section 3, and continuing with the proofs of the size and cancellation properties in Sections 5 and 6.

It will be seen that the proof of the theorem relies mostly on the reproducing formula, the homogeneous structure of the balls on ∂Ω, and the symmetry properties of a certain choice of coordinates (see Section 2.5). This might allow an extension to other domains in C^{n} where these can be achieved.

This paper is essentially the doctoral dissertation which I wrote at Princeton University under the direction of E. M. Stein, to whom I am very grateful for suggesting this problem and for many useful discussions concerning it.

2 Preliminary Definitions

Before we begin let us establish some notation. Coordinates in C^{n}, n ≥ 2, will be denoted by (z, w), with z = (z_{1}, …, z_{n−1}) and w = z_{n}. The real and imaginary parts of the z_{i} will be denoted by x_{i} and y_{i}, respectively, and of w by t and τ. We will write |z| for the quantity (|z_{1}|^{2} + … + |z_{n−1}|^{2})^{1/2}, and given a multiindex α = (α_{1}, …, α_{n−1}), z^{α} for z_{1}^{α_{1}} ⋯ z_{n−1}^{α_{n−1}}. Occasionally we will abuse the accepted notation by using dz for the volume form dx_{1} … dx_{n}dy_{1} … dy_{n} and dw for the area form dtdτ.

2.1 Nonisotropic Balls

Let Ω be a smoothly bounded pseudoconvex domain in C^{n}, which is either strongly pseudoconvex, or of finite type m when n = 2. Let U be an open set containing the boundary ∂Ω. Fix a C^{∞} function r : U → R (to be called the defining function for Ω) such that

U ∩ Ω = {Q ∈ U : r(Q) < 0}

and ∇r ≠ 0 on ∂Ω.

For the nonisotropic metric on Ω, we choose one of the several equivalent metrics defined in [NSW1] and [NSW2], which is the metric ρ used in [NRSW] for n = 2. Working in a sufficiently small open set on ∂Ω so that we can take a basis L_{1}, …, L_{n−1} for the complex tangential holomorphic vector fields, we write L^{j} = \frac{1}{2}(X_{1}^{j} − iX_{2}^{j}), where the X_{i}^{j} are real vector fields. Then for P_{0} and P_{1} on ∂Ω we say there is a curve of “length” δ between P_{0} and P_{1} if there exists a continuous piecewise smooth map ϕ : [0, 1] → ∂Ω with ϕ(0) = P_{0}, ϕ(1) = P_{1}, and almost everywhere

ϕ′(t) = ∑_{j=1}^{n−1} a_{1}^{j}(t)X_{1}^{j} + a_{2}^{j}(t)X_{2}^{j}

with |a_{1}^{j}(t)| < δ and |a_{2}^{j}(t)| < δ for all j. Define ρ(P_{0}, P_{1}) to be the infimum of the “lengths” of all such curves. The nonisotropic ball of radius R > 0 is defined by

B(P_{0}, R) = {P_{1} ∈ ∂Ω : ρ(P_{0}, P_{1}) < R}.

Let us define what we mean by the tent B̂(P, R) over the ball B(P, R). Let π : U → ∂Ω be the normal projection, i.e. π(Q) − Q is normal to the boundary at π(Q). This is a well-defined, smooth mapping when U is sufficiently small. Let us set

B̂(P, R) = \{Q ∈ \overline{Ω} ∩ U : π(Q) ∈ B(P, R) and |r(Q)| ≤ C_{T} Λ(P, R)\},

where the “tent constant” C_{T} will be taken sufficiently large later on. Here Λ(P, δ) is as defined in [NRSW] (p. 116) for the finite type case in C^{2}, while for the case of strongly pseudoconvex domains in C^{n}, we will use Λ(P, δ) to denote simply δ^{2} for all P. Fixing P, we see that Λ(P, δ) is strictly increasing in δ, so it is invertible as a function of δ. We denote the inverse function by μ(P, δ). Again, in the strongly pseudoconvex case, μ(P, δ) = \sqrt{δ} for all P.

We shall also need to use the concept of a normalized bump function on ∂Ω. As in [NRSW] (p. 122, (1.3)), this is a smooth function φ supported on a ball B(P_{0}, R) and satisfying

sup |X_{i_{1}}^{j} … X_{i_{k}}^{j}φ(P)| ≤ C_{k}R^{−k},

where the X_{i}^{j} are the vector fields X_{1}^{j} and X_{2}^{j} defined above.

2.2 Admissible Approach Regions, Maximal Function, and H^{p}(Ω)

In Section 2.4 we will introduce, for P ∈ ∂Ω, special coordinates based at P, denoted (z_{P} , w_{P}) or (z_{P} , t_{P} + iτ_{P}). In these coordinates r defines implicitly a smooth function h^{P} : C × R → R which satisfies the conditions of [NRSW], pp. 118-119, (1.11)-(1.13). That is, in these coordinates ∂Ω is the graph (over its tangent space at P) of h^{P}:

Ω ∩ V = {Q ∈ V : τ_{P} (Q) > h^{P} (z_{P} (Q), t_{P} (Q))}

for V a sufficiently small neighborhood of P, h^{P} has no pure z terms (up to order m), and the mixed z terms are dominated by Λ(P, |z|). Because of the uniform esimates on h^{P}, we can find a constant C_{Ω} such that

h^{P} (z, t) < C_{Ω}Λ(P, |z|) + C_{Ω}|t|

for all P ∈ ∂Ω and all (z, t) in some sufficiently small neighborhood of the origin. This means that if we make τ_{P} (Q) > 2C_{Ω}Λ(P, |z_{P} (Q)|), and τ_{P} (Q) > 2C_{Ω}|t_{P} (Q)|, then Q ∈ Ω. Set α_{Ω} = 1/(2C_{Ω}).

Definition 2.1 For P ∈ ∂Ω and 0 < α < α_{Ω}, the (truncated) admissible approach region of aperture α at P is defined by

A_{α}(P) = {Q : Λ(P, |z_{P} (Q)|) < ατ_{P} (Q), |t_{P} (Q)| < ατ_{P} (Q), 0 < τ_{P} (Q) < τ_{0}}

for some fixed small τ_{0} > 0. For 0 < ϵ < τ_{0} we also define the region

A^{ϵ}_{α}(P) = {Q : Λ(P, |z_{P} (Q)|) < ατ_{P} (Q), |t_{P} (Q)| < ατ_{P} (Q), ϵ < τ_{P} (Q) < τ_{0}}

which is truncated below.

Note that the choice of α_{Ω} above guarantees that A_{α}(P) lies inside Ω for all P. We can now define the admissible maximal function.

Definition 2.2 For F defined in Ω and 0 < α ≤ 1, the admissible maximal function of aperture α is defined by

F_{α}^{∗}(P) = sup_{Q∈A_{α}(P)} |F (Q)|.

Recall the definition of the holomorphic Hardy space H^{p}(Ω), where Ω ⊂ C^{n} is any bounded domain with smooth boundary. For small δ > 0, let

S_{δ} = {Q ∈ Ω : r(Q) = −δ}.

Definition 2.3 For 0 < p < ∞, H^{p}(Ω) is the class of all functions F holomorphic in Ω and satisfying

sup_{δ>0} ∫_{S_{δ}} |F |^{p}dσ_{δ} < ∞,

where dσ_{δ} is the induced measure on S_{δ}.

As shown in [S], this definition is independent on the choice of the defining function r. Thus we can use the notation

∥F ∥_{H^{p}} = (sup_{δ>0} ∫_{S_{δ}} |F |^{p}dσ_{δ})^{1/p} .

Having defined the notion of an admissible maximal function, we can state the following useful characterization of H^{p}.

Theorem 2.4 If F ∈ H^{p}(Ω), 0 < p < ∞, and F_{α}^{∗} is defined as above for 0 < α ≤ 1, then

∫_{∂Ω} |F_{α}^{∗}|^{p}dσ ≤ C_{p,α}∥F ∥_{H^{p}}^{p},

where we use σ to denote the induced volume measure on ∂Ω.

This theorem is essentially proved in [S], pp. 32-40. However, the balls and admissible approach regions used there are the ones appropriate for the strongly pseudoconvex case. Nonetheless, the proof carries through for finite type in C^{2} with the nonisotropic balls defined above. This follows from the “engulfing” and “doubling” properties of the balls ([NRSW], (1.2) and (1.3)), which allow us to show the usual mapping properties of the Hardy-Littlewood maximal function associated to these balls.

2.3 Atoms and h^{p}(∂Ω)

We define a notion of atoms on ∂Ω.

Definition 2.5 A nonisotropic p-atom, p ≤ 1, is a function a on ∂Ω which is supported in a nonisotropic ball B(P_{0}, R) and satisfies a size condition

∥a∥_{L^{2}(∂Ω)} ≤ σ(B(P_{0}, R))^{\frac{1}{2} − \frac{1}{p}}

and a cancellation condition

|∫_{∂Ω} a(P)g_{P_{0}}(P)dσ(P)| ≤ CR^{η},

for some fixed positive η (depending only on p, m, and n), all functions g_{P_{0}} which are polynomials in the coordinates based at P_{0}, i.e.

g_{P_{0}}(P) = z^{α}t^{β}

for P = h^{P_{0}}(z, t), and C independent of a and g_{P_{0}}.

Note that we are taking the monomial with respect to the coordinates based at the center of the ball in which the atom is supported. This is more convenient and shows immediately the existence of atoms. However, we could have taken the monomial with respect to any other point on the boundary, and reduced it to the above case by taking its Taylor expansion in the coordinates based at P_{0}, modulo an error which gives a positive power of R. In the following the functions g_{P_{0}}(P) above will be called monomials with respect to P_{0}.

Using these atoms, we define local Hardy spaces on ∂Ω.

Definition 2.6 For 0 < p ≤ 1, h^{p}(∂Ω) is the class of all distributions on ∂Ω which admit a decomposition

f = ∑_{j} λ_{j}a_{j},

where the the λ_{j} are real numbers satisfying

∑ |λ_{j}|^{p} < ∞,

and the a_{j} are nonisotropic p-atoms.

We write

∥f∥_{h^{p}} = (inf ∑ |λ_{j}|^{p})^{1/p} ,

where the infimum is taken over all such decompositions.

The notion of local Hardy spaces was first introduced by Goldberg for R^{n}, and atomic decomposition was proved with atoms satisfying exact vanishing moment conditions when supported in small balls ([Gol]). For the case of manifolds like ∂Ω, exact vanishing moment conditions are not convenient because they are not invariant under diffeomorphisms, and so one requires the approximate cancellation conditions above. One can show (see [D], Appendix B) that the two notions of atoms are equivalent.

2.4 Coordinates

We now introduce our choice of holomorphic coordinates based at each point P ∈ U. In the strongly pseudoconvex case, these are the normal coordinates descibed in [FS], pp. 500-507. In the finite type case these are the coordinates defined by Catlin (see [Cat], pp. 432-433) and McNeal (see [Mc], pp. 500-501). For P ∈ ∂Ω they also agree with the biholomorphisms H_{P} of [NRSW] (p. 118) and [BDN] (p. 215). Let us describe the construction of these coordinates in C^{2}.

The construction for n > 2 can be done analogously, with z = (z^{1}, …, z^{n−1}) and w = z^{n}.

Let the ambient coordinates be denoted by (z_{0}, w_{0}). We begin by making a complex-linear change of variables

z_{1} = 2i\frac{∂r}{∂\overline{w_{0}}} (P)(z_{0} − z_{0}(P)) − 2i\frac{∂r}{∂\overline{z_{0}}} (P)(w_{0} − w_{0}(P))  (1)
w_{1} = −2i\frac{∂r}{∂z_{0}} (P)(z_{0} − z_{0}(P)) − 2i\frac{∂r}{∂w_{0}} (P)(w_{0} − w_{0}(P))

which places the origin at P and points Im (w_{1}) in the direction of the inward normal at P.

We want to eliminate all pure z terms up to order m in the Taylor expansion of r around P. We define recursively

z_{j+1} = z_{j}
w_{j+1} = w_{j} − 2i\frac{1}{(j + 1)!} \frac{∂^{j+1}r}{∂z_{j}^{j+1}} (P)z_{j}^{j+1}

until j + 1 = m. Let z = z_{m}, w = w_{m}. Then in the coordinates (z, w) the expansion of r is

r(z, w) = r(P) − Im (w) + γ_{P} (z, \overline{z}) + O(|w||z| + |w|^{2} + |z|^{m+1}),  (2)

where γ_{P} (z, \overline{z}) is a real-valued polynomial of degree m with no pure terms in z or \overline{z}.

Note that all these changes of coordinates depend smoothly on P and that the Jacobian determinant is |∇r(P)|^{2}, which is nonvanishing for P ∈ U and can be taken to be = 1 for all P ∈ ∂Ω by an appropriate choice of the defining function r.

Definition 2.7 The coordinates (z, w) defined above will be called the coordinates based at P and shall be denoted (z_{P} , w_{P}) = (z_{P} , t_{P} + iτ_{P}). The inverse map, taking a neighborhood of the origin to a neighborhood of P, shall be denoted by H^{P}.

Before we proceed, let us explain some notation for denoting the order of a function with respect to the base point. We will use the notation O^{1}, O^{2}, etc., to denote the usual notion of order, i.e. a function f is said to be O^{1} if

f(Q) = O(|z_{P} (Q)| + |w_{P} (Q)|) as Q → P,

and f = O^{k} if f = O(O^{1} · O^{k−1}). This means that the function vanishes to the same order in all directions as Q approaches P. However, for a certain class of errors we shall want to take into account the different homogeneities inherent in the construction of the coordinates based at P. Thus if

f(Q) = O(|w_{P} ||z_{P} | + |w_{P} |^{2} + |z_{P} |^{m+1}) as Q → P,

then f will be said to be an admissible error with respect to P, denoted by O_{P}^{A}.

2.5 Symmetry

Let us see how the coordinates vary as we vary the base point. The following lemma shows that the errors we get by doing this can be controlled. This will be important since the atoms will be defined relative to each point on the boundary, while the cancellation conditions for the atoms involve integrating along the boundary. The errors are thus what accounts for the approximate cancellation conditions.

The following lemma is the analogue of Theorem 18.6 in [FS], which applies only to the strongly pseudoconvex case.

Lemma 2.8 If P and Q are two points in U, and the coordinates with respect to each are denoted (z_{P} , w_{P}) and (z_{Q}, w_{Q}) respectively, then

z_{P} (Q) = −z_{Q}(P) + O^{2}
Re w_{P} (Q) = −Re w_{Q}(P) + O_{P}^{A}(Q) + O_{Q}^{A}(P),

and

Im w_{P} (Q) = −Im w_{Q}(P) + 2γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P))
+ O_{P}^{A}(Q) + O_{Q}^{A}(P).

Here O^{2} could be taken with respect to either set of coordinates and γ_{Q} is a real-valued polynomial of degree m with no pure terms.

Proof: We can assume that Q is the origin, and the coordinates with respect to it are (z_{0}, w_{0}) as above. For ease of notation we shall also resume writing (z, w) instead of (z_{P} , w_{P}).

The first equation in the lemma follows immediately upon expanding the coefficients on the right-hand side of ( 1) around the origin.

To prove the second and third equations we need to look at the Taylor expansion of r. Let r_{m}^{h}(P)(ζ, ω) denote the holomorphic expansion of r around P up to order m in the coordinates (ζ, ω), that is

r_{m}^{h}(P)(ζ, ω) = ∑_{0<k+l≤m} \frac{1}{k!l!} \frac{∂^{k+l}r}{∂ζ^{k}ω^{l}} (P)(ζ − ζ(P))^{k}(ω − ω(P))^{l}

(note that we are not including constant terms). Then in the coordinates based at P, we have

r_{m}^{h}(P)(z, w) = \frac{i}{2}w + O(|z||w| + |w|^{2}),

or

w = −2ir_{m}^{h}(P)(z, w) + O(|z||w| + |w|^{2}).

Our aim is to express r_{m}^{h}(P)(z, w) in terms of the coordinates (z_{0}, w_{0}) and thus find the relation between w and w_{0}.

Let us write the Taylor expansion of r around P in the coordinates (z, w) as

r = r(P) + r_{m}^{h}(P)(z, w) + \overline{r_{m}^{h}(P)(z, w)}
+ mixed terms up to order m in z, \overline{z}, w, \overline{w} + O^{m+1}(z, w).

Recalling the coordinates (z_{1}, w_{1}) defined in the process of constructing the coordinates based at P (see equations 1), we also have the expansion:

r = r(P) + r_{m}^{h}(P)(z_{1}, w_{1}) + \overline{r_{m}^{h}(P)(z_{1}, w_{1})}
+ mixed terms up to order m in z_{1}, \overline{z_{1}}, w_{1}, \overline{w_{1}} + O^{m+1}(z_{1}, w_{1}).

Since the coordinate changes from (z_{1}, w_{1}) to (z_{m}, w_{m}) = (z, w) were holomorphic, we must have

r_{m}^{h}(P)(z, w) = r_{m}^{h}(P)(z_{1}, w_{1}) + O^{m+1}(z, w) + O^{m+1}(z_{1}, w_{1}).

Moreover, since the change from (z_{0}, w_{0}) to (z_{1}, w_{1}) was linear,

r_{m}^{h}(P)(z_{1}, w_{1}) = r_{m}^{h}(P)(z_{0}, w_{0}).

Thus

w = −2ir_{m}^{h}(P)(z_{0}, w_{0}) + O(|z||w| + |w|^{2} + |z|^{m+1}),

and at 0

w(0) = −2ir_{m}^{h}(P)(0, 0) + O(|z(0)||w(0)| + |w(0)|^{2} + |z(0)|^{m+1}).  (3)

All that remains is to express r_{m}^{h}(P)(0, 0) in terms of (z_{0}(P), w_{0}(P)).

Claim 2.9

r_{m}^{h}(P)(0, 0) = \frac{−i}{2}w_{0}(P) − γ_{0}(z_{0}(P), \overline{z_{0}}(P))
+ O(|z_{0}(P)||w_{0}(P)| + |w_{0}(P)|^{2} + |z_{0}(P)|^{m+1}).

To prove the claim we begin by writing the Taylor expansion of r around the origin in the coordinates (z_{0}, w_{0}) in the form

r = r(0) + ∑_{0<α+β+γ+δ≤m} C_{αβγδ}z_{0}^{α}\overline{z_{0}}^{β}w_{0}^{γ} \overline{w_{0}}^{δ} + O^{m+1}.

Then the expansion around P can be written as

r_{m}^{h}(P)(z_{0}, w_{0}) = ∑_{0<α+β+γ+δ≤m} C_{αβγδ}z_{0}^{α}\overline{z_{0}}(P)^{β}w_{0}^{γ} \overline{w_{0}}(P)^{δ}

− ∑_{0<α+β+γ+δ≤m} C_{αβγδ}z_{0}(P)^{α}\overline{z_{0}}(P)^{β}w_{0}(P)^{γ} \overline{w_{0}}(P)^{δ}.

This can be easily verified by differenting each term and using the binomial theorem. Therefore

r_{m}^{h}(P)(0, 0) = ∑_{0<β+δ≤m} C_{0β0δ}\overline{z_{0}}(P)^{β}\overline{w_{0}}(P)^{δ}

− ∑_{0<α+β+γ+δ≤m} C_{αβγδ}z_{0}(P)^{α}\overline{z_{0}}(P)^{β}w_{0}(P)^{γ} \overline{w_{0}}(P)^{δ}.

However, since the coordinates (z_{0}, w_{0}) are the coordinates based at the origin, we actually have (see equation 2):

r(z_{0}, w_{0}) = r(0) − Im (w_{0}) + γ_{0}(z_{0}, \overline{z_{0}}) + O(|w_{0}||z_{0}| + |w_{0}|^{2} + |z_{0}|^{m+1}),

where γ_{0} has no pure terms, so the first sum in the equation above yields only \frac{−i}{2} \overline{w_{0}}(P) + O(|z_{0}(P)||w_{0}(P)|) (all the coefficients C_{0β00} are zero), while the second sum is just

−Im w_{0}(P) + γ_{0}(z_{0}(P), \overline{z_{0}}(P)) + O(|z_{0}(P)||w_{0}(P)| + |w_{0}(P)|^{2} + |z_{0}(P)|^{m+1}).

Thus

r_{m}^{h}(P)(0, 0) = \frac{−i}{2}\overline{w_{0}}(P) + Im w_{0}(P) − γ_{0}(z_{0}(P), \overline{z_{0}}(P))
+ O(|z_{0}(P)||w_{0}(P)| + |w_{0}(P)|^{2} + |z_{0}(P)|^{m+1})
= \frac{−i}{2}w_{0}(P) − γ_{0}(z_{0}(P), \overline{z_{0}}(P))
+ O(|z_{0}(P)||w_{0}(P)| + |w_{0}(P)|^{2} + |z_{0}(P)|^{m+1}),

which proves the claim.

Plugging this into equation 3, we get

w(0) + w_{0}(P) = 2iγ_{0}(z_{0}(P), \overline{z_{0}}(P))
+ O(|z_{0}(P)||w_{0}(P)| + |w_{0}(P)|^{2} + |z_{0}(P)|^{m+1})
+ O(|z(0)||w(0)| + |w(0)|^{2} + |z(0)|^{m+1}),

and the second and third equations of the lemma follow by taking real and imaginary parts.

Corollary 2.10 A function is O(z_{Q}(P), w_{Q}(P)) whenever it is O(z_{P} (Q), w_{P} (Q)). If Q is in A_{α}(P), and f is O(z_{P} (Q), w_{P} (Q)), then f is O(|r(Q)|^{1/m}), and if f is O_{P}^{A}(Q) or O_{Q}^{A}(P), then f is O(|r(Q)|^{1+1/m}).

Proof: The first statement follows immediately from Lemma 2.8. Now assume Q ∈ A_{α}(P). Recall that t_{P} = Re w_{P} and τ_{P} = Im w_{P}. By the definition of A_{α}(P),

C|z_{P} (Q)|^{m} ≤ Λ(P, |z_{P} (Q)|) ≤ ατ_{P} (Q)

and

|t_{P} (Q)| ≤ ατ_{P} (Q),

so an O(z_{P} (Q), w_{P} (Q)) function is O(τ_{P} (Q)). Moreover, an O_{P}^{A} function is

O(|w_{P} (Q)||z_{P} (Q)| + |w_{P} (Q)|^{2} + |z_{P} (Q)|^{m+1}) = O(τ_{P} (Q)^{1+1/m}).

But also

τ_{P} (Q) = −r(Q) + γ_{P} (z_{P} (Q), \overline{z_{P} (Q)}) + O_{P}^{A}(Q),

and

|γ_{P} (z_{P} (Q), \overline{z_{P} (Q)})| ≤ Λ(P, |z_{P} (Q)|) ≤ ατ_{P} (Q),

so we have

τ_{P} (Q) ≈ |r(Q)|

and thus an O(z_{P} (Q), w_{P} (Q)) function is O(|r(Q)|^{1/m}) and an O_{P}^{A}(Q) function is O(|r(Q)|^{1+1/m}).

For O_{Q}^{A}(P), we have to use the estimates from lemma 2.8:

|z_{Q}(P)| = |z_{P} (Q)| + O^{2}
= O(|r(Q)|^{1/m}),
|t_{Q}(P)| = |t_{P} (Q)| + O_{P}^{A}(Q) + O_{Q}^{A}(P)
= O(|r(Q)|) + O_{Q}^{A}(P),
and |τ_{Q}(P)| ≤ |τ_{P} (Q)| + 2|γ_{P} (Q)| + O_{P}^{A}(Q) + O_{Q}^{A}(P)
= O(|r(Q)|) + O_{Q}^{A}(P).

Thus

|w_{Q}(P)||z_{Q}(P)| + |w_{Q}(P)|^{2} + |z_{Q}(P)|^{m+1} =
O(|r(Q)|^{1+1/m}) + O(|r(Q)|^{1/m})O_{Q}^{A}(P) + O_{Q}^{A}(P)O_{Q}^{A}(P)

which gives that O_{Q}^{A}(P) is O(|r(Q)|^{1+1/m}).

3 Reproducing Formula

Let F ∈ H^{p}(Ω). In order to prove Theorem 1.1, we shall want to express the boundary values of F (in the sense of distributions) in terms of its values in the approach regions A_{α}(P).

We begin with a variant of the one-variable reproducing formula of Calderón ([Cal]) which was used by J. M. Wilson to prove atomic decomposition for H^{p}(R_{+}^{2}) ([W]; see also [GCRF], Section III.3.11). Let ψ ∈ C^{∞}(R) be real, even, supported in (−1, 1), having vanishing moments up to some order N_{p} ≥ 0, and satisfying

∫_{0}^{∞} e^{−2πξ}ψ̂(ξ)dξ = \frac{−1}{2π}.

Note that this can be written as

(K_{1} ∗ ψ)(0) = \frac{−1}{2π},

where

K_{y}(s) = \frac{−1}{2πi(s + iy)}

is the Cauchy kernel for the upper half-plane. Since ψ is even, this is also equivalent to P_{1} ∗ ψ(0) = −1/π, where P is the Poisson kernel for the upper half-plane. Let ψ_{τ} (x) = τ^{−1}ψ(x/τ). Then for u ∈ H^{2} ∩H^{p}(R_{+}^{2}) with boundary distribution f (i.e. u = P_{y} ∗ f), it can be shown (using Fourier transforms) that

f = ∫_{R_{+}^{2}} \frac{∂u}{∂τ}(t, τ)ψ_{τ} (x − t)dtdτ

in S′(R).

For Ω a strongly pseudoconvex domain in C^{n} or a weakly pseudoconvex domain of finite type m in C^{2}, let us assume that α_{Ω} > 1, so that the approach regions A(P) = A_{1}(P) lie in Ω for all P (otherwise the following definitions and calculations would have to be scaled by α for some α < α_{Ω}). For τ ≤ τ_{0}, let c_{n} denote the volume of the unit ball in R^{2n−2}, and set

V_{P} (τ) = c_{n}(μ(P, τ))^{2n−2}τ.

Note that V_{P} (τ) is the volume of the cross-section of A(P) at height τ. Now set, for Q ∈ A(P),

Ψ_{P} (Q) = \frac{1}{V_{P} (τ_{P} (Q))}ψ (\frac{t_{P} (Q)}{τ_{P} (Q)}) .

Theorem 3.1 Let F be holomorphic in Ω. For P ∈ ∂Ω and ϵ > 0, define F^{ϵ}(P) = F (H^{P} (0, iϵ)), and let ν_{P} be the inward unit normal at P. Then

1.

F^{ϵ}(P) = ∫_{A^{ϵ}(P)} \frac{∂F}{∂ν_{P}} (Q)Ψ_{P}^{α} (Q)dQ − G^{ϵ}(P) + G^{τ_{0}}(P),

where G^{ϵ}(P) and G^{τ_{0}}(P) are functions which depend on F and its derivatives at the points H^{P} (0, iϵ) and H^{P} (0, iτ_{0}), respectively;

2. If F ∈ H^{p}(Ω) then G^{ϵ} → 0 in the sense of distributions on ∂Ω and ∥G^{τ_{0}}∥_{∞} ≤ C∥F ∥_{H^{p}}.

Proof of Part 1: For ease of notation, let us use (z, t + iτ) instead of (z_{P} , t_{P} + iτ_{P}), and F instead of F ∘ H^{P}. Since |J_{H^{P}} | = 1, we can write the integral in these coordinates as:

∫_{ϵ}^{τ_{0}} ∫_{|t|≤τ} ∫_{Λ(P,|z|)≤τ} \frac{∂F}{∂τ} (z, t + iτ)\frac{1}{c_{n}(μ(P, τ))^{2n−2}τ} ψ (\frac{t}{τ} ) dzdtdτ.

Since \frac{∂F}{∂τ} is holomorphic, and hence harmonic, in z, we can average over \{Λ(P, |z|) ≤ τ} (this is a ball of volume c_{n}(μ(P, τ))^{2n−2}) to get

∫_{ϵ}^{τ_{0}} ∫_{|t|≤τ} \frac{∂F}{∂τ} (0, t + iτ)\frac{1}{τ} ψ (\frac{t}{τ} ) dtdτ.

Now \frac{∂F}{∂τ} is also holomorphic in w, and the disk \{(0, w) : |w − iτ| ≤ τ\} lies in Ω for every τ ≤ τ_{0} (since we assumed α_{Ω} > 1), so we can write the power series expansion:

\frac{∂F}{∂τ} (0, t + iτ) = ∑_{k=0}^{∞} \frac{1}{k!} (\frac{∂F}{∂τ} )_{w}^{(k)} (0, iτ)t^{k},

and make a change of variables s = t/τ so the integral becomes:

∫_{ϵ}^{τ_{0}} ∫_{|s|≤1} ∑_{k=0}^{∞} \frac{1}{k!} (\frac{∂F}{∂τ} )_{w}^{(k)} (0, iτ)τ^{k}s^{k}ψ(s)dsdτ.

But

(\frac{∂F}{∂τ} )_{w}^{(k)} = (−i)^{k} \frac{∂^{k+1}F}{∂τ^{k+1}} ,

so by interchanging the summation and the integration (which we can do since the derivatives of F are bounded inside Ω_{ϵ}), we get:

∑_{k=0}^{∞} \frac{(−i)^{k}}{k!} ∫_{ϵ}^{τ_{0}} \frac{∂^{k+1}F}{∂τ^{k+1}} (0, iτ)τ^{k}dτ ∫_{|s|≤1} ψ(s)s^{k}ds.

For each k we can integrate by parts in τ:

∫_{ϵ}^{τ_{0}} \frac{∂^{k+1}F}{∂τ^{k+1}} (0, iτ)τ^{k}dτ = (−1)^{k}k![−F^{ϵ}(0) − G^{ϵ}_{k}(0) + G^{τ_{0}}_{k}(0)],

where

G^{ϵ}_{k}(0) = ∑_{j=1}^{k} \frac{(−1)^{j}}{j!} \frac{∂^{j}F}{∂τ^{j}} (0, iϵ)ϵ^{j}

and

G^{τ_{0}}_{k}(0) = ∑_{j=0}^{k} \frac{(−1)^{j}}{j!} \frac{∂^{j}F}{∂τ^{j}} (0, iτ_{0})τ_{0}^{j}.

Inserting this into the sum, we have

∑_{k=0}^{∞} i^{k}[−F^{ϵ}(0) − G^{ϵ}_{k}(0) + G^{τ_{0}}_{k}(0)] ∫_{|s|≤1} ψ(s)s^{k}ds.

Again, since we are in Ω_{ϵ}, we can exchange the summation and the integration to get three terms, the first of which is

= −F^{ϵ}(0) ∫_{|s|≤1} ψ(s) [∑_{k=0}^{∞}(is)^{k}] ds

= −F^{ϵ}(0) ∫_{|s|≤1} \frac{ψ(s)}{1 − is}ds
= F^{ϵ}(0),

by the condition on ψ. The second and third terms are

G^{ϵ}(P) = − ∫_{|s|≤1} ψ(s) [∑_{k=0}^{∞}(is)^{k}G^{ϵ}_{k}(0)] ds

and

−G^{τ_{0}}(P) = ∫_{|s|≤1} ψ(s) [∑_{k=0}^{∞}(is)^{k}G^{τ_{0}}_{k}(0)] ds.

This concludes the proof of part 1.

Before we can continue with the proof of Part 2, we need the following

Lemma 3.2 Let F ∈ H^{p}(Ω). For P ∈ ∂Ω, let ν_{P} denote the inward unit normal to ∂Ω at P, and consider the function

\frac{∂^{j}F}{∂ν^{j}} (Q) = \frac{∂^{j}F}{∂ν_{π(Q)}^{j}} (Q)

on S_{δ} = {Q : r(Q) = −δ}, j > 0. Then

∥\frac{∂^{j}F}{∂ν^{j}} ∥_{∞} ≤ C∥F ∥_{H^{p}}j!δ^{−j−n/p},

and considered as a distribution on ∂Ω,

|< \frac{∂^{j}F}{∂ν^{j}} , φ >| ≤ C∥F ∥_{H^{p}}∥φ∥_{C^{j}}

for all φ ∈ C^{∞}(∂Ω).

The first estimate in the lemma follows by writing the derivative as an average over a polydisk sandwiched between shells S_{c_{1}δ} and S_{c_{2}δ}. The second can be proved by converting the normal derivatives to tangential derivatives and integrating by parts. The details are standard and will be omitted so we can continue with the proof of Part 2 of the theorem.

Proof of Part 2: Suppose F ∈ H^{p}, and let ϕ ∈ C^{∞}(∂Ω). Then the action of G^{ϵ} on ϕ is

< G^{ϵ}, ϕ >= − ∫_{|s|≤1} ψ(s) [∑_{k=0}^{∞}(is)^{k} < G^{ϵ}_{k}, ϕ >] ds

since everything is bounded. But

< G^{ϵ}_{k}, ϕ >= ∑_{j=1}^{k} \frac{(−1)^{j}}{j!} < (\frac{∂^{j}F}{∂τ^{j}} )^{ϵ} , ϕ > ϵ^{j}

and

|< (\frac{∂^{j}F}{∂τ^{j}} )^{ϵ} , ϕ >| ≤ C∥F ∥_{H^{p}}∥ϕ∥_{C^{j}}

(from Lemma 3.2 above), so

| < G^{ϵ}, ϕ > | ≤ C∥F ∥_{H^{p}}∥ϕ∥_{C^{∞}}[e^{ϵ} − 1] ∫_{|s|≤1} |ψ(s)| ∑_{k=0}^{∞} |s|^{k}ds
≤ C̃∥F ∥_{H^{p}}∥ϕ∥_{C^{∞}}[e^{ϵ} − 1]

since |s| < 1 in the support of ψ. Thus < G^{ϵ}, ϕ >→ 0 as ϵ → 0.

As for G^{τ_{0}}, we have

|G^{τ_{0}}(P)| ≤ ∫_{|s|≤1} |ψ(s)| [∑_{k=0}^{∞} |s|^{k}|G^{τ_{0}}_{k}(0)|] ds,

and

|G^{τ_{0}}_{k}(0)| ≤ ∑_{j=0}^{k} \frac{1}{j!}|\frac{∂^{j}F}{∂τ^{j}} (0, iτ_{0})|τ_{0}^{j}.

However

|\frac{∂^{j}F}{∂τ^{j}} (0, iτ_{0})| ≤ Cj!τ_{0}^{−j−n/p}∥F ∥_{H^{p}}

(from Lemma 3.2 above), so

|G^{τ_{0}}(P)| ≤ C∥F ∥_{H^{p}} ∫_{|s|≤1} |ψ(s)| [∑_{k=0}^{∞} k|s|^{k}] ds
≤ C∥F ∥_{H^{p}}.

This completes the proof of Theorem 3.1.

4 Atomic Decomposition

After the preliminaries and the reproducing formula, we are ready to state and prove the main theorem on atomic decomposition. Here we restate Theorem 1.1 in full detail.

Theorem 4.1 Let Ω be a strongly pseudoconvex domain in C^{n} or a pseudoconvex domain of finite type m in C^{2}. If F is in H^{p}(Ω), then then the boundary values f of F, considered in the sense of distributions on ∂Ω, admit an atomic decomposition

f = ∑ λ_{j}a_{j},

where the λ_{j} are real numbers satisfying

∑ |λ_{j}|^{p} ≤ C_{p,α}∥F ∥_{H^{p}}^{p},

and the a_{j} are nonisotropic p-atoms on ∂Ω.

Proof: The existence of the boundary distribution f can be shown in the general case of a harmonic function u satisfying

∫_{S_{δ}} |u|dσ_{δ} ≤ Cδ^{−α_{0}},

for some real number α_{0} (for technical details see [D], Appendix A). Here we will be concerned with the proof of the atomic decomposition. The general outline of this proof is the one used in [W] and [GCRF] to prove atomic decomposition in R^{n}. We first note that by taking a partition of unity it suffices to prove the decomposition in V ∩ ∂Ω, where V is some open set which will be chosen sufficiently small when the need arises.

Let us look at the admissible maximal function F_{α}^{∗}, with α to be determined later, and set

E^{k} = {P ∈ V ∩ ∂Ω : F_{α}^{∗}(P) > 2^{k}}.

We can decompose E^{k} into balls B_{j}^{k}(P_{j}^{k}, R_{j}^{k}) satisfying

1. ∪B_{j}^{k} = E^{k},

2. the balls B(P_{j}^{k}, K^{−1}R_{j}^{k}) are pairwise disjoint, for some constant K,

3. not more than M balls intersect at each point, and

4. the distance of B_{j}^{k}(P_{j}^{k}, R_{j}^{k}) from ∂Ω \ E^{k} is propotional to its radius, i.e. there exist constants C′, C′′ such that for every P ∈ B_{j}^{k} we have

C′R_{j}^{k} ≤ inf_{P̃∉E^{k}}ρ(P, P̃) ≤ C′′R_{j}^{k}

(cf. [CW]). Consider the tents B̂_{j}^{k}, and associate to them a rough partition of unity consisting of step functions {ϕ_{j}^{k}} defined by

ϕ_{j}^{k}(Q) = χ_{B_{j}^{k}}(Q)/ (∑_{l} χ_{B_{l}^{k}}(Q))

for Q ∈ Ω. Also set

Ê^{k} = ∪B̂_{j}^{k}

and

T_{j}^{k} = B̂_{j}^{k} \ Ê^{k+1}.

Motivated by the reproducing formula (Theorem 3.1), we now define the pre-atoms g_{j}^{k}^{(ϵ)} by

g_{j}^{k}^{(ϵ)}(P) = ∫_{T_{j}^{k}∩A^{ϵ}(P)} ϕ_{j}^{k}(Q)\frac{∂F}{∂ν_{P}} (Q)Ψ_{P} (Q)dQ.

Because the {ϕ_{j}^{k}} form a partition of unity and ∪E^{k} = V ∩ ∂Ω, we have

∑ g_{j}^{k}^{(ϵ)}(P) = ∑_{k} ∑_{j} ∫_{T_{j}^{k}∩A^{ϵ}(P)} ϕ_{j}^{k}(Q)\frac{∂F}{∂ν_{P}} (Q)Ψ_{P} (Q)dQ

= ∑_{k} ∫_{(_{E^{k}}\_{E^{k+1}})∩A^{ϵ}(P)} \frac{∂F}{∂ν_{P}} (Q)Ψ_{P} (Q)dQ

= ∫_{A^{ϵ}(P)} \frac{∂F}{∂ν_{P}} (Q)Ψ_{P} (Q)dQ

The reproducing formula (Theorem 3.1) shows that the right-hand side converges, as ϵ → 0, to the distribution f, modulo an error G^{τ_{0}} whose L^{∞} norm is bounded by ∥F ∥_{H^{p}} (so that it can be added to the atomic decomposition as an atom supported in a large ball). Thus it remains to show that the left-hand side converges, in the sense of distributions, to the desired atomic decomposition.

We will first show that g_{j}^{k}^{(ϵ)} are supported in balls B̃_{j}^{k}(P_{j}^{k}, R̃_{j}^{k}) where R̃_{j}^{k} =

CR_{j}^{k} for some C > 1. We want to show that if g_{j}^{k}^{(ϵ)}(P) ≠ 0, then P ∈

B(P_{j}^{k}, CR_{j}^{k}). The condition g_{j}^{k}^{(ϵ)} ≠ 0 means that the set T_{j}^{k} ∩ A^{ϵ}(P) is nonempty. Thus we can find Q ∈ Ω such that π(Q) ∈ B_{j}^{k}, |r(Q)| ≤ Λ(P_{j}^{k}, R_{j}^{k}), and Q ∈ A(P). Using the definition of the approach regions, we can show that this implies that π(Q) ∈ B(P, δ) with Λ(P, δ) ≤ C|r(Q)|, which by the properties of the balls implies that B(P, δ) ⊂ B(P_{j}^{k}, CR_{j}^{k}) for some C > 1 independent of

the balls, and in particular P ∈ B(P_{j}^{k}, CR_{j}^{k}) = B̃_{j}^{k}(P_{j}^{k}, R̃_{j}^{k}).

In Section 5 (see Lemma 5.1) we will show that g_{j}^{k}^{(ϵ)} is uniformly bounded

in the L^{2}(∂Ω)-norm by C_{0}2^{k}σ(B̃_{j}^{k})^{1/2} and, in fact (see Corollary 5.4),

∥g_{j}^{k}^{(ϵ_{1})} − g_{j}^{k}^{(ϵ_{2})}∥_{L^{2}} → 0

as ϵ_{1} and ϵ_{2} tend to 0, so that the sequence {g_{j}^{k}^{(ϵ)}} converges in L^{2}(∂Ω) to a

function g_{j}^{k} which is also supported (in the closure of) B̃_{j}^{k} and has L^{2} norm

bounded by 2^{k}σ(B̃_{j}^{k})^{1/2}. Furthermore, in Section 6 we will show that g_{j}^{k}^{(ϵ)} satisfy the cancellation condition

| ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)g_{P_{j}^{k}}(P)dσ(P)| ≤ C2^{k}σ(B̃_{j}^{k})^{1/p}(R̃_{j}^{k})^{η}

for some fixed positive η, and all monomials g_{P_{j}^{k}}(P) based at P_{j}^{k}. From the L^{2} convergence, this condition carries over to g_{j}^{k}.

Thus if we define

a_{j}^{k}(P) = C_{0}^{−1}2^{−k}σ(B̃_{j}^{k})^{−1/p}g_{j}^{k}(P),

where C_{0} is the constant that comes up in the L^{2} estimate, then the a_{j}^{k} are nonisotropic p-atoms. We can also set

λ_{j}^{k} = C_{0}2^{k}σ(B̃_{j}^{k})^{1/p},

and then

∑_{k,j} |λ_{j}^{k}|^{p} = ∑_{k,j} C_{0}^{p}2^{kp}σ(B̃_{j}^{k})
≤ C_{0}^{p} ∑_{k} 2^{kp} ∑_{j} Aσ(B(P_{j}^{k}, K^{−1}R_{j}^{k}))
≤ C_{p} ∑_{k} 2^{kp}σ(E^{k})
= C_{p}2 ∑_{k} 2^{k−1}2^{k(p−1)}σ({P ∈ V ∩ ∂Ω : F_{α}^{∗}(P) > 2^{k}})

≤ C_{p} ∫_{0}^{∞} λ^{p−1}σ({P ∈ V ∩ ∂Ω : F_{α}^{∗}(P) > λ})dλ
≤ C_{p}∥F_{α}^{∗}∥_{L^{p}(∂Ω)}^{p}
≤ C_{p,α}∥F ∥_{H^{p}}.

This will give us the desired atomic decomposition if we can show that

∑ g_{j}^{k}^{(ϵ)} → ∑ g_{j}^{k}

as ϵ → 0 in the sense of distributions on ∂Ω. In fact, we can show a stronger convergence, namely in h^{p}(∂Ω). It is clear from the decomposition above that ∑ g_{j}^{k} = ∑ λ_{j}^{k}a_{j}^{k} is in h^{p}(∂Ω) and

∥ ∑ g_{j}^{k}∥_{h^{p}}^{p} ≤ C ∑ ∥g_{j}^{k}∥_{h^{p}}^{p} ≤ C∥F ∥_{H^{p}}.

The same thing holds for ∑ g_{j}^{k}^{(ϵ)}, and so we can write

∥ ∑ g_{j}^{k} − ∑ g_{j}^{k}^{(ϵ)}∥_{h^{p}}^{p} ≤ C ∑ ∥g_{j}^{k} − g_{j}^{k}^{(ϵ)}∥_{h^{p}}^{p}.

However,

∥g_{j}^{k} − g_{j}^{k}^{(ϵ)}∥_{h^{p}} ≤ Cσ(B̃_{j}^{k})^{\frac{1}{p} − \frac{1}{2}} ∥g_{j}^{k} − g_{j}^{k}^{(ϵ)}∥_{L^{2}}

and the right-hand side is dominated by ∥F ∥_{H^{p}} while ∥g_{j}^{k} − g_{j}^{k}^{(ϵ)}∥_{L^{2}} → 0 as

ϵ → 0, so we have that ∑ g_{j}^{k}^{(ϵ)} converges to ∑ g_{j}^{k} in h^{p}(∂Ω), and thus

f = ∑ λ_{j}^{k}a_{j}^{k}

as distibutions on ∂Ω, and we have proved the theorem.

Before we go on to show the size and cancellation properties, let us prove two lemmas which will be used in both. The first of these is in effect the essence of the use of tents for atomic decomposition. It is in the proof of this lemma that we choose the “tent constant” C_{T} and the aperture α.

Lemma 4.2 In the notation used in the proof of the theorem above, let T_{j}^{k}^{(ϵ)} = T_{j}^{k} ∩ Ω_{ϵ}, where Ω_{ϵ} = {Q : r(Q) < −ϵ}. Then

∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)|^{2}|r(Q)|dQ ≤ C2^{2k}σ(B_{j}^{k})

with a constant C independent of ϵ.

Proof: Up to some bounded positive function, we can replace r by any other defining function in U, so we choose the Green’s function for Ω with a pole at some point in Ω \ U, thus making r harmonic in U. We then write |∇F |^{2} = \frac{1}{2}Δ|F |^{2}, since F is harmonic, and apply Green’s theorem to the left hand side:

∫_{T_{j}^{k}_{(ϵ)}} |∇F |^{2}|r(Q)|dQ = \frac{1}{2} ∫_{∂T_{j}^{k}^{(ϵ)}} (\frac{∂|F |^{2}}{∂ν} r − |F |^{2} \frac{∂r}{∂ν} ) dσ,

where here ν is the outward unit normal to ∂T_{j}^{k}^{(ϵ)} and dσ is the induced volume measure on that surface.

Now from the definition of T_{j}^{k} we have

σ(∂T_{j}^{k}^{(ϵ)}) ≤ σ(∂(B̂_{j}^{k} ∩ Ω_{ϵ})) + σ(∂(Ê^{k+1} ∩ B̂_{j}^{k} ∩ Ω_{ϵ}))
≤ σ(∂B̂_{j}^{k} ∩ Ω_{ϵ}) + ∑_{l} σ(∂B̂_{l}^{k+1} ∩ B̂_{j}^{k} ∩ Ω_{ϵ})
≤ Cσ(B_{j}^{k}) + ∑_{l} Cσ(B_{l}^{k+1} ∩ B_{j}^{k})
≤ (C + C_{M})σ(B_{j}^{k})

(since not more than M balls intersect at each point), so what we need to show is

|\frac{∂|F |^{2}}{∂ν} | |r| + |F |^{2} |\frac{∂r}{∂ν} | ≤ C2^{2k}

on ∂T_{j}^{k}^{(ϵ)}. Since |∇r| is bounded in U, and |\frac{∂|F |^{2}}{∂ν} | ≤ 2|F ||∇F |, it remains to show that |F | and |∇F ||r| are both bounded by C2^{k}.

Suppose Q ∈ ∂T_{j}^{k}^{(ϵ)}. Then either π(Q) ∈ B_{j}^{k} \ E^{k+1} or π(Q) ∈ E^{k+1}. If π(Q) is in B_{j}^{k} \ E^{k+1}, then F_{α}^{∗}(π(Q)) ≤ 2^{k+1}, so since Q ∈ A_{α}(π(Q)), we have |F (Q)| ≤ 2 · 2^{k}. Furthermore, we can insert into A_{α}(π(Q)) a polydisk D_{α|r(Q)|}(Q) centered at Q with radii α|r(Q)| in all directions. Since

|∇F |^{2} = 2 ∑ |\frac{∂F}{∂z_{j}} |^{2} ,

we can estimate each derivative as an average over the polydisk and get

|∇F (Q)||r(Q)| ≤ C sup_{Q̃∈D_{α|r(Q)|}(Q)} |F (Q̃)|
≤ C2^{k+1}.

Now if π(Q) is in E^{k+1}, then

π(Q) ∈ B(P, R) = B_{l}^{k+1}(P_{l}^{k+1}, R_{l}^{k+1})

for some ball B_{l}^{k+1}(P_{l}^{k+1}, R_{l}^{k+1}) in the decomposition of E^{k+1}, and also

|r(Q)| ≥ C_{T} Λ(P, R),

since Q lies on the boundary of the tent B̂ or on the boundary of some “larger” tent “overhanging” it (we can always take the “smallest” choice of B_{l}^{k+1} for B). Note that the case Q ∈ ∂Ω_{ϵ} can only occur when ϵ ≥ C_{T} Λ(P, R).

We want to show that we can choose C_{T} and α sufficiently large so that Q ∈ A_{α}(P̃) for some P̃ ∉ E^{k+1}, and so that a polydisk D_{c|r(Q)|}(Q) with radii c|r(Q)|, for some uniformly small c, can be inserted into A_{α}(P̃). Once we have done that, we can repeat the argument above to show that both |F (Q)| and |∇F (Q)||r(Q)| are bounded by C2^{k}.

So let us proceed to choose α. From property 4 of the balls in the decomposition, we have a constant C′′ such that for some P̃ ∉ E^{k+1},

ρ(P, P̃) ≤ C′′R.

According to [NRSW] p. 117, (1.4), this implies that

Λ(P̃, R) ≤ C_{2}Λ(P, R)

for some fixed positive constant C_{2}. Thus

Λ(P̃, R) ≤ \frac{C_{2}}{C_{T}} |r(Q)|.

Furthermore, since P ∈ B(P̃, C′′R) and π(Q) ∈ B(P, R), we have

π(Q) ∈ B(P̃, AR)

where again the constant A is fixed uniformly. This implies that |zP̃(Q)| < C̃R and |tP̃(Q)| < Λ(P̃, C̃R) for another uniform constant C̃. Since we also have that

τP̃(Q) ≈ |r(Q)| + Λ(P̃, |zP̃(Q)|) + OA ; P̃(Q),

with a uniform constant, we finally get

τP̃(Q) ≥ CC_{T} Λ(P̃, |zP̃(Q)|)

and

τP̃(Q) ≥ CC_{T} |tP̃(Q)|.

Thus we need to take C_{T} sufficiently large so that C · C_{T} > 4C_{Ω}, and set α = 2/(CC_{T}), so α < α_{Ω}. This gives us that Q ∈ A_{α/2}(P̃), which allows us to insert the polydisk D_{c|r(Q)|}(Q) into A_{α}(P̃) with c ≈ 1/2.

Corollary 4.3 Let N > mn(1/p−1) and set η = 2/m(N −mn(1/p−1)). Then

∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)||r(Q)|^{N/m}dQ ≤ C2^{k}σ(B_{j}^{k})^{1/p}(R_{j}^{k})^{η}.

Proof: Applying the Cauchy-Schwarz inequality and using Lemma 4.2, we get

∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)||r(Q)|^{N/m}dQ ≤ (∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)|^{2}|r(Q)|dQ)^{1/2}

× (∫_{T_{j}^{k}_{(ϵ)}} |r(Q)|^{\frac{2N}{m} −1}dQ)^{1/2}

≤ C2^{k}σ(B_{j}^{k})^{1/2} (∫_{B_{j}^{k}} |r(Q)|^{\frac{2N}{m} −1}dQ)^{1/2}

≤ C2^{k}σ(B_{j}^{k})^{1/2} (σ(B_{j}^{k}) ∫_{0}^{Λ(P_{j}^{k},R_{j}^{k})} s^{\frac{2N}{m} −1}ds)^{1/2}
≤ C2^{k}σ(B_{j}^{k})Λ(P_{j}^{k}, R_{j}^{k})^{N/m}
≤ C2^{k}σ(B_{j}^{k})^{1/p}σ(B_{j}^{k})^{1−1/p}Λ(P_{j}^{k}, R_{j}^{k})^{N/m}
≤ C2^{k}σ(B_{j}^{k})^{1/p}(R_{j}^{k})^{(2n−2)(1−1/p)}Λ(P_{j}^{k}, R_{j}^{k})^{(1−1/p)+N/m}.

Now since N > mn(1/p − 1), we have 1/p < 1 + N/m, so we can use the fact that Λ(P_{j}^{k}, R_{j}^{k}) ≤ C(R_{j}^{k})^{2} to get a power of R_{j}^{k} equal to

(2n − 2)(1 − 1/p) + 2(1 − 1/p) + 2N/m = 2n(1 − 1/p) + 2N/m = η.

The second lemma is a useful application of the compatibility in volumes of balls which are sufficiently close.

Lemma 4.4 Let B(Q) = {P ∈ ∂Ω : Q ∈ A(P)}. Then

∫_{B(Q)} |Ψ_{P} (Q)|dσ(P)dQ ≤ C.

Proof: Let P ∈ B(Q). Since Q ∈ A(P), we can take R_{P} = C|z_{P} (Q)| for some C > 0 such that π(Q) ∈ B(P, R_{P}), Λ(P, R_{P}) ≈ |r(Q)|, and V_{P} (τ_{P} (Q)) ≈ σ(B(P, R_{P})). Thus P ∈ B(π(Q), R_{P}), Λ(π(Q), R_{P}) ≈ |r(Q)| and V_{P} (τ_{P} (Q) ≈ σ(B(π(Q), R_{P})). Let

R^{∗} = sup_{P ∈B(Q)} R_{P}

and

R_{0} = inf_{P ∈B(Q)} R_{P} .

Since for all P, Λ(π(Q), R_{P}) ≈ |r(Q)| > 0, we have that R_{0} > 0 and furthermore

\frac{R^{∗}}{R_{0}} ≤ C.

Therefore

∫_{B(Q)} |Ψ_{P} (Q)|dσ(P)dQ ≤ C∥ψ∥_{∞} ∫_{B(Q)} \frac{1}{|V_{P} (τ_{P} (Q))|}dσ(P)
≤ C ∫_{B(π(Q),R^{∗})} \frac{1}{σ(B(π(Q), R_{P}))}dσ(P)
≤ C \frac{1}{σ(B(π(Q), R_{0}))}σ(B(π(Q), R^{∗}))
≤ C.

5 Size Estimate

In this section we will show the size estimate for the pre-atoms g_{j}^{k}^{(ϵ)}:

Lemma 5.1 The functions g_{j}^{k}^{(ϵ)} defined in Section 4 are in L^{2}(∂Ω) and satisfy

∥g_{j}^{k}^{(ϵ)}∥_{L^{2}(∂Ω)} ≤ C2^{k}σ(B̃_{j}^{k})^{1/2}.

Proof: Let h ∈ L^{2}(∂Ω). We want to show

< g_{j}^{k}^{(ϵ)}, h >= ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)h(P)dσ(P) ≤ C2^{k}σ(B̃_{j}^{k})^{1/2}∥h∥_{L^{2}}.

Using the definition of g_{j}^{k}^{(ϵ)}, we can write the left-hand side as

∫_{∂Ω} ∫_{T_{j}^{k}∩A^{ϵ}(P)} ϕ_{j}^{k}(Q)\frac{∂F}{∂ν_{P}} (Q)Ψ_{P} (Q)h(P)dQdσ(P).

We change the order of integration and get rid of the dependence on P in the derivative of F by means of an error E_{1}:

< g_{j}^{k}^{(ϵ)}, h > = ∫_{T_{j}^{k}_{(ϵ)}} ϕ_{j}^{k}(Q)(∇F · ∇(−r))(Q) ∫_{B^{ϵ}(Q)} Ψ_{P} (Q)h(P)dσ(P)dQ
+ E_{1},  (4)

where B^{ϵ}(Q) = {P ∈ ∂Ω : Q ∈ A^{ϵ}(P)}. Note that the ϵ in T_{j}^{k}^{(ϵ)} (see Lemma 4.2)

is actually not the ϵ in g_{j}^{k}^{(ϵ)} but a somewhat smaller ϵ′ taken to allow ∪_{P} A^{ϵ}(P) ⊂ Ω_{ϵ′}. For convenience we will use the same letter to denote both, since for the relevant estimates we only need to know 0 < ϵ′ ≤ ϵ.

Claim 5.2

|E_{1}| ≤ C2^{k}σ(B̃_{j}^{k})^{1/2}∥h∥_{L^{2}}.

This claim will be proved later on. For now let us denote the main term on the right-hand side of equation 4 above by I_{1}, and estimate it using the Cauchy-Schwarz inequality:

|I_{1}|^{2} ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)|^{2}|r(Q)|dQ · ∫_{Ω_{ϵ}} |∫_{B^{ϵ}(Q)} Ψ_{P} (Q)h(P)dσ(P)|^{2} \frac{dQ}{|r(Q)|}.

By Lemma 4.2 we have that the first factor is bounded by C2^{2k}σ(B_{j}^{k}), and since

σ(B_{j}^{k}) ≤ Cσ(B̃_{j}^{k}), it remains to show that the second factor is bounded by a constant times ∥h∥_{L^{2}}^{2}. Let us denote it by I_{2} and expand out the absolute values to get

I_{2} = ∫_{Ω_{ϵ}} \{∫_{B^{ϵ}(Q)} Ψ_{P_{1}}(Q)h(P_{1})dσ(P_{1})\} \{∫_{B^{ϵ}(Q)} \overline{Ψ_{P_{2}}(Q)}\overline{h(P_{2})}dσ(P_{2})\} \frac{dQ}{|r(Q)|}.

Changing the order of integration, we get

I_{2} = ∫_{∂Ω} ∫_{∂Ω} K^{ϵ}(P_{1}, P_{2})\overline{h(P_{2})}h(P_{1})dσ(P_{2})dσ(P_{1}),

where

K^{ϵ}(P_{1}, P_{2}) = ∫_{A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} Ψ_{P_{1}}(Q)\overline{Ψ_{P_{2}}(Q)}\frac{dQ}{|r(Q)|}.

Since |K^{ϵ}| < ∞ for ϵ > 0, we can define an operator T^{ϵ} on g ∈ L^{2}(∂Ω) by

T^{ϵ}(h)(P_{1}) = ∫_{∂Ω} K^{ϵ}(P_{1}, P_{2})g(P_{2})dσ(P_{2}),

P_{1} ∈ ∂Ω.

Claim 5.3 The operator T^{ϵ} defined above is bounded on L^{2}(∂Ω), with a bound independent of ϵ.

Again the proof of the claim will follow shortly. Assuming the claim, however, we have by Cauchy-Schwarz that

|I_{2}| ≤ (∫_{∂Ω} |T^{ϵ}(\overline{h})(P_{1})|^{2}dσ(P_{1}))^{1/2} (∫_{∂Ω} |h(P_{1})|^{2}dσ(P_{1}))^{1/2}
≤ C∥h∥_{L^{2}}∥h∥_{L^{2}}
= C∥h∥_{L^{2}}^{2},

and so the lemma is proved.

We now come to the proof of the claims.

Proof of Claim 5.2: We want to esimate the error caused by replacing \frac{∂F}{∂ν_{P}} (Q) = \frac{∂F}{∂τ_{P}} (Q) by (∇F ·∇(−r))(Q). From the construction of the coordinates based at P, we see that

τ_{P} (Q) + r(Q) = O^{2}(z_{P} (Q), w_{P} (Q)),

so

|∇τ_{P} (Q) − ∇(−r)(Q)| = O(z_{P} (Q), w_{P} (Q)).

Now if Q ∈ A(P) we have Λ(P, |z_{P} (Q)|) < τ_{P} (Q) and |t_{P} (Q)| < τ_{P} (Q), so

|∇τ_{P} (Q) − ∇(−r)(Q)| = O(|r(Q)|^{\frac{1}{m}})

and

|\frac{∂F}{∂ν_{P}} (Q) − (∇F · ∇(−r))(Q)| ≤ C|∇F ||r(Q)|^{\frac{1}{m}} .

Thus an estimate for the first error is

|E_{1}| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |ϕ_{j}^{k}(Q)||∇F ||r(Q)|^{\frac{1}{m}} ∫_{B^{ϵ}(Q)} |Ψ_{P} (Q)||h(P)|dσ(P)dQ.

Applying the Cauchy-Schwarz inequality, we get

|E_{1}|^{2} ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F |^{2}|r(Q)|dQ·∫_{Ω_{ϵ}} |∫_{B^{ϵ}(Q)} |Ψ_{P} (Q)||h(P)|dσ(P)|^{2} |r(Q)|^{\frac{2}{m} −1}dQ.

As before, from Lemma 4.2 we get that the first factor is bounded by C2^{2k}σ(B_{j}^{k}),

and since σ(B_{j}^{k}) ≤ Cσ(B̃_{j}^{k}), it remains to show that the second factor is bounded by ∥h∥_{L^{2}}^{2}. Let us denote it by I_{2}.

Now since

∫_{B^{ϵ}(Q)} |Ψ_{P} (Q)|dσ(P) ≤ C

(see Lemma 4.4), we have (by applying Cauchy-Schwarz to the integral with respect to dσ(P)):

I_{2} ≤ C ∫_{Ω_{ϵ}} ∫_{B^{ϵ}(Q)} |Ψ_{P} (Q)||h(P)|^{2}dσ(P)|r(Q)|^{\frac{2}{m} −1}dQ

≈ ∫_{∂Ω} \{∫_{A^{ϵ}(P)} |Ψ_{P} (Q)||r(Q)|^{\frac{2}{m} −1}dQ\} |h(P)|^{2}dσ(P)
≤ C∥h∥_{L^{2}}^{2},

where the last estimate follows by integrating in the coordinates based at P, to get that

∫_{A^{ϵ}(P)} |Ψ_{P} (Q)||r(Q)|^{\frac{2}{m} −1}dQ ≈ ∫_{ϵ}^{τ_{0}} τ_{P} (Q)^{\frac{2}{m} −1}dτ_{P} ≤ C_{τ_{0}}.

Thus Claim 5.2 is proved.

Proof of Claim 5.3: To show T^{ϵ} is bounded on L^{2}(∂Ω), we use the general form of the T (1) theorem in [DJS]. We first have to show that K^{ϵ} is a “standard kernel”, i.e.

|K^{ϵ}(P_{1}, P_{2})| ≤ Cδ(P_{1}, P_{2})^{−1}  (5)

for P_{1} ≠ P_{2}, where C > 0 is independent of ϵ and δ(P_{1}, P_{2}) = inf_{B} σ(B), the infimum taken over all balls B containing P_{1} and P_{2}, and

|K^{ϵ}(P, P_{2}) − K^{ϵ}(P_{1}, P_{2})| ≤ C \frac{δ(P, P_{1})^{η}}{(δ(P, P_{2}) + δ(P_{1}, P_{2}))^{1+η}}  (6)

for all P ∈ ∂Ω such that δ(P, P_{1}) ≤ δ(P_{1}, P_{2})/2, and some C > 0 and η > 0 independent of ϵ.

To show the estimate 5, we just take absolute values in the definition of K^{ϵ}:

|K^{ϵ}(P_{1}, P_{2})| ≤ ∫_{A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} |Ψ_{P_{1}}(Q)||Ψ_{P_{2}}(Q)|\frac{dQ}{|r(Q)|}.

Recall that

|Ψ_{P} (Q)| ≤ C \frac{1}{V_{P} (τ_{P} (Q))},

where V_{P} (τ) is the volume of the cross-section of A(P) at height τ. Now for Q ∈ A^{ϵ}(P_{1}) ∩ A^{ϵ}(P_{2}), we have that π(Q) ∈ B(P_{1}, R_{1}) ∩ B(P_{2}, R_{2}) with

Λ(P_{1}, R_{1}) ≈ |r(Q)| ≈ Λ(P_{2}, R_{2}),

and so

V_{P_{1}}(τ_{P_{1}}(Q)) ≈ σ(B(P_{1}, R_{1})) ≈ σ(B(P_{2}, R_{2})) ≈ V_{P_{2}}(τ_{P_{2}}(Q)).

Since τ_{P_{1}}(Q) ≈ |r(Q)| for Q ∈ A^{ϵ}(P_{1}), we can write

|K^{ϵ}(P_{1}, P_{2})| ≤ C ∫_{A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} \frac{1}{V_{P_{1}}(τ_{P_{1}}(Q))^{2}} \frac{dQ}{τ_{P_{1}}(Q)}.

Furthermore, we see that for all Q ∈ A^{ϵ}(P_{1}) ∩ A^{ϵ}(P_{2}), τ_{P_{1}}(Q) ≥ CΛ(P_{1}, R) for some R, and P_{2} ∈ B(P_{1}, AR) for some constant A. Let R_{0} be the infimum of these R, and note that R_{0} > 0 if P_{2} ≠ P_{1}. Then

δ(P_{1}, P_{2}) ≤ σ(B(P_{1}, AR_{0})) ≈ R_{0}^{2n−2}Λ(P_{1}, R_{0}).

Thus if we integrate in the coordinates based at P_{1}, doing the integration first in t_{P_{1}} and z_{P_{1}} to get a factor of V_{P_{1}}(τ_{P_{1}}(Q)), we have

|K^{ϵ}(P_{1}, P_{2})| ≤ ∫_{CΛ(P_{1},R_{0})}^{τ_{0}} \frac{dτ_{P_{1}}}{V_{P_{1}}(τ_{P_{1}})τ_{P_{1}}}
= ∫_{CΛ(P_{1},R_{0})}^{τ_{0}} \frac{dτ_{P_{1}}}{c_{n}μ(P_{1}, τ_{P_{1}})^{2n−2}τ_{P_{1}}^{2}}
≤ CR_{0}^{−(2n−2)} ∫_{CΛ(P_{1},R_{0})}^{τ_{0}} \frac{dτ_{P_{0}}}{τ_{P_{0}}^{2}}
≤ CR_{0}^{−(2n−2)}Λ(P_{1}, R_{0})^{−1}
≤ Cδ(P_{1}, P_{2})^{−1}.

Note that we did not use ϵ in any of the estimates.

As for estimate 6, let us write the difference |K^{ϵ}(P, P_{2}) − K^{ϵ}(P_{1}, P_{2})| as

|∫_{A^{ϵ}(P)∩A^{ϵ}(P_{2})} Ψ_{P} (Q)Ψ_{P_{2}}(Q)\frac{dQ}{|r(Q)|} − ∫_{A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} Ψ_{P_{1}}(Q)Ψ_{P_{2}}(Q)\frac{dQ}{|r(Q)|}| .

We have three sets to integrate over: A^{ϵ}(P)∩A^{ϵ}(P_{1})∩A^{ϵ}(P_{2}), (A^{ϵ}(P)∩A^{ϵ}(P_{2}))\ A^{ϵ}(P_{1}), and (A^{ϵ}(P_{1}) ∩ A^{ϵ}(P_{2})) \ A^{ϵ}(P). Let us start with the first:

I_{1} = ∫_{A^{ϵ}(P)∩A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})}[Ψ_{P} (Q) − Ψ_{P_{1}}(Q)]Ψ_{P_{2}}(Q)\frac{dQ}{|r(Q)|}.

Then

|I_{1}| ≤ ∫_{A^{ϵ}(P)∩A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} |Ψ_{P} (Q) − Ψ_{P_{1}}(Q)||Ψ_{P_{2}}(Q)|\frac{dQ}{|r(Q)|}.

From the smoothness of the coordinates with respect to the base point, we can show that

|Ψ_{P} (Q) − Ψ_{P_{1}}(Q)| ≈ |Ψ_{P_{1}}(Q)|O(|z_{P_{1}}(P)|)

(for details see the proof of Step II in Section 6). Thus

|I_{1}| ≤ C|z_{P_{1}}(P)| ∫_{A^{ϵ}(P)∩A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} |Ψ_{P_{1}}(Q)||Ψ_{P_{2}}(Q)|\frac{dQ}{|r(Q)|}
≤ Cδ(P, P_{1})^{η}(δ(P, P_{2}) + δ(P_{1}, P_{2}))^{−1},

where η = 1/(2n) for the strongly pseudoconvex case, and η = 1/(2 + m) for the finite type case in C^{2}. The second factor is obtained from the integral by the same argument used in the proof of 5 above.

Let us now do the integration over (A^{ϵ}(P_{1})∩A^{ϵ}(P_{2}))\A^{ϵ}(P). Here Ψ_{P} (Q) = 0, so we only have to estimate

I_{2} = ∫_{(A^{ϵ}(P_{1})∩A^{ϵ}(P_{2}))\A^{ϵ}(P)} Ψ_{P_{1}}(Q)Ψ_{P_{2}}(Q)\frac{dQ}{|r(Q)|}.

If we integrate in the coordinates based at P_{1}, as before, we need to estimate the volume of a cross section of (A^{ϵ}(P_{1}) ∩ A^{ϵ}(P_{2})) \ A^{ϵ}(P) at some height τ. The volume of this region can be bounded by σ(B(P_{1}, R) \ B(P, CR)) for R ≈ μ(P_{1}, τ) and some constant C. However,

B(P_{1}, R) \ B(P, CR) ⊂ B(P, R̃),

where R̃ = CR + C′ρ(P, P_{1}), so

σ(B(P_{1}, R) \ B(P, CR)) ≤ σ(B(P, R̃)) − σ(B(P, CR))
≤ C \frac{σ(B(P, R))}{R}ρ(P, P_{1})
≤ C \frac{V_{P_{1}}(τ)}{μ(P_{1}, τ)}δ(P, P_{1})^{η}

since V_{P_{1}}(τ) ≈ σ(B(P_{1}, R)) ≈ σ(B(P, R)) for ρ(P, P_{1}) ≤ R. Thus an integration in the coodinates based at P_{1}, like the one performed above to prove 5, gives

|I_{2}| ≤ ∫_{(A^{ϵ}(P_{1})∩A^{ϵ}(P_{2}))\A^{ϵ}(P)} |Ψ_{P_{1}}(Q)||Ψ_{P_{2}}(Q)|\frac{dQ}{|r(Q)|}
≤ C ∫_{CΛ(P_{1},R_{0})}^{τ_{0}} \frac{1}{V_{P_{1}}(τ)τ_{P_{0}}μ(P_{1}, τ)}δ(P, P_{1})^{η}dτ_{P_{1}}
≤ Cδ(P, P_{1})^{η} ∫_{CΛ(P_{1},R_{0})}^{τ_{0}} \frac{dτ_{P_{1}}}{[μ(P_{1}, τ_{P_{1}})]^{2n−1} τ_{P_{1}}^{2}}

≤ Cδ(P, P_{1})^{η}R_{0}^{−(2n−1)} ∫_{CΛ(P_{1},R_{0})}^{τ_{0}} \frac{dτ_{P_{0}}}{τ_{P_{0}}^{2}}
≤ Cδ(P, P_{1})^{η}R_{0}^{−(2n−1)}Λ(P_{1}, R_{0})^{−1}
≤ Cδ(P, P_{1})^{η}δ(P_{1}, P_{2})^{−(1+η)}
≤ Cδ(P, P_{1})^{η}(δ(P, P_{2}) + δ(P_{1}, P_{2}))^{−(1+η)},

since δ(P, P_{2}) ≈ δ(P_{1}, P_{2}) when δ(P, P_{1}) ≤ δ(P_{1}, P_{2})/2. The integration over (A^{ϵ}(P) ∩ A^{ϵ}(P_{2})) \ A^{ϵ}(P_{1}) is estimated in an analogous fashion.

Next we must show the “weak-boundedness” property for T^{ϵ}. Let φ be a bump function associated to a ball B(P_{0}, R). We want to show

∥T^{ϵ}(φ)∥_{L^{2}} ≤ C(σ(B(P_{0}, R))^{1/2},

with C independent of φ. Let us write

∥T^{ϵ}(φ)∥_{L^{2}}^{2} = ∫_{B(P_{0},2R)} |T^{ϵ}(φ)(P_{1})|^{2}dσ(P_{1}) + ∫_{P_{1}∉B(P_{0},2R)} |T^{ϵ}(φ)(P_{1})|^{2}dσ(P_{1}).

The second integral is easier so let us deal with it first. Suppose P_{1} ∉ B(P_{0}, 2R). Then

|T^{ϵ}(φ)(P_{1})| = |∫_{B(P_{0},R)} K^{ϵ}(P_{1}, P_{2})φ(P_{2})dσ(P_{2})|

≤ ∫_{B(P_{0},R)} |K^{ϵ}(P_{1}, P_{2})||φ(P_{2})|dσ(P_{2})
≤ C ∫_{B(P_{0},R)} δ(P_{1}, P_{2})^{−1}dσ(P_{2})
≤ Cδ(P_{1}, P_{0})^{−1}σ(B(P_{0}, R)).

Thus

∫_{P_{1}∉B(P_{0},2R)} |T^{ϵ}(φ)(P_{1})|^{2}dσ(P_{1}) ≤ C(σ(B(P_{0}, R)))^{2} ∫_{P_{1}∉B(P_{0},2R)} δ(P_{1}, P_{0})^{−2}dσ(P_{1})
≤ Cσ(B(P_{0}, R)).

Now assume P_{1} ∈ B(P_{0}, 2R). Here we must use some cancelation. Let us write

T^{ϵ}(φ)(P_{1}) = ∫ K^{ϵ}(P_{1}, P_{2})φ(P_{2})dσ(P_{2})
= ∫ K^{ϵ}(P_{1}, P_{2})[φ(P_{2}) − φ(P_{1})]dσ(P_{2}) + φ(P_{1}) ∫ K^{ϵ}(P_{1}, P_{2})dσ(P_{2}).

We will show that both terms are bounded by a constant. Let us call the first term I_{1} and take absolute values:

|I_{1}| ≤ ∫ |K^{ϵ}(P_{1}, P_{2})||φ(P_{2}) − φ(P_{1})|dσ(P_{2}).

Since both P_{1} and P_{2} are in B(P_{0}, 2R), we can connect them by piecewise smooth curves whose tangent vectors are linear combinations of the X_{i}^{j}, and the infimum of the “lengths” of these curves is ρ(P_{1}, P_{2}) (see Section 2.1). By applying the mean value theorem along these curves and using the properties of the normalized bump function φ, we get that

|φ(P_{2}) − φ(P_{1})| ≤ CR^{−1}ρ(P_{1}, P_{2}).

Now we know |K^{ϵ}(P_{1}, P_{2})| ≤ Cδ(P_{1}, P_{2})^{−1}, and since B(P_{0}, R) ⊂ B(P_{1}, AR) we get

|I_{1}| ≤ CR^{−1} ∫_{B(P_{1},AR)} \frac{ρ(P_{1}, P_{2})}{δ(P_{1}, P_{2})}dσ(P_{2}).

Using the coordinates based at P_{1}, we can write

|I_{1}| ≤ CR^{−1} ∫_{|t_{P_{1}} (P_{2})|≤Λ(P_{1},AR)}∫_{|z_{P_{1}} (P_{2})|≤AR} \frac{ρ(P_{1}, P_{2})}{δ(P_{1}, P_{2})}dz_{P_{1}}dt_{P_{1}}
≤ CR^{−1} ∫_{|t|≤Λ(P_{1},AR)}∫_{Λ(P_{1},|z|)≤|t|} \frac{|z|}{(μ(P_{1}, |t|)^{2n−2}|t|}dzdt

+ CR^{−1} ∫_{|z|≤AR}∫_{|t|≤Λ(P_{1},|z|)} \frac{|z|}{|z|^{2n−2}Λ(P_{1}, |z|)}dzdt

≤ CR^{−1} ∫_{|t|≤Λ(P_{1},AR)} \frac{μ(P_{1}, |t|)}{(μ(P_{1}, |t|)^{2n−2}|t|} × (μ(P_{1}, |t|)^{2n−2}dt

+ CR^{−1} ∫_{|z|≤AR} \frac{|z|}{|z|^{2n−2}Λ(P_{1}, |z|)} × Λ(P_{1}, |z|)dz

≤ CR^{−1} ∫_{|t|≤Λ(P_{1},AR)} \frac{μ(P_{1}, |t|)}{|t|}dt

+ CR^{−1} ∫_{|z|≤AR} \frac{dz}{|z|^{2n−1}}

≤ C.

Now we show that the second term, I_{2}, is bounded by a constant. Since

|I_{2}| = |φ(P_{1})| |∫ K^{ϵ}(P_{1}, P_{2})dσ(P_{2})|

and φ is bounded, we must show the integral of K^{ϵ} is bounded. Let us write

∫ K^{ϵ}(P_{1}, P_{2})dσ(P_{2}) = ∫_{∂Ω} \{∫_{A^{ϵ}(P_{1})∩A^{ϵ}(P_{2})} Ψ_{P_{1}}(Q)Ψ_{P_{2}}(Q)\frac{dQ}{|r(Q)|}\} dσ(P_{2})

= ∫_{A^{ϵ}(P_{1})} Ψ_{P_{1}}(Q) \{∫_{B^{ϵ}(Q)} Ψ_{P_{2}}(Q)dσ(P_{2})\} \frac{dQ}{|r(Q)|}.

The inner integral can be shown to equal zero except for an error of the order of |r(Q)|^{1/m} (see the proofs of Steps II and III in Section 6). Thus we have

|∫ K^{ϵ}(P_{1}, P_{2})dσ(P_{2})| ≤ C ∫_{A^{ϵ}(P_{1})} |Ψ_{P_{1}}(Q)||r(Q)|^{1/m} \frac{dQ}{|r(Q)|}
≈ C ∫_{ϵ}^{τ_{0}} \frac{dτ_{P_{1}}}{|τ_{P_{1}}(Q)|^{(1−1/m)}}
≤ C_{τ_{0}},

where the second estimate is made by integrating in the coordinates based at P_{1} in the z_{P_{1}} and t_{P_{1}} directions and using the fact that the size of |Ψ_{P_{1}}(Q)| is the reciprocal of the volume of the cross-section.

This concludes the estimates for the “weak-boundedness” condition for T^{ϵ}. Finally, in order to use the “T (1)” theorem, we must show T^{ϵ}(1) and (T^{ϵ})^{∗}(1) are in BMO(∂Ω) with a bound independent of ϵ. It suffices to show T^{ϵ}(1) ∈ L^{∞}(∂Ω). But this is what we have just shown, i.e.

|∫ K^{ϵ}(P_{1}, P_{2})dσ(P_{2})| ≤ C,

with C independent of ϵ. As for (T^{ϵ})^{∗}, we have \overline{K^{ϵ}(P_{2}, P_{1})} = K^{ϵ}(P_{1}, P_{2}) by the definition of K^{ϵ}, and so we are done.

Corollary 5.4 With g_{j}^{k}^{(ϵ)} as above, we have

∥g_{j}^{k}^{(ϵ_{1})} − g_{j}^{k}^{(ϵ_{2})}∥_{L^{2}} → 0

as ϵ_{1} and ϵ_{2} tend to 0.

Proof: This is basically a corollary of the proof of the lemma. That is, in computing the L^{2} norm of g_{j}^{k}^{(ϵ_{1})} − g_{j}^{k}^{(ϵ_{2})}, we can follow the proof above, except now the relevant intgeral (assuming ϵ_{1} < ϵ_{2}) will be

∫_{T_{j}^{k}^{(ϵ_{1})}\T_{j}^{k}^{(ϵ_{2})}} |∇F (Q)|^{2}|r(Q)|dQ.

In light of Lemma 4.2, this clearly tends to 0 as ϵ_{1} and ϵ_{2} tend to 0.

6 Cancellation Condition

In this section we want to show the cancellation condition for the pre-atoms g_{j}^{k}^{(ϵ)}.

Lemma 6.1 For the functions g_{j}^{k}^{(ϵ)} defined as in Section 4, we have

| ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)g_{P_{j}^{k}}(P)dσ(P)| ≤ C2^{k}σ(B̃_{j}^{k})^{1/p}(R̃_{j}^{k})^{η}

for some fixed positive η, and all function g_{P_{j}^{k}} which are monomials with respect to P_{j}^{k}, i.e.

g_{P_{j}^{k}}(P) = z_{P_{j}^{k}}(P)^{α}Re w_{P_{j}^{k}}(P)^{β}.

Proof: The cancellation follows from the vanishing moments of ψ in the reproducing formula (see Section 3). Let us now specify the order of vanishing required, N_{p}, to be [mn(\frac{1}{p} − 1)], where [·] denotes the greatest integer function. The proof will be divided into two cases: p near 1, and p small.

6.1 p near 1

We first deal with the case N_{p} = 0, i.e. 1 ≥ p > \frac{mn}{mn+1}. Since the higher moments can be bounded using the size estimate (with η = 1 − (2n − 2 + m)(\frac{1}{p} − 1) > 0), we need only show the zero-th moment condition:

| ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)dσ(P)| ≤ C2^{k}σ(B̃_{j}^{k})^{1/p}(R̃_{j}^{k})^{η}.  (7)

Using the definition of the pre-atoms g_{j}^{k}^{(ϵ)}, we have:

∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)dσ(P) = ∫_{∂Ω} ∫_{T_{j}^{k}∩A^{ϵ}(P)} ϕ_{j}^{k}(Q)\frac{∂F}{∂ν_{P}} (Q)Ψ_{P} (Q)dQdσ(P).

We will show estimate 7 in three steps.

STEP I: Let I denote the integral of g_{j}^{k}^{(ϵ)} above. We change the order of integration and get rid of the dependence on P in the derivative of F by means of an error E_{1}:

I = ∫_{T_{j}^{k}_{(ϵ)}} ϕ_{j}^{k}(Q)(∇F · ∇(−r))(Q) ∫_{{P ∈∂Ω:Q∈A^{ϵ}(P)}} Ψ_{P} (Q)dσ(P)dQ
+ E_{1},

where

|E_{1}| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)||r(Q)|^{1/m}.

STEP II: Let B^{ϵ}(Q) = {P ∈ ∂Ω : Q ∈ A^{ϵ}(P)}. We want to estimate

I_{Ψ} = ∫_{B^{ϵ}(Q)} Ψ_{P} (Q)dσ(P) = ∫_{B^{ϵ}(Q)} \frac{1}{V_{P} (τ_{P} (Q))}ψ (\frac{t_{P} (Q)}{τ_{P} (Q)}) dσ(P).

We rewrite this in terms of the coordinates at Q modulo a second error as:

I_{Ψ} = ∫_{B^{ϵ}(Q)} \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(P))}ψ (\frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)}) dσ(P)
+ E_{2},

where γ_{Q}(P) = γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P)) is the real valued polynomial of lemma 2.8, and

|E_{2}| ≤ C|r(Q)|^{1/m}.

STEP III: We change to integration in the coordinates based at Q and integrate, using the vanishing zero-th moment of ψ, to get:

I_{Ψ} = 0 + E_{2} + E_{3},

with

|E_{3}| ≤ C|r(Q)|^{1/m}.

From steps I-III, and using Corollary 4.3 with N = 1 > mn(1/p − 1) and η = 2n(1 − 1/p) + 2/m > 0, we see that

| ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)dσ(P)| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)||r(Q)|^{1/m}
≤ C2^{k}σ(B̃_{j}^{k})^{1/p}(R̃_{j}^{k})^{η}.

Proof of Step I: We want to esimate the error caused by replacing \frac{∂F}{∂ν_{P}} (Q) = \frac{∂F}{∂τ_{P}} (Q) by (∇F ·∇(−r))(Q). From the proof of Claim 5.2 in Section 5, we get that

|\frac{∂F}{∂ν_{P}} (Q) − (∇F · ∇(−r))(Q)| ≤ C|∇F ||r(Q)|^{\frac{1}{m}} .

Thus the estimate for the first error is

|E_{1}| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |ϕ_{j}^{k}(Q)||∇F ||r(Q)|^{\frac{1}{m}} ∫_{\{P ∈∂Ω:Q∈A(P)\}} |Ψ_{P} (Q)|dσ(P)dQ

≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F ||r(Q)|^{\frac{1}{m}}

by Lemma 4.4.

Proof of Step II: First we estimate the error caused by replacing t_{P} (Q) by t_{Q}(P) and τ_{P} (Q) by τ_{Q}(P) in ψ(t/τ). Recall that by lemma 2.8,

|t_{P} (Q) − t_{Q}(P)| = O_{P}^{A}(Q) + O_{Q}^{A}(P)

and

τ_{P} (Q) = −τ_{Q}(P) + 2γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P)) + O_{P}^{A}(Q) + O_{Q}^{A}(P),

which when combined with the expansion

0 = r(P) = r(Q) − τ_{Q}(P) + γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P)) + O_{Q}^{A}(P)

gives

|τ_{P} (Q) − (−r(Q) + γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P))| = O_{P}^{A}(Q) + O_{Q}^{A}(P).

For Q ∈ A(P), this implies (see Corollary 2.10) that

|t_{P} (Q) − t_{Q}(P)| = O(|r(Q)|^{1+1/m})

and

|τ_{P} (Q) − (−r(Q) + γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P))| = O(|r(Q)|^{1+1/m}).

Furthermore, since

|γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P))| ≤ CΛ(π(Q), |z_{Q}(P)|)

and

Λ(π(Q), |z_{Q}(P)|) ≈ Λ(P, |z_{P} (Q)|) ≤ τ_{P} (Q),

we have

τ_{P} (Q) ≈ |r(Q)|

and

| − r(Q) + γ_{Q}(P)| ≈ |r(Q)|.

These estimates combine to give

|\frac{t_{P} (Q)}{τ_{P} (Q)} − \frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)}| = O(|r(Q)|^{1/m}).

Since ψ is a smooth function,

|ψ (\frac{t_{P} (Q)}{τ_{P} (Q)}) − ψ (\frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)})| ≤ ∥ψ′∥_{∞} |\frac{t_{P} (Q)}{τ_{P} (Q)} − \frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)}|
≤ C∥ψ′∥_{∞}|r(Q)|^{1/m}.

As for the change from V_{P} (τ_{P} (Q)) to V_{π(Q)}(−r(Q) + γ_{Q}(P)), we have to estimate

|\frac{1}{V_{P} (τ_{P} (Q))} − \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(P))}| = |\frac{V_{π(Q)}(−r(Q) + γ_{Q}(P)) − V_{P} (τ_{P} (Q))}{V_{P} (τ_{P} (Q))V_{π(Q)}(−r(Q) + γ_{Q}(P))}| .

Now

|V_{π(Q)}(−r(Q) + γ_{Q}(P)) − V_{P} (τ_{P} (Q))|

is dominated by

|V_{π(Q)}(−r(Q) + γ_{Q}(P)) − V_{π(Q)}(τ_{P} (Q))| + |V_{π(Q)}(τ_{P} (Q)) − V_{P} (τ_{P} (Q))|.

Since V_{π(Q)}(δ) is smooth in δ,

|V_{π(Q)}(| − r(Q) + γ_{Q}(P)|) − V_{π(Q)}(τ_{P} (Q))| ≈ \frac{V_{π(Q)}(−r(Q) + γ_{Q}(P))}{|r(Q)|}
× |(−r(Q) + γ_{Q}(P)) − τ_{P} (Q)|
≈ V_{π(Q)}(−r(Q) + γ_{Q}(P))|r(Q)|^{1/m}.

On the other hand, the change in V_{P} as the base point P varies can be approximated by

|V_{π(Q)}(τ_{P} (Q)) − V_{P} (τ_{P} (Q))| ≤ CV_{π(Q)}(τ_{P} (Q))O^{1}
≈ V_{π(Q)}(−r(Q) + γ_{Q}(P))|r(Q)|^{1/m},

and so

|\frac{1}{V_{P} (τ_{P} (Q))} − \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(P))}| ≤ C \frac{|r(Q)|^{1/m}}{V_{P} (τ_{P} (Q))}.

Thus the final estimate for the second error is

|E_{2}| ≤ ∫_{B^{ϵ}(Q)} \{\frac{C∥ψ′∥_{∞}|r(Q)|^{1/m}}{|V_{P} (τ_{P} (Q))|} + \frac{C|r(Q)|^{1/m}}{|V_{P} (τ_{P} (Q))|}∥ψ∥_{∞}\} dσ(P)
≤ C|r(Q)|^{1/m}∥ψ∥_{C^{1}} ∫_{B^{ϵ}(Q)} \frac{1}{|V_{P} (τ_{P} (Q))|}dσ(P)
≤ C|r(Q)|^{1/m}

by Lemma 4.4.

Proof of Step III: We want to estimate the error caused by replacing integration over B^{ϵ}(Q) ⊂ ∂Ω with respect to the volume element dσ with integration over some set in R^{2n−1} with respect to the volume element dzdt (we use dz to denote the Euclidean volume elment on R^{2n−2}).

We assume that the coordinates based at Q are well-defined and invertible in a neighborhood V of Q which includes B^{ϵ}(Q). Thus the map Θ : V ∩ ∂Ω → R^{2n−1} defined by Θ(P) = (z_{Q}(P), t_{Q}(P)) is invertible, and we can perform a change of variables on

I_{3} = ∫_{B^{ϵ}(Q)} \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(P))}ψ (\frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)}) dσ(P)

to get

I_{3} = ∫_{Θ(B^{ϵ}(Q))} \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(z, \overline{z}))}ψ (\frac{t}{−r(Q) + γ_{Q}(z, \overline{z})}) J_{Θ}^{−1}dzdt.

We can approximate the Jacobian J_{Θ}^{−1}(z_{Q}(P), t_{Q}(P)) by the constant |∇r(Q)|^{2n−1} with an error which is O^{1}(z_{Q}(P), t_{Q}(P)) = O(|r(Q)|^{1/m}) when P ∈ B^{ϵ}(Q).

It remains to extend Θ(B^{ϵ}(Q)) to a set \overline{Θ(B^{ϵ}(Q))} so that whenever (z_{0}, t_{0}) ∈ Θ(B^{ϵ}(Q)) we have (z_{0}, t) ∈ \overline{Θ(B^{ϵ}(Q))} for every t satisfying |t| ≤ −r(Q) + γ_{Q}(z_{0}, \overline{z_{0}}). Now Θ(B^{ϵ}(Q)) contains all points (z_{Q}(P), t_{Q}(P)) corresponding to P such that

Λ(P, |z_{P} (Q)|) ≤ τ_{P} (Q)

and

|t_{P} (Q)| ≤ τ_{P} (Q).

Furthermore, for these P,

τ_{P} (Q) = −r(Q) + γ_{Q}(z_{Q}(P), \overline{z_{Q}(P)}) + O(|r(Q)|^{1+1/m}).

Also,

t_{P} (Q) = −t_{Q}(P) + O(|r(Q)|^{1+1/m}).

By continuity, we can say that if (z_{0}, t_{0}) ∈ Θ(B^{ϵ}(Q)) then Θ(B^{ϵ}(Q)) contains all points (z_{0}, t) with t ranging up to some value within O(|r(Q)|^{1+1/m}) of −r(Q) + γ_{Q}(z_{0}, \overline{z_{0}}). Thus the error can be estimated by

C ∫_{z∈Θ(B^{ϵ}(Q))} \frac{|r(Q)|^{1+1/m}}{V_{π(Q)}(−r(Q) + γ_{Q}(z, \overline{z}))} ≤ C|r(Q)|^{1/m},

as in the proof of Lemma 4.4.

So finally, modulo the errors, we can write

I_{3} = |r(Q)|^{2n−1} ∫\overline{Θ(B^{ϵ}(Q))} \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(z, \overline{z}))}ψ (\frac{t}{−r(Q) + γ_{Q}(z, \overline{z})}) dtdz
= |r(Q)|^{2n−1} ∫_{z∈Θ(B^{ϵ}(Q))} \frac{1}{V_{π(Q)}(−r(Q) + γ_{Q}(z, \overline{z}))}

× ∫_{|t|≤−r(Q)+γ_{Q}(z,\overline{z})} ψ (\frac{t}{−r(Q) + γ_{Q}(z, \overline{z})}) dtdz

= |r(Q)|^{2n−1} ∫_{z∈Θ(B^{ϵ}(Q))} \frac{−r(Q) + γ_{Q}(z, \overline{z})}{V_{π(Q)}(−r(Q) + γ_{Q}(z, \overline{z}))} ∫_{|s|≤1} ψ(s)dsdz
= 0,

since ψ has a vanishing zero-th moment.

6.2 p small

Now we want to allow for N_{p} > 0, i.e. we want to estimate the higher moments of g_{j}^{k}^{(ϵ)}. Let g_{P_{j}^{k}} be a monomial with respect to P_{j}^{k}. We can assume the degree of the monomial is bounded (for example, by [(2n − 2 + m)(\frac{1}{p} − 1)]), since for sufficiently high moments we can just use the size estimate. We want to show that

| ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)g_{P_{j}^{k}}(P)dσ(P)| ≤ C2^{k}σ(B̃_{j}^{k})^{1/p}(R̃_{j}^{k})^{η}

for some η > 0. Again, we will do this in three steps.

STEP I′: Let us call the integral on left-hand-side of the equation above I(g_{P_{j}^{k}}). Then by getting rid of the dependence on P in the derivative of F in

g_{j}^{k}^{(ϵ)}, and changing the order of integration, we can write:

I(g_{P_{j}^{k}}) = ∫_{T_{j}^{k}_{(ϵ)}} ϕ_{j}^{k}(Q)(∇F · ∇(−r))(Q) ∫_{B^{ϵ}(Q)} Ψ_{P} (Q)dσ(P)dQ  (8)

+∫_{T_{j}^{k}_{(ϵ)}} ϕ_{j}^{k}(Q) ∑_{1}^{2n−1} F_{u_{i}}(Q) ∫_{B(Q)} P_{i}(z_{P} (Q), w_{P} (Q))Ψ_{P} (Q)g_{P_{j}^{k}}(P)dσ(P)dQ+E_{1},

where the F_{u_{i}} denote the derivatives of F with respect to the variables x_{1}, …, x_{n−1}, y_{1}, …, y_{n−1}, and t, the P_{i} are polynomials of degrees ≤ N_{p}, and

|E_{1}| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)||r(Q)|^{(N_{p}+1)/m}.

STEP II′: Let P = P_{i} for some i. We want to estimate

I_{Ψ}(P, g_{P_{j}^{k}}) = ∫_{B^{ϵ}(Q)} P(z_{P} (Q), w_{P} (Q))Ψ_{P} (Q)g_{P_{j}^{k}}(P)dσ(P)
= ∫_{B^{ϵ}(Q)} P(z_{P} (Q), w_{P} (Q))\frac{1}{V_{P} (τ_{P} (Q))}ψ (\frac{t_{P} (Q)}{τ_{P} (Q)}) g_{P_{j}^{k}}(P)dσ(P).

We rewrite this in terms of the coordinates at Q modulo a second error as:

I_{Ψ}(P, g_{P_{j}^{k}}) = ∫_{B^{ϵ}(Q)} P(P, Q) ∑_{i=0}^{N_{p}} \frac{(d(P, Q))^{i}}{Ṽ_{i}(Q, P)} ∑_{l=0}^{N_{p}} ψ̃_{l}(Q, P)(e(P, Q))^{l}g_{P_{j}^{k}}(P)dσ(P)
+ E_{2},  (9)

where P(P, Q) is shorthand for P(z_{P} (Q), w_{P} (Q)), d(P, Q) and e(P, Q) are functions which are O(z_{Q}(P), w_{Q}(P)), Ṽ_{i}(Q, P) depend only on r(Q) and z_{Q}(P) and satisfy

Ṽ_{i}(Q, P) ≈ V_{π(Q)}(−r(Q) + γ_{Q}(P))

(γ_{Q}(P) = γ_{Q}(z_{Q}(P), \overline{z_{Q}}(P)) is the real valued polynomial of lemma 2.8),

ψ̃_{l}(Q, P) = ψ^{(l)} (\frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)}) ,

and

|E_{2}| ≤ C|r(Q)|^{(N_{p}+1)/m}.

STEP III′: We express all factors in terms of the coordinates based at Q and integrate, using the vanishing moments of ψ, to get:

I_{Ψ}(P, g_{P_{j}^{k}}) = 0 + E_{2} + E_{3},

with

|E_{3}| ≤ C|r(Q)|^{(N_{p}+1)/m}.

From steps I′-III′, and using Corollary 4.3 with N = N_{p} + 1 > mn(1/p − 1) and η = 2n(1 − 1/p) + 2/m(N_{p} + 1) > 0, we see that

| ∫_{∂Ω} g_{j}^{k}^{(ϵ)}(P)dσ(P)| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F (Q)||r(Q)|^{(N_{p}+1)/m}
≤ C2^{k}σ(B̃_{j}^{k})^{1/p}(R̃_{j}^{k})^{η}.

Proof of Step I′: We write the expansion of r around P in the coordinates z_{P} , w_{P} up to degree N_{p}:

r(Q) = −τ_{P} (Q) + P_{N_{p}+1}(z_{P} (Q), w_{P} (Q)) + R_{N_{p}+2}(z_{P} (Q), w_{P} (Q)),

where P_{N_{p}} is a polynomial of degree N_{p} + 1 and R_{N_{p}+2} is the remainder. Then when we take the gradients on both sides and apply them to ∇F, we get

(∇F ·∇r)(Q) = −\frac{∂F}{∂τ_{P}} (Q)+ ∑_{1}^{2n−1} \frac{∂F}{∂u_{i}} (Q)P_{i}(z_{P} (Q), w_{P} (Q))+O^{N_{p}+1}(z_{P} (Q), w_{P} (Q)).

This gives us the integrals in equation 8, and since for Q ∈ A(P)

O^{N_{p}+1}(z_{P} (Q), w_{P} (Q)) = O(|r(Q)|^{(N_{p}+1)/m}),

we can estimate the error by

|E_{1}| ≤ C ∫_{T_{j}^{k}_{(ϵ)}} |ϕ_{j}^{k}(Q)||∇F ||r(Q)|^{\frac{Np+1}{m}} ∫_{B(Q)} |Ψ_{P} (Q)||g_{P_{j}^{k}}(P)|dσ(P)dQ

≤ C ∫_{T_{j}^{k}_{(ϵ)}} |∇F ||r(Q)|^{\frac{Np+1}{m}}

(see Lemma 4.4).

Proof of Step II′: For τ > 0, we treat 1/(VP̃(τ)) as a smooth function in both the base point P̃ and the height τ, and so we expand it around the base point P̃ = π(Q) and the height τ = −r(Q) + γ_{Q}(P). Evaluating this expansion at base point P̃ = P and height τ = τ_{P} (Q), we can write

\frac{1}{V_{P} (τ_{P} (Q))} = ∑_{i=0}^{N_{p}} \frac{(d(P, Q))^{i}}{Ṽ_{i}(Q, P)} + R_{N_{p}+1}(Q, P).

As in the proof of Step II in the p near 1 case above, we can show that

d(P, Q) = O(|r(Q)|^{1/m}),
Ṽ_{i}(Q, P) ≈ V_{π(Q)}(−r(Q) + γ_{Q}(P)),

and

R_{N_{p}+1}(Q, P) ≤ C \frac{|r(Q)|^{(N_{p}+1)/m}}{V_{P} (τ_{P} (Q))} .

We also expand ψ(s) around the value s = t_{Q}(P)/(−r(Q) + γ_{Q}(P)) and evaluate at s = t_{P} (Q)/τ_{P} (Q):

ψ (\frac{t_{P} (Q)}{τ_{P} (Q)}) = ∑_{l=0}^{N_{p}} ψ̃_{l}(Q, P)(e(P, Q))^{l} + R′_{N_{p}+1}(Q, P).

Here

ψ̃_{l}(Q, P) = ψ^{(l)} (\frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)}) ,

e(P, Q) = \frac{t_{P} (Q)}{τ_{P} (Q)} − \frac{t_{Q}(P)}{−r(Q) + γ_{Q}(P)} = O(|r(Q)|^{1/m})

(see the proof of Step II in the p near 1 case above), and

R′_{N_{p}+1}(Q, P) = O(|r(Q)|^{(N_{p}+1)/m}).

Now when we replace both \frac{1}{V_{P} (τ_{P} (Q))} and ψ (\frac{t_{P} (Q)}{τ_{P} (Q)}) by their expansions, we get the integral in equation 9, while the remainders give the error

|E_{2}| ≤ C ∫_{B^{ϵ}(Q)} |P(P, Q)|\frac{|r(Q)|^{(N_{p}+1)/m}}{V_{P} (τ_{P} (Q))} |g_{P_{j}^{k}}(P)|dσ(P)
≤ C|r(Q)|^{(N_{p}+1)/m} ∫_{B^{ϵ}(Q)} \frac{1}{V_{P} (τ_{P} (Q))}dσ(P)
≤ C|r(Q)|^{(N_{p}+1)/m}

by Lemma 4.4. Note here that the bound on P(P, Q) depends only on the derivatives of the defining function r at P and the coordinates z_{P} (Q) and w_{P} (Q) which are bounded for P ∈ B^{ϵ}(Q) by powers of r(Q).

Proof of Step III′: The factors in I_{Ψ}(P, g_{P_{j}^{k}}) that still depend on z_{P} (Q) and w_{P} (Q) are P(z_{P} (Q), w_{P} (Q)), g_{P_{j}^{k}}(P), d(P, Q) and e(P, Q), which depend smoothly on P, so we can expand them around Q in the coordinates z_{Q}(P) and w_{Q}(P) up to order N_{p}. Note that since P ∈ ∂Ω, we can express τ_{Q}(P) in terms of z_{Q}(P) and t_{Q}(P) up to sufficiently high order. The remainders are all O^{N_{p}+1}(z_{Q}(P), w_{Q}(P)) = O(|r(Q)|^{(N_{p}+1)/m}). This gives a total error

≤ C_{N_{p}} ∫_{B^{ϵ}(Q)} |r(Q)|^{(N_{p}+1)/m} \frac{∥ψ∥_{C^{Np}}}{V_{π(Q)}(−r(Q) + γ_{Q}(P))}dσ(P)
≤ C_{N_{p}}∥ψ∥_{C^{Np}} |r(Q)|^{(N_{p}+1)/m} ∫_{B^{ϵ}(Q)} \frac{1}{V_{P} (t_{P} (Q)}dσ(P)
≤ C|r(Q)|^{(N_{p}+1)/m}

by Lemma 4.4.

The other contribution to the error comes from making the change of variables and integrating over R^{2n−1} with respect to dz_{Q}dt_{Q} instead of over B^{ϵ}(Q) with respect to dσ(P). As in the proof of Step III in the case p near 1 above, this involves a certain Jacobian factor J_{Θ}^{−1}(z_{Q}(P), w_{Q}(P)), which can be approximated by |∇r(Q)|^{2n−1}. We expand this Jacobian around this value up to order N_{p} in z_{Q}(P), w_{Q}(P), and again get an error of O(|r(Q)|^{N_{p}+1}). Furthermore, as in Step III above, we need to change the limits of integration in the t_{Q} variable to |t| ≤ −r(Q) + γ_{Q}(z, \overline{z}). The original error in doing this just involved ∥ψ∥_{∞} and |r(Q)|^{1+1/m}. Now we must expand this error to higher order. This involves taking derivatives of the integral with respect to the limits of integration, which of course turn into one lower order derivatives of ψ. Fortunately, however, ψ is C^{∞} and it and all its derivatives vanish at the endpoints (i.e. for |t| = −r(Q) + γ_{Q}(z, \overline{z}), since that is what we are scaling by). Thus we are just left with the remainder of this expansion, which is bounded by ∥ψ∥_{C^{Np}} |r(Q)|^{(N_{p}+1)(1+1/m)}, so as in the proof of Step III above, the error

≤ C ∫_{z∈Θ(B^{ϵ}(Q))} \frac{|r(Q)|^{(N_{p}+1)(1+1/m)}}{V_{π(Q)}(| − r(Q) + γ_{Q}(z, \overline{z})|)}
≤ C|r(Q)|^{(N_{p}+1)/m} ∫_{z∈Θ(B^{ϵ}(Q))} \frac{|r(Q)|}{V_{π(Q)}(| − r(Q) + γ_{Q}(z, \overline{z})|)}
≤ C|r(Q)|^{(N_{p}+1)/m}.

Finally, after accounting for all the errors, we can integrate in the coordinates z_{Q}(P), t_{Q}(P). We fix z, factor out all terms involving only z, and integrate first in t over the interval |t| ≤ −r(Q) + γ_{Q}(z, \overline{z}). The integrand is a summation of expressions involving

ψ^{(l)} (\frac{t}{−r(Q) + γ_{Q}(z, \overline{z})}) ,

for l ≤ N_{p}, multiplied by polynomials in t of degree ≥ l (respectively) and ≤ N_{p}. This allows us to integrate by parts l times (respectively) and then use the vanishing moments of ψ to get that all the integrals are zero. This concludes the proof of Step III′ and of the whole cancellation condition for g_{j}^{k}^{(ϵ)}.

42 CONTENTS

Appendix A

Existence of Boundary Values

Let F ∈ H^{p}(Ω). We want to show that F has boundary values in the sense of distributions.

We can do this in a more general setting. Let Ω be a bounded domain in R^{N} with smooth boundary, defined by a defining function r. For δ > 0, define the shell S_{δ} = {r = −δ}, and let π_{δ} : S_{δ} → ∂Ω be the normal projection, which is well-defined and invertible for sufficiently small δ. Consider a function u in Ω. We can define the action of u_{δ}, the function u restricted to S_{δ}, on a smooth function φ on ∂Ω as follows:

< u_{δ}, φ >= ∫_{∂Ω} u(π_{δ}^{−1}(w))φ(w)dσ.

Lemma A.1 If u is harmonic in Ω and satisfies

∫_{S_{δ}} |u|dσ_{δ} ≤ Cδ^{−α_{0}},

for some real number α_{0}, then the family of distributions u_{δ} on ∂Ω, defined as above, has a limit as δ → 0.

Proof: Given φ ∈ C^{∞}(∂Ω), we want to show < u_{δ}, φ > has a limit as δ → 0. It suffices to show < u_{δ_{1}}, φ > − < u_{δ_{2}}, φ >→ 0 as δ_{1}, δ_{2} → 0.

We first show that < u_{δ}, φ > is bounded by C∥φ∥_{C^{M_{0}}} for some M_{0} depending on α_{0}, with a constant independent of φ. To do this, we show that if for some α ≠ 1 we have, for all small δ > 0 and all φ ∈ C^{∞}(∂Ω):

| < u_{δ}, φ > | ≤ Cδ^{−α}∥φ∥_{C^{M}} ,

then

| < u_{δ}, φ > | ≤ C′(δ^{1−α} + C′′)∥φ∥_{C^{M+1}}

for all small δ > 0 and all φ ∈ C^{∞}(∂Ω). Since the first inequality is clearly satisfied for α_{0} (if α_{0} is an integer, we take α = α_{0} + ϵ), we see that eventually we will get a positive power of δ, and the desired boundedness.

If we restrict φ to have support in some coordinate neighborhood U on ∂Ω, then in some small tubular neighborhood Ũ of U in Ω we can have coordinates (ζ, δ), where ζ are just coordinates on U, and δ = −r (see [NRSW], p. 139). Thus, with some abuse of notation, we can write

< u_{δ}, φ >= ∫_{∂Ω} u(ζ, δ)φ(ζ)dσ(ζ).

For a fixed small δ_{0} and δ < δ_{0}, we want to bound

< u_{δ}, φ > − < u_{δ_{0}}, φ > = ∫_{∂Ω}[u(ζ, δ) − u(ζ, δ_{0})]φ(ζ)dσ(ζ)
= ∫_{∂Ω}[∫_{δ}^{δ_{0}} −\frac{∂u}{∂δ}(ζ, δ)dδ]φ(ζ)dσ(ζ)
= ∫_{δ}^{δ_{0}} ∫_{∂Ω} −\frac{∂u}{∂δ}(ζ, δ)φ(ζ)dσ(ζ)dδ
= ∫_{δ}^{δ_{0}} ∫_{S_{δ}}(\frac{∂u}{∂ν})(φ ∘ π_{δ})J_{π_{δ}}dσ_{δ}dδ,

where ν is the outward unit normal.

Let

v_{δ}(z) = ∫_{S_{δ}} P_{δ}(z, w)φ(π_{δ}(w))J_{π_{δ}}dσ_{δ},

where P_{δ}(z, w) is the Poisson kernel for Ω_{δ}. Applying Green’s theorem for Ω_{δ}, we get

∫_{S_{δ}}(\frac{∂u}{∂ν})(φ ∘ π_{δ})J_{π_{δ}}dσ_{δ} = ∫_{S_{δ}} u\frac{∂v_{δ}}{∂ν} dσ_{δ}
= ∫_{∂Ω}(u ∘ π_{δ}^{−1}) (\frac{∂v_{δ}}{∂ν} ∘ π_{δ}^{−1}) J_{π_{δ}}^{−1}dσ
= < u_{δ}, φ̃_{δ} >,

where φ̃_{δ}(w) = \frac{∂v_{δ}}{∂ν} (π_{δ}^{−1}(w))J_{π_{δ}}^{−1} for w ∈ ∂Ω.

Taking absolute values,

| < u_{δ}, φ > | ≤ ∫_{δ}^{δ_{0}} | < u_{δ}, φ̃_{δ} > |dδ + | < u_{δ_{0}}, φ > |
≤ ∫_{δ}^{δ_{0}} Cδ^{−α}∥φ̃_{δ}∥_{C^{M} (∂Ω)}dδ + Cδ_{0}^{−α}∥φ∥_{C^{M}} .

However, since

∥φ̃_{δ}∥_{C^{M} (∂Ω)} = ∥(\frac{∂v_{δ}}{∂ν} ∘ π_{δ}^{−1}) J_{π_{δ}}^{−1}∥_{C^{M} (∂Ω)} ≤ C∥φ∥_{C^{M+1}(∂Ω)}

with a constant independent of φ, we have

| < u_{δ}, φ > | ≤ C′(δ^{1−α} + C′′)∥φ∥_{C^{M+1}(∂Ω)},

where the constants depend on α and δ_{0}.

Now that we have shown < u_{δ}, φ > is bounded by C∥φ∥_{C^{M_{0}}} , for all φ ∈ C^{∞}(∂Ω), we can apply the same method to get

| < u_{δ_{1}}, φ > − < u_{δ_{2}}, φ > | ≤ ∫_{δ_{1}}^{δ_{2}} | < u_{δ}, φ̃_{δ} > |dδ

≤ ∫_{δ_{1}}^{δ_{2}} C∥φ̃_{δ}∥_{C^{M_{0}} (∂Ω)}dδ
≤ C∥φ∥_{C^{M_{0}+1}(∂Ω)}(δ_{2} − δ_{1})

for 0 < δ_{1} < δ_{2} < δ_{0}. Clearly this tends to 0 as δ_{1} and δ_{2} tend to 0.

Using this lemma, we can show the following

Corollary A.2 Let F ∈ H^{p}(Ω), Ω strongly pseudoconvex in C^{n} or weakly pseudoconvex of finite type in C^{2}. Then F has boundary values in the sense of distributions on ∂Ω.

Proof: For Q on S_{δ}, we can fit inside Ω a polydisk D_{δ}(Q) with radii cδ in the complex normal direction and c\sqrt{δ} in the complex tangential directions. This polydisk will lie between two shells S_{c_{1}δ} and S_{c_{2}δ}. Then, since |F |^{p} is subharmonic,

|F (Q)|^{p} ≤ \frac{1}{V (P_{δ}(Q))} ∫_{D_{δ}(Q)} |F |^{p}dV

≤ Cδ^{−(n+1)} ∫_{c_{1}δ}^{c_{2}δ} ∫_{S_{δ}} |F |^{p}dσ_{δ}dδ
≤ C∥F ∥_{H^{p}}^{p}δ^{−n}.

This implies

∫_{S_{δ}} |F (Q)|dσ_{δ} ≤ ∫_{S_{δ}} |F (Q)|^{p}(C∥F ∥_{H^{p}}δ^{−n/p})^{1−p}dσ_{δ}
≤ C∥F ∥_{H^{p}}δ^{−n( \frac{1}{p} −1)},

which together with the lemma above (for α_{0} = n(\frac{1}{p} − 1)) proves the existence of the boundary distribution.

46 APPENDIX A. EXISTENCE OF BOUNDARY VALUES

Appendix B

Moment Conditions for h^{p}

In this appendix we will discuss the equivalence of local Hardy spaces defined using approximate cancellation conditions for atoms (as in Section 2.3) with the local Hardy spaces defined by Goldberg ([Gol]). We will do this in three cases: for R^{n}, for spaces of homogeneous type (with p near 1), and for the domains discussed in this paper, i.e. strongly pseudoconvex domains in C^{n} and pseudoconvex domains of finite type in C^{2}.

1 h^{p}(R^{n})

In [Gol], Goldberg gives several charcterizations of the local Hardy spaces h^{p}. We shall use the following one. Let φ ∈ S with ∫ φ ≠ 0. Then a tempered distribution f is in h^{p} iff sup_{0<t<1} |φ_{t} ∗ f(x)| is in L^{p}, and we can write

∥f∥_{h^{p}} = ∥ sup_{0<t<1} |φ_{t} ∗ f(x)|∥_{L^{p}}.

We want to show that this is equivalent to the space happrox^{p} of tempered distributions f which can be written as

f = ∑ λ_{j}a_{j}

with real numbers λ_{j} satisfying

∑ |λ_{j}|^{p} < ∞

and “approximate atoms” a_{j}. An “approximate atom” is a function a supported in a ball B(x_{0}, R) and satisfying a size condition:

∥a∥_{L^{2}} ≤ |B|^{\frac{1}{2} − \frac{1}{p}}

and an approximate cancellation condition:

|∫ a(x)(x − x_{0})^{α}dx| ≤ CR^{η}

48 APPENDIX B. MOMENT CONDITIONS FOR H^{P}

for all |α| ≤ [n(1/p − 1)], and some fixed η > 0. We will write

∥f∥_{happrox^{p}} = (inf ∑ |λ_{j}|^{p})^{1/p},

the infimum being taken over all such decompositions.

Lemma B.1 For 0 < p ≤ 1,

happrox^{p}(R^{n}) = h^{p}(R^{n}),

and the norms are equivalent.

Proof: The fact that h^{p}(R^{n}) ⊂ happrox^{p}(R^{n}) (with the appropriate bounding of norms) follows from the atomic decomposition exhibited by Goldberg ([Gol], Lemma 5) with atoms supported in cubes Q, satisfying an L^{∞} size condition

∥a∥_{∞} ≤ |Q|^{−1/p},

and having exact vanishing moments

∫ a(x)x^{α}dx = 0

for all |α| ≤ [n(1/p − 1)], whenever |Q| < 1. Clearly such atoms satisfy the L^{2} size condition and the approximate cancellation conditions with respect to balls.

Conversely, let us take φ as above, and f ∈ happrox^{p}. We want to show

∥ sup_{0<t<1} |φ_{t} ∗ f(x)|∥_{L^{p}} ≤ C∥f∥_{happrox^{p}}.

By the atomic decomposition, and since p ≤ 1, it is enough to show that for an “approximate atom” a,

∥ sup_{0<t<1} |φ_{t} ∗ a(x)|∥_{L^{p}} ≤ C.

It suffices to show this when a is supported in a ball B(0, R) centered at the origin. Let us write

∫ ( sup_{0<t<1} |φ_{t} ∗ a(x)|)^{p}dx ≤ ∫_{x∈2B} + ∫_{x∉2B},

where 2B is the ball B(0, 2R). Since 2/p > 1, we have by Cauchy-Schwarz that

∫_{2B}( sup_{0<t<1} |φ_{t} ∗ a(x)|)^{p}dx ≤ |2B|^{1−p/2} (∫_{2B}( sup_{0<t<1} |φ_{t} ∗ a(x)|)^{2}dx)^{p/2} .

The operator f → sup_{0<t<1} |φ_{t} ∗ f| is bounded on L^{2}, so

∫_{2B}( sup_{0<t<1} |φ_{t} ∗ a(x)|)^{p}dx ≤ C|2B|^{1−p/2}∥a∥_{L^{2}}^{p}
≤ C2^{n(1−p/2)}|B|^{1−p/2}|B|^{p/2−1}
= C_{n,p}.

1. H^{P} (R^{N}) 49

Now assume x ∉ 2B, and look at

φ_{t} ∗ a(x) = ∫_{y∈B} φ_{t}(x − y)a(y)dy.

Let N = [n(1/p − 1)]. Using the Taylor expansion of φ up to order N:

φ(x + h) = ∑_{|α|≤N} C_{α}φ^{(α)}(x)h^{α} + R_{N+1}(x, h),

where

φ^{(α)}(x) = \frac{∂^{|α|}φ}{∂x_{1}^{α_{1}} … ∂x_{n}^{α_{n}}} (x_{1}, …, x_{n})

and

R_{N+1}(x, h) = ∑_{|α|=N+1} C_{α}φ^{(α)}(x_{h})h^{α}

for some x_{h} on the line between x and x + h, we can write

φ_{t}(x − y) = t^{−n} ∑_{|α|≤N} C_{α}φ^{(α)}(x/t)(−y/t)^{α} + t^{−n}R_{N+1}(x/t, −y/t).

Here x/t denotes (x_{1}/t, …, x_{n}/t). Since φ ∈ S, η > 0, and

ϵ := N + 1 − n(1/p − 1) > 0,

there exists a constant C_{N} such that for all x,

sup_{|α|≤N} |C_{α}φ^{(α)}(x)| ≤ C_{N}|x|^{−n/p−η}

and

∑_{|α|=N+1} |C_{α}φ^{(α)}(x)| ≤ C_{N}|x|^{−n/p−ϵ}.

Thus

|φ_{t} ∗ a(x)| ≤ t^{−n} ∑_{|α|≤N} C_{α}|φ^{(α)}(x/t)|t^{−|α|} |∫ a(y)(−y)^{α}dy|

+ t^{−n} |∫ R_{N+1}(x/t, −y/t)a(y)dy|
≤ C_{N} ∑_{|α|≤N} t^{−n−|α|}|x/t|^{−n/p−η}R^{η}

+ C_{N} ∫ t^{−n}|x_{y}/t|^{−n/p−ϵ}|y/t|^{N+1}|a(y)|dy,

where x_{y} lies on the line between x and x − y. Now for |α| ≤ n(1/p − 1) and t < 1 we have t^{−n−|α|+n/p+η} ≤ 1 and by definition of ϵ, t^{−n+n/p+ϵ−N−1} = 1. Furthermore, if y ∈ B and x ∉ 2B, then |x_{y}| ≥ |x|/2. Thus

|φ_{t} ∗ a(x)| ≤ C|x|^{−n/p−η}R^{η} + C|x|^{−n/p−ϵ}R^{N+1}|B|^{1−1/p}
≤ C|x|^{−n/p−η}R^{η} + C|x|^{−n/p−ϵ}R^{ϵ}.

So finally

∫_{x∉2B}( sup_{0<t<1} |φ_{t} ∗ a(x)|)^{p}dx ≤ CR^{ηp} ∫_{|x|≥2R} |x|^{−n−ηp}dx + CR^{ϵp} ∫_{|x|≥2R} |x|^{−n−ϵp}dx
≤ C_{p}.

2 h^{p} on Spaces of Homogeneous Type

Let (X, ρ, σ) be a space of homogeneous type, where d is a quasi-distance and σ is a doubling measure. In this context we can only discuss moments of order zero, so we must restrict p to be near 1 (for the exact range of p, see [MS]). Here we write the approximate cancellation condition for an atom a supported in a ball B as

|∫_{X} adσ| ≤ Cσ(B)^{η}

for some fixed η > 0. The exact cancellation condition is

∫_{X} adσ = 0

whenever σ(B) < 1. Clearly an atom satisfying the exact cancellation condition also satisfies the approximate one (for σ(B) ≥ 1 we just use the size estimate on a), so in order to show the equivalence of atomic decomposition with “approximate atoms” and atomic decomposition with “exact atoms”, we just need to show the following lemma.

Lemma B.2 Suppose a is an atom satisfying the approximate cancellation condition defined above. Then

a = ∑ λ_{j}a_{j},

where the a_{j} are atoms satisfying the exact cancellation condition, and

∑ |λ_{j}|^{p} ≤ C

for a constant C independent of a.

Proof: Consider the ball B_{a} in which a is supported. If σ(B_{a}) ≥ 1, there is nothing to prove, so assume σ(B_{a}) < 1. Recall that there are finite constants A_{2} ≥ A_{1} > 1 such that for all balls B,

A_{1}σ(B) ≤ σ(2B) ≤ A_{2}σ(B),

where 2B is the ball with the same center as B and twice the radius. Let us take an integer k ≥ 1 such that

A_{1}^{k−1}σ(B_{a}) < 1 ≤ A_{1}^{k}σ(B_{a}),

2. H^{P} ON SPACES OF HOMOGENEOUS TYPE 51

i.e.

k − 1 < − log_{A_{1}} σ(B_{a}) ≤ k.

Now let

I_{a} = ∫_{X} adσ,

and set, with χ_{B} being the characteristic function of B,

b_{0} = a − I_{a}σ(B_{a})^{−1}χ_{B_{a}},
b_{j} = I_{a}(σ(2^{j−1}B_{a})^{−1}χ_{2^{j−1}B_{a}} − σ(2^{j}B_{a})^{−1}χ_{2^{j}B_{a}})

for 1 ≤ j ≤ k, and

b_{k+1} = I_{a}σ(2^{k}B_{a})^{−1}χ_{2^{k}B_{a}}.

Then clearly

a = ∑_{j=0}^{k+1} b_{j},

b_{j} is supported in 2^{j}B_{a} for 0 ≤ j ≤ k, b_{k+1} is supported in 2^{k}B_{a} with σ(2^{k}B_{a}) ≥ A_{1}^{k}σ(B_{a}) ≥ 1, and

∫_{X} b_{j}dσ = 0

for 0 ≤ j ≤ k. Let us look at the L^{2} norms of the b_{j}.

For j = 0,

∥b_{0}∥_{L^{2}} ≤ ∥a∥_{L^{2}} + |I_{a}|σ(B_{a})^{−1}σ(B_{a})^{1/2}
≤ σ(B_{a})^{\frac{1}{2} − \frac{1}{p}} + σ(B_{a})^{η−1/2}
≤ 2σ(B_{a})^{\frac{1}{2} − \frac{1}{p}} ,

since p ≤ 1, η > 0, and σ(B_{a}) < 1.

For 1 ≤ j ≤ k,

∥b_{j}∥_{L^{2}} ≤ |I_{a}|(σ(2^{j−1}B_{a})^{−1/2} + σ(2^{j}B_{a})^{−1/2})
≤ |I_{a}|(A_{2}^{1/2} + 1)σ(2^{j}B_{a})^{−1/2}
≤ |I_{a}|C_{1}σ(2^{j}B_{a})^{\frac{1}{2} − \frac{1}{p}} ,

where C_{1} depends only on p and on the constants A_{1} and A_{2}.

Finally,

∥b_{k+1}∥_{L^{2}} ≤ |I_{a}|σ(2^{k}B_{a})^{−1/2}
≤ |I_{a}|C_{2}σ(2^{k}B_{a})^{\frac{1}{2} − \frac{1}{p}} ,

where again C_{2} depends only on p, A_{1} and A_{2}.

Thus if we let

a_{0} = \frac{b_{0}}{2} , λ_{0} = 2,

a_{j} = \frac{b_{j}}{|I_{a}|C_{1}} , λ_{j} = |I_{a}|C_{1},

for 1 ≤ j ≤ k, and

a_{k+1} = \frac{b_{k+1}}{|I_{a}|C_{2}} , λ_{k+1} = |I_{a}|C_{2},

then

a = ∑_{j=0}^{k+1} λ_{j}a_{j},

the a_{j} are atoms satisfying the exact cancellation condition, and

∑_{j=0}^{k+1} |λ_{j}|^{p} = 2^{p} + k(|I_{a}|C_{1})^{p} + (|I_{a}|C_{2})^{p}
≤ 2^{p} + σ(B_{a})^{ηp}(kC_{1}^{p} + C_{2}^{p})
< 2^{p} + σ(B_{a})^{ηp}((− log_{A_{1}} σ(B_{a}) + 1)C_{1}^{p} + C_{2}^{p})
≤ C_{p}

when σ(B_{a}) < 1, since η > 0.

3 h^{p}(∂Ω)

Let Ω be a strongly pseudoconvex domain in C^{n} or a pseudoconvex domain of finite type in C^{2}. In Section 2.3, we defined a nonisotropic p-atom a supported in a ball B(P_{0}, R) ⊂ ∂Ω using the approximate cancellation conditions:

|∫_{∂Ω} a(P)g_{P_{0}}(P)dσ(P)| ≤ CR^{η},

for some fixed positive η (depending only on p, m, and n) and all functions g_{P_{0}} which are of the form

g_{P_{0}}(P) = M(z, t)

for P = h^{P_{0}}(z, t), where M(z, t) = x^{α}y^{β}t^{γ} is a (monic) monomial of degree |α| + |β| + γ ≤ N_{p}, and C is independent of a and g_{P_{0}}.

We will show here that an atomic decomposition with atoms satisfying these conditions is equivalent to an atomic decomposition with atoms satisfying the following exact moment conditions: whenever a is supported in B(P_{0}, R) with σ(B(P_{0}, R)) < 1, then

∫_{∂Ω} a(P)g_{P_{0}}(P)dσ(P) = 0

for all functions g_{P_{0}} which are of the form

g_{P_{0}}(P) = M(z, t)

3. H^{P} (∂Ω) 53

for P = h^{P_{0}}(z, t), where M(z, t) = x^{α}y^{β}t^{γ} is a (monic) monomial of nonisotropic degree |α| + |β| + mγ ≤ [2n(1/p − 1)]. This definition is not canonical and can be made with other choices of coordinates. Here we will show the equivalence of the two atomic decompositions specifically for this choice, but the proof can be modified for other choices. The constants will depend on the coordinates.

As in the previous section, we only need to show the following

Lemma B.3 Suppose a is an atom satisfying the approximate cancellation condition defined above. Then

a = ∑ λ_{j}a_{j},

where the a_{j} are atoms satisfying the exact moment conditions, and

∑ |λ_{j}|^{p} ≤ C

for a constant C independent of a.

Proof: Let B_{a} = B(P_{0}, R) be the ball in which a is supported. Again we only need to consider the case σ(B_{a}) < 1. Since we have dealt with the zero order monomial in the previous section, we can use a sort of induction on the monomials. Let us list all the “monomials based at P_{0}” considered for the exact moment conditions above in some arbitrary order, starting with g_{0} = 1:

g_{0}, g_{1}, …, g_{L}.

For an integer l ≥ 1, we assume that a is already orthogonal to g_{0}, …, g_{l−1}, and we show that we can decompose

a = ∑ λ_{j}a_{j},

where the a_{j} are orthogonal to g_{0}, …, g_{l}, and

∑ |λ_{j}|^{p} ≤ C.

Since there are finitely many steps, we can combine all the decompositions to get a decomposition with a_{j} satisfying all the exact moment conditions.

We will use a process analogous to the zero-th order case. As before, we can find constants A_{1} and A_{2} as above (see [NRSW]), and we can take an integer k ≥ 1 such that

A_{1}^{k−1}σ(B_{a}) < 1 ≤ A_{1}^{k}σ(B_{a}).

We want to define b_{0}, …, b_{k+1}. In order to define b_{0}, since a is already orthogonal to g_{0}, …, g_{l−1}, we need only make it orthogonal to g_{l}, but first we must modify g_{l} by subtracting its projection onto the subspace of L^{2}(B_{a}, dσ) spanned by g_{0}, …, g_{l−1}. Let us call this new function g_{l}^{0}. Note that

∥g_{l}^{0}∥_{L^{2}(B_{a},dσ)} ≥ C∥g_{l}∥_{L^{2}(B_{a},dσ)}

with a constant C ≤ 1 depending on the inner products of the monomials in L^{2}(B_{a}, dσ). Using estimates on the coordinates z_{P_{0}}, t_{P_{0}} found in the body of this thesis and in [NRSW], one can show that C is bounded below by a positive constant not depending on B_{a}. Furthermore, if g_{l}(P) = M(z, t) = x^{α}y^{β}t^{γ} for P = h^{P_{0}}(z, t), with |α| + |β| + mγ ≤ 2n(1/p − 1), then

∥g_{l}∥_{L^{2}(B_{a},dσ)} ≈ R^{|α|+|β|+n−1}Λ(P_{0}, R)^{γ+1/2}
≥ CR^{|α|+|β|+mγ}R^{n−1}Λ(P_{0}, R)^{1/2}
≥ CR^{2n(1/p−1)}R^{n−1}Λ(P_{0}, R)^{1/2}
≥ CR^{(2n−2)(1/p−1/2)}R^{2(1/p−1)}Λ(P_{0}, R)^{1/2}
≥ CR^{(2n−2)(1/p−1/2)}Λ(P_{0}, R)^{1/p−1/2}
≈ σ(B_{a})^{1/p−1/2}.

Set M_{l} = ∫ a(P)g_{l}^{0}(P)dσ(P), and define

b_{0} = a − M_{l} \frac{g_{l}^{0}}{∥g_{l}^{0}∥_{L^{2}(B_{a},dσ)}^{2}} χ_{B_{a}}.

Then b_{0} is orthogonal to g_{0}, …, g_{l}, b_{0} is supported in B_{a}, and

∥b_{0}∥_{L^{2}(B_{a},dσ)} ≤ ∥a∥_{L^{2}} + |M_{l}|∥g_{l}^{0}∥_{L^{2}(B_{a},dσ)}^{−1}
≤ σ(B_{a})^{1/2−1/p} + CR^{η}σ(B_{a})^{1/2−1/p}
≤ C_{0}σ(B_{a})^{1/2−1/p}

since R ≤ 1.

In order to define b_{j}, 1 ≤ j ≤ k, we need to produce auxiliary functions g_{l}^{j} which are the result of subtracting from g_{l} its projection onto the subspaces spanned by g_{0}, …, g_{l−1} in L^{2}(2^{j}B_{a}, dσ), respectively. Again, as for g_{l}^{0},

∥g_{l}^{j}∥_{L^{2}(2^{j}B_{a},dσ)} ≥ Cσ(2^{j}B_{a})^{1/2−1/p}.

Now we can set

b_{j} = M_{l} \frac{g_{l}^{j−1}}{∥g_{l}^{j−1}∥_{L^{2}(2^{j−1}B_{a},dσ)}^{2}} χ_{2^{j−1}B_{a}} − M_{l} \frac{g_{l}^{j}}{∥g_{l}^{j}∥_{L^{2}(2^{j}B_{a},dσ)}^{2}} χ_{2^{j}B_{a}}

for 1 ≤ j ≤ k, and

b_{k+1} = M_{l} \frac{g_{l}^{k}}{∥g_{l}^{k}∥_{L^{2}(2^{k}B_{a},dσ)}^{2}} χ_{2^{k}B_{a}}.

Then by definition b_{j} is orthogonal to g_{0}, …, g_{l} and is supported in 2^{j}B_{a} for 1 ≤ j ≤ k, and b_{k+1} is supported in 2^{k}B_{a} with σ(2^{k}B_{a}) ≥ 1. Furthermore,

∥b_{j}∥_{L^{2}} ≤ |M_{l}|∥g_{l}^{j−1}∥_{L^{2}(2^{j}B_{a})}^{−1} + |M_{l}|∥g_{l}^{j}∥_{L^{2}(2^{j}B_{a})}^{−1}
≤ C|M_{l}|σ(2^{j−1}B_{a})^{1/2−1/p} + C|M_{l}|σ(2^{j}B_{a})^{1/2−1/p}
≤ C_{1}|M_{l}|σ(2^{j}B_{a})^{1/2−1/p}

3. H^{P} (∂Ω) 55 for 1 ≤ j ≤ k, and similarily

∥b_{k}∥_{L^{2}(2^{k}B_{a})} ≤ C_{2}|M_{l}|σ(2^{k}B_{a})^{1/2−1/p}.

Thus if we do as in the previous section and let

a_{0} = \frac{b_{0}}{C_{0}} , λ_{0} = C_{0},

a_{j} = \frac{b_{j}}{|M_{l}|C_{1}} , λ_{j} = |M_{l}|C_{1},

for 1 ≤ j ≤ k, and

a_{k+1} = \frac{b_{k+1}}{|M_{l}|C_{2}} , λ_{k+1} = |M_{l}|C_{2},

then

a = ∑_{j=0}^{k+1} λ_{j}a_{j},

the a_{j} are atoms orthogonal to g_{0}, …, g_{l}, and

∑_{j=0}^{k+1} |λ_{j}|^{p} = C_{0}^{p} + k(|M_{l}|C_{1})^{p} + (|M_{l}|C_{2})^{p}
≤ C_{0}^{p} + σ(B_{a})^{ηp}(kC_{1}^{p} + C_{2}^{p})
< C_{0}^{p} + σ(B_{a})^{ηp}((− log_{A_{1}} σ(B_{a}) + 1)C_{1}^{p} + C_{2}^{p})
≤ C_{p}

when σ(B_{a}) < 1, since η > 0. This concludes the induction step and therefore proves the lemma.

56 APPENDIX B. MOMENT CONDITIONS FOR H^{P}

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