Boundary Traces of Holomorphic Functions on the Unit Ball in $\mathbb {C}^n$

Abstract

It is a classical theorem that if a function on the unit circle is integrable, then it is the nontangential limit of a holomorphic function on the open disc (subject to a certain growth condition) if and only if its Fourier coefficients for nonnegative integers are zero. In this article we generalize this result to higher complex dimensions by proving that for an integrable function on the unit sphere, it is a “boundary trace” of a holomorphic function on the open unit ball if and only if two particular families of integral equations are satisfied. To do this, we use the theory of Hardy spaces as well as the invariant Poisson and Cauchy integrals.

1 Introduction

Holomorphic functions are special. This is the major theme of an undergraduate course in complex analysis. Whereas the stars in a real analysis class are the counterexamples and the theorems you thought should be true but are actually false (in full generality at least), the stars in a complex analysis class are facts you would never dream to be true but are, such as the Cauchy Integral formula, Liouville’s Theorem, and Hadamard’s Three Lines Theorem. Now you can make an argument that lots of families of functions are special, like smooth functions and harmonic functions. But those functions are in many ways ubiquitous and holomorphic functions are much more rare. Put another way, most integrable functions on a reasonable domain can be approximated by smooth functions, but not by holomorphic functions. If an integrable function saw a smooth function in public they wouldn’t be surprised, but if they saw a holomorphic function they would be tempted to ask it for a selfie. In the same vein, most integrable functions along a boundary of some reasonable subset of $\mathbb {R}^{n}$ are the boundary values of a harmonic function. Only a select few of those integrable functions are the boundary values of a holomorphic function (in the case n is even). This paper is about this latter topic: what does it take for an integrable function on a boundary to be a boundary value of a holomorphic function when the “reasonable subset” is the unit ball of $\mathbb {C}^n$? (Admittedly, the unit ball is about as nice of a subset you can find, but it will still lead to an interesting problem.)

Let’s make the question above more formal. Fix $n\in \mathbb {N}$. Let $B^n$ denote the open unit ball and $S^n$ denote the unit sphere in $\mathbb {C}^n$. That is, for $z=(z_1,\ldots ,z_n)\in \mathbb {C}^n$, we have $z\in B^n$ if and only if $|z_1|^2+\cdots +|z_n|^2<1$ and $z\in S^n$ if and only if $|z_1|^2+\cdots +|z_n|^2=1$. When the dimension n is understood we will write these sets as B and S, respectively. We say that a function $F:B\rightarrow \mathbb {C}$ is holomorphic if F is continuous and is holomorphic in each of its n complex variables separately. As we noted above, we are going to look at integrable functions on S. This is a large space of functions that will allow us to consider integral formulas for our boundary functions. Thus, we will consider functions in $L^p(S,\sigma )$, where $1\le p <\infty$, $\sigma$ is the standard surface measure of the sphere S normalized so that $\sigma (S)=1$, and

$$\begin{aligned}L^p(S,\sigma )=\left\{ f:S\rightarrow \mathbb {C}\left| \int _{S} |f(\zeta )|^p d\sigma (\zeta ) < \infty \right. \right\} .\end{aligned}$$

Furthermore, we have the norm $\Vert f\Vert _p:=(\int _{S} |f(\zeta )|^p d\sigma (\zeta ))^{1/p}$ for $f\in L^p(S,\sigma )$ under which $L^p(S,\sigma )$ is complete.

Next we address what it means for f to be “the boundary value of a function F that is holomorphic on B.” Let A(S) denote the set of functions on S that are restrictions to S of functions F that are continuous on the closure of B and holomorphic on B. While A(S) is naturally a set of “boundary traces” of holomorphic functions on B that also lie in $L^p(S,\sigma )$ for all $p\ge 1$, it turns out that we can expand it: for $1\le p<\infty$ let $H^p(S)$ denote the closure of A(S) in $L^p(S,\sigma )$. While we are mainly interested in the case that $1\le p<\infty$, $H^p(S)$ can also be defined in this way for $0<p<1$. Theorem 1 below will show that $H^p(S)$ can be thought of boundary traces of holomorphic functions. But before we can state this theorem, we need to define the Hardy spaces $H^p(B)$ of holomorphic functions on B:

Definition 1

Let $0<p<\infty$. Define the Hardy space $H^p(B)$ as the set of all holomorphic functions F on B such that

$$\begin{aligned}\sup _{0\le r< 1} \left( \int _{S} |F(r\zeta )|^p d\sigma (\zeta ) \right) ^{1/p}<\infty .\end{aligned}$$

If $p\ge 1$, the supremum above can be used as a norm on $H^p(B)$ and under this norm $H^p(B)$ is a Banach space (see [1]). It turns out that every element of $H^p(B)$ admits a special type of limit for almost every point on the boundary S. Korányi first defined these limits and called them “admissible limits” in [2] but we will follow [1] and call them K-limits. We will give a more detailed definition of a K-limit in Sect. 2, but we note here that it is a limit to points on S taken within B but with certain trajectories disallowed; in the case that $n=1$, these limits are usually called nontangential limits but this nomenclature is inaccurate for $n>1$ as some tangential trajectories are allowed. It turns out that all elements of $H^p(B)$ have K-limits that lie in $H^p(S)$ (including the case where $0<p<1$):

Theorem 1

Let $0< p <\infty$. For a function F with domain B and $\zeta \in S$, let $F^*(\zeta )$ denote the K-limit of F at $\zeta$, if that limit exists.

• 1. For any $F\in H^p(B)$, the K-limit $F^*$ exists almost everywhere with respect to $\sigma$ on S and $F^*\in H^p(S)$. Furthermore, the mapping $F\mapsto F^*$ is an isometric isomorphism.

• 2. If $p\ge 1$ and $f\in H^p(S)$, then $C[f]\in H^p(B)$, $C[f]=P[f]$, and $C[f]^*=f$.

For a proof, see [1] where the above theorem is stated as Theorem 5.6.8. Since $H^p(B)$ and $H^p(S)$ are isometrically isomorphic (part 1 above), we also call $H^p(S)$ a Hardy space. In the statement of Theorem 1, we note that P[f] is the invariant Poisson integral of f and C[f] is the Cauchy integral of f, and we will give detailed definitions of these integrals in Sect. 2. At the moment, though, we note that part 1 of Theorem 1 implies that the elements of $H^p(S)$ are K-limits of holomorphic functions on B and thus $H^p(S)$ is an appropriate choice to serve as our set of boundary traces of holomorphic functions when $p\ge 1$. Thus we can now restate our original question rigorously:

Fix $1\le p<\infty$ and suppose $f\in L^p(S,\sigma )$. We want to find necessary and sufficient integral formulas for f to lie in $H^p(S)$.

Classical Hardy space theory provides an answer to the above question in the case that $n=1$:

Theorem 2

Let $1\le p<\infty$ and suppose $f\in L^p(S^1,\sigma )$. Then $f\in H^p(S^1)$ if and only if for every $j\in \mathbb {N}$ we have

$$\begin{aligned}\int _{S^1} \zeta ^jf(\zeta )d\sigma (\zeta ) = 0.\end{aligned}$$

Put another way, $f\in H^p(S^1)$ if and only if its Fourier coefficients for negative integers are all 0. This result is classical and can be found, for example, as [3, Thm. 3.4]. This result is related to the celebrated theorem of F. and M. Riesz which states that any complex Borel measure on $S^1$ whose negative Fourier coefficients vanish must be absolutely continuous with respect to Lebesgue measure (see, for example, [3, Thm. 3.8] or [4, Thm. 17.13]).

The main result of this paper (Theorem 4) is a generalization of Theorem 2 for $n>1$. Before we can state this generalization, we will need some definitions: first if $x+iy$ is a complex number with x and y real, then $\overline{x+iy}=x-iy$ is its complex conjugate. For $z=(z_1,z_2,\ldots ,z_n)\in \mathbb {C}^n$, we define $\overline{z}=(\overline{z}_1, \overline{z}_2,\ldots , \overline{z}_n)$. We also need to be able to take “powers” of elements of $\mathbb {C}^n$. To that end, we define a multi-index $\alpha =(\alpha _1,\ldots ,\alpha _n)$ to be any element of $(\mathbb {N}\cup \{0\})^n$. Then for $z=(z_1,z_2,\ldots ,z_n)\in \mathbb {C}^n$ and multi-index $\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)$, we define

$$\begin{aligned}z^\alpha = z_1^{\alpha _1} z_2^{\alpha _2}\cdots z_n^{\alpha _n}.\end{aligned}$$

Note that $z^\alpha$ is holomorphic. Furthermore, we define $|\alpha |$ and $\alpha !$ as

$$\begin{aligned} |\alpha | = \alpha _1+\alpha _2+\cdots +\alpha _n,\, \alpha ! = \alpha _1!\alpha _2!\cdots \alpha _n!. \end{aligned}$$

Finally, for any multi-index $\omega \in (\mathbb {N}\cup \{0\})^n$ we define the constant $c_{\omega }$ as

$$\begin{aligned} c_{\omega }:= \frac{(n-1)!\omega !}{(n-1+|\omega |)!}. \end{aligned}$$

(1)

One reason the constants $c_{\omega }$ are significant is due to the following result (found as [1, Prop. 1.4.8, Prop. 1.4.9]):

Theorem 3

Let $\omega ,\upsilon \in (\mathbb {N}\cup \{0\})^n$ be multi-indices. Then

$$\begin{aligned} \int _{S} \zeta ^\omega \overline{\zeta }^\upsilon d\sigma (\zeta )=\left\{ \begin{array}{lr} 0 &{} \text{ if } \omega \ne \upsilon \\ c_{\omega } &{} \text{ if } \omega =\upsilon \end{array}\right. . \end{aligned}$$

(2)

With these prerequisites in place, we can state the main result of the paper:

Theorem 4

Suppose $1\le p<\infty$ and $f\in L^p(S,\sigma )$. Then $f\in H^p(S)$ if and only if for any pair of multi-indices $\alpha$ and $\beta$ we have the following:

• (a) If $\alpha _j>\beta _j$ for some $1\le j\le n$, then we have

  $$\begin{aligned} \int _{S} \zeta ^\alpha \overline{\zeta }^\beta f(\zeta )d\sigma (\zeta )=0. \end{aligned}$$

  (3)

• (b) If $\alpha _j\le \beta _j$ for all $1\le j\le n$ and we define $\beta -\alpha =(\beta _1-\alpha _1,\beta _2-\alpha _2,\ldots ,\beta _n-\alpha _n)$, then

  $$\begin{aligned} c_{\beta }^{-1}\int _{S}\zeta ^\alpha \overline{\zeta }^{\beta }f(\zeta )d\sigma (\zeta )= c_{\beta -\alpha }^{-1}\int _{S}\overline{\zeta }^{\beta -\alpha }f(\zeta )d\sigma (\zeta ). \end{aligned}$$

  (4)

Admittedly, Theorem 4 is not as elegant as Theorem 2. One reason for this inelegance is due to the fact that when $n>1$, for $\zeta \in S$ and multi-index $\alpha$ the product $\zeta ^\alpha \overline{\zeta }^\alpha =|\zeta _1|^{\alpha _1}\cdots |\zeta _n|^{\alpha _n}$ can be strictly less than 1 and thus not a constant (whereas when $n=1$ this product will always equal 1). That said, Theorem 4 reduces to Theorem 2 in the case that $n=1$. Indeed, if $\alpha _1>\beta _1$ and $\zeta _1\in S^1$, then $\zeta _1^{\alpha _1}\overline{\zeta _1}^{\beta _1} = \zeta _1^{\alpha _1-\beta _1}$ and so Equation (3) is equivalent to the condition in Theorem 2. In the case that $\alpha _1\le \beta _1$ and $\zeta _1\in S^1$, then $\zeta _1^{\alpha _1}\overline{\zeta _1}^{\beta _1}=\overline{\zeta _1}^{\beta _1-\alpha _1}$. Furthermore, as $n-1=0$ in this case, the coefficients $c_{\beta }$ and $c_{\beta -\alpha }$ both equal 1. Thus, Equation (4) holds for any $f\in L^1(S^1,\sigma )$ (which is why there is not a second case present in Theorem 2). To be clear, Equation (4) does not generally hold for every $f\in L^1(S,\sigma )$ when $n>1$. For a counterexample, let $n=2$ and consider $\alpha =\beta =(1,1)$ with the (non-holomorphic) function $f(\zeta _1,\zeta _2)=(\zeta _1\zeta _2)\overline{(\zeta _1\zeta _2)}$. Then by Theorem 3 the left-hand side of Equation (4) equals $c_{(1,1)}^{-1}c_{(2,2)}=1/5$, whereas the right-hand side of Equation (4) equals $c_{(0,0)}^{-1}c_{(1,1)}=1/6$.

The rest of the paper is organized as follows: in Sect. 2 we will give the definitions and known results of Hardy spaces, so-called $\mathcal {M}$-harmonic functions, and the invariant Poisson and Cauchy integrals that play a large role in the proof of Theorem 4, in Sect. 3 we will prove Theorem 4, and in Sect. 4 we make some concluding remarks. As many of the background theorems that we will cite come from [1], we have and will continue to mostly adopt its notation to make it easier to look up results while reading this paper.

2 Background

In this section we will give the background theory that our proof of Theorem 4 relies on. First, we establish some basic notation. Let $z=(z_1,z_2,\ldots ,z_n)$ and $w=(w_1,w_2,\ldots ,w_n)$ be elements of $\mathbb {C}^n$. Define the inner product $\langle z,w\rangle$ and length |z| as

$$\begin{aligned}\langle z,w\rangle =z_1\overline{w_1}+ z_2\overline{w_2}+\cdots +z_n\overline{w_n},\quad |z| = \langle z,z\rangle ^{1/2}.\end{aligned}$$

Next we provide the promised definition of a K-limit. For $\xi \in S$ and $\alpha >1$, define the set $D_\alpha (\xi )$ as

$$\begin{aligned}D_\alpha (\xi ):= \left\{ z\in B: |1-\langle z,\xi \rangle |< \frac{\alpha }{2}(1-|z|^2)\right\} .\end{aligned}$$

For a function $F:B\rightarrow \mathbb {C}$, we say that F has the K-limit L at $\xi \in S$ if for every $\alpha >1$ and every sequence $\{z_j\}$ in $D_\alpha (\xi )$, the sequence $\{F(z_j)\}$ converges to L. In this case we write $F^*(\xi )=L$. We note that if $n>1$, the approach region $D_\alpha (\xi )$ does allow some tangential directions (see [1, Sect. 5.4] for justification of this fact).

Next we make a brief digression into real analysis: let $k>1$ and let $\Omega \subseteq \mathbb {R}^k$ be an open set. We say that $F:\Omega \rightarrow \mathbb {C}$ is harmonic if $\Delta F (x) = 0$ for all $x\in \Omega$, where $\Delta$ is the Laplacian operator defined as

$$\begin{aligned}\Delta =\sum _{j=1}^k \frac{\partial ^2}{\partial x_j^2}.\end{aligned}$$

In turns out that we will be interested not in harmonic functions, but the so-called $\mathcal {M}$-harmonic functions. To define $\mathcal {M}$-harmonic functions, we move back to the complex setting, next defining a family of mappings of the n-ball B onto itself. Fix $a\in B$. Define the function $\phi _a:B\rightarrow B$ as

$$\begin{aligned}\phi _a(z) = \frac{a - \frac{\langle z, a\rangle }{\langle a, a\rangle }a-\frac{1}{\sqrt{1-|a|^2}}\left( z-\frac{\langle z, a\rangle }{\langle a, a\rangle }a\right) }{1-\langle z, a\rangle }\end{aligned}$$

$\phi _a$ is holomorphic and an involution with $\phi _a(0)=a$ (see, for example, [1, Thm. 2.2.2] for properties of $\phi _a$). Indeed, the family $\phi _a$ generalizes fractional linear transformation when $n=1$. We can use this family to define $\mathcal {M}$-harmonic functions: if $F:B\rightarrow \mathbb {C}$ is a twice continuously differentiable function, we define the operator $\tilde{\Delta }$ as

$$\begin{aligned}(\tilde{\Delta }F)(a):= \Delta (F\circ \phi _a)(0)\quad \text{ for } \text{ all } a\in B\text{. }\end{aligned}$$

We call $\tilde{\Delta }$ the invariant Laplacian as it commutes with all the automorphisms of B (that is, holomorphic functions from B onto B who inverse is also holomorphic; see [1, Thm. 4.1.2]). Finally, we say that F is $\mathcal {M}$-harmonic if $\tilde{\Delta } F(z)= 0$ for all $z\in B$. We note that harmonic functions and $\mathcal {M}$-harmonic functions coincide when $n=1$ but are different classes of functions when $n>1$; indeed, if F is both harmonic and $\mathcal {M}$-harmonic on B, then F is pluriharmonic on B (see [1, Thm. 4.4.9]). We also note here that any holomorphic function on B is both harmonic and $\mathcal {M}$-harmonic.

Next let $z\in B$ and $\zeta \in S$. Define the invariant Poisson kernel $P(z,\zeta )$ and invariant Poisson integral P[f] for any $f\in L^1(S,\sigma )$ as

$$\begin{aligned}P(z,\zeta )=\frac{(1-|z|^2)^n}{|1-\langle z,\zeta \rangle |^{2n}},\, P[f](z)=\int _{S} P(z,\zeta ) f(\zeta )d\sigma (\zeta ).\end{aligned}$$

Let $z,w\in \mathbb {C}^n$ with $\langle z,w\rangle \ne 1$. Define the Cauchy kernel C(z, w) and Cauchy integral C[f] for any $f\in L^1(S,\sigma )$ as

$$\begin{aligned}C(z,w)=\frac{1}{(1-\langle z, w\rangle )^n},\quad C[f](z)=\int _{S} C(z,\zeta ) f(\zeta )d\sigma (\zeta ) \quad \text{ for } z\in B.\end{aligned}$$

The invariant Poisson and Cauchy integrals have very nice properties. In particular, for any $f\in L^p(S,\sigma )$, C[f] is holomorphic on B and P[f] is $\mathcal {M}$-harmonic. Since we are looking for holomorphic functions on B one may wonder why we consider the invariant Poisson integral and $\mathcal {M}$-harmonic functions at all. It is because the invariant Poisson integral has nice convergence properties at the boundary S as Theorem 5 will show, and because the invariant Poisson kernel and Cauchy kernel are related (as we will see in the first equality in Equation (7) below). For Theorem 5, we need a quick definition: If $u:B\rightarrow \mathbb {C}$, for any $0\le r<1$ we define the function $u_r:S\rightarrow \mathbb {C}$ as

$$\begin{aligned}u_r(\zeta ) = u(r\zeta ).\end{aligned}$$

Theorem 5

Let $1\le p <\infty$ and let $f\in L^p(S,\sigma )$.

• 1. For any $0\le r<1$, we have $\Vert P[f]_r\Vert _p\le \Vert f\Vert _p$ and $\lim _{r\rightarrow 1^-}\Vert P[f]_r-f\Vert _p=0$. ([1, Thm. 3.3.4 (b)])

• 2. For almost all $\zeta \in S$ (with respect to $\sigma$), the K-limit of $P[f](\zeta )$ exists and equals $f(\zeta )$ (more succinctly, $P[f]^*= f$). ( [1, Thm. 5.4.8])

Thus P[f] converges to f on S in both an $L^p$-sense and in terms of K-limits. Note that Theorem 5 implies that, loosely speaking, all functions in $L^p(S,\sigma )$ are boundary traces of a $\mathcal {M}$-harmonic functions on B (that is, $\mathcal {M}$-harmonic are special but not special).¹ In contrast, the Cauchy integral C[f] is not as well-behaved; while C[f] does have K-limits at almost all points of S (see the corollary to [1, Thm. 6.2.3]), these K-limits need not equal the original function f. Indeed, for $1<p<\infty$, the mapping $f\mapsto C[f]^*$ is a linear projection from $L^p(S,\sigma )$ onto $H^p(S)$ (see the corollary to [1, Thm. 6.3.1]).

If the condition $P[f]=C[f]$ is satisfied and we let F equal both, then we have the best of both worlds. Indeed, in this case F converges to f on S in terms of K-limits (by part (b) of Theorem 5 since $F=P[f]$), F is holomorphic (since $F=C[f]$), and $F\in H^p(B)$ (by part (a) of Theorem 5 since $F=P[f]$). By part (a) of Theorem 1, we see that $f=F^*$ must lie in $H^p(S)$. That is, we have proven the following lemma:

Lemma 6

Let $1\le p <\infty$. If $f\in L^p(S,\sigma )$ and $P[f]=C[f]$, then $f\in H^p(S)$.

Lemma 6 will be central to our proof of Theorem 4 which we are now in a position to prove.

3 Proof of Theorem 4

Proof of Theorem 4

First we prove the forward direction. Suppose $f\in H^p(S)$ and let $F=P[f]$. Our strategy here will be to prove Equations (3) and (4) for monomials, use uniform convergence of power series to show that the equations then hold for $F_r$, and then use the $L^p$ convergence of $F_r$ to f to show the equations hold for f. By part 2 of Theorem 1, $F\in H^p(B)$. Thus F is holomorphic on B, so we can write it as a power series of the form $\sum _{\gamma } a_\gamma z^\gamma$ where the series is taken over all multi-indices $\gamma$ and converges to F uniformly on compact subsets of B (see Remark (i) in [1, Subsect. 1.2.6]). Let $\alpha$ and $\beta$ be a pair of multi-indices. First, to prove Equation (3) suppose that $\alpha _j>\beta _j$ for some $1\le j\le n$. Then for any other multi-index $\gamma$ we have $\alpha _j+\gamma _j>\beta _j$. Thus, $\alpha +\gamma \ne \beta$ and so by Theorem 3 we have

$$\begin{aligned}\int _{S} \zeta ^{\alpha +\gamma }\overline{\zeta }^\beta d\sigma (\zeta )=0.\end{aligned}$$

Let $0<r<1$. By the uniform convergence of $F(z)=\sum _{\gamma } a_\gamma z^\gamma$ on the sphere of radius r, the above implies that

$$\begin{aligned}\int _{S} \zeta ^{\alpha }\overline{\zeta }^\beta F(r\zeta )d\sigma (\zeta )=0.\end{aligned}$$

Thus, using the fact that $|\zeta ^{\alpha }\overline{\zeta }^\beta |\le 1$, we have

$$\begin{aligned} \left| \int _{S} \zeta ^{\alpha }\overline{\zeta }^\beta f(\zeta )d\sigma (\zeta )\right|= & {} \left| \int _{S} \zeta ^{\alpha }\overline{\zeta }^\beta (f(\zeta )-F(r\zeta ))d\sigma (\zeta )\right| \\\le & {} \int _{S}|f(\zeta )-F(r\zeta )|d\sigma (\zeta )\\\le & {} \left( \int _{S}|f(\zeta )-F(r\zeta )|^pd\sigma (\zeta )\right) ^{1/p}. \end{aligned}$$

As $r\rightarrow 1^-$, the right-hand expression goes to 0 by part 1 of Theorem 5 and so $\int _{S} \zeta ^{\alpha }\overline{\zeta }^\beta f(\zeta )d\sigma (\zeta )=0$, proving Equation (3). Now to prove Equation (4) suppose $\alpha _j\le \beta _j$ for all $1\le j\le n$ and define the multi-index $\lambda =\beta -\alpha$. Note that for any multi-index $\gamma$, again by Theorem 3 we have

$$\begin{aligned}c_{\beta }^{-1}\int _{S}\zeta ^\alpha \overline{\zeta }^{\beta }\zeta ^\gamma d\sigma (\zeta )=\left\{ \begin{array}{lr}0 &{} \text{ if } \gamma \ne \lambda \\ 1 &{} \text{ if } \gamma =\lambda \end{array}\right\} = c_{\lambda }^{-1}\int _{S}\overline{\zeta }^{\lambda }\zeta ^\gamma d\sigma (\zeta ).\end{aligned}$$

Again by the uniform convergence of $F(z)=\sum _{\gamma } a_\gamma z^\gamma$ on the sphere of radius r, the above implies that

$$\begin{aligned} c_{\beta }^{-1}\int _{S}\zeta ^\alpha \overline{\zeta }^{\beta }F(r\zeta )d\sigma (\zeta )= c_{\lambda }^{-1}\int _{S}\overline{\zeta }^{\lambda }F(r\zeta )d\sigma (\zeta ). \end{aligned}$$

Using an analogous argument as we used in to prove Equation (3), the above implies that Equation (4) holds, completing the forward direction.

For the backward direction, suppose $f\in L^p(S,\sigma )$ and satisfies Equations (3) and (4). By Lemma 6, to show that $f\in H^p(S)$ it suffices to show that $P[f](z)=C[f](z)$ for all $z\in B$. Our strategy to show $P[f]=C[f]$ will be to use an equation that relates the Poisson kernel with the Cauchy kernel (the first equality in Equation (7) below) and show that Equations (3) and (4) cause many terms in the resulting expansion to drop out, leaving us with only the terms that make up the Cauchy integral. To that end, we will first show that for any z, w in the closure of B where at least one of z or w lies in B (that is, $|z|,|w|\le 1$ and $|z||w|<1$) we have

$$\begin{aligned} C(z,w) = \frac{1}{(1-\langle z,w\rangle )^n} = \sum _{\omega }c_{\omega }^{-1} z^\omega \overline{w}^\omega , \end{aligned}$$

(5)

where the sum is taken over all multi-indices $\omega$ and $c_\omega$ is defined as in Theorem 3. Using the power series $1/(1-x)^n=\sum _{j=0}^\infty \left( {\begin{array}{c}j+n-1\\ n-1\end{array}}\right) x^j$ and the multinomial theorem, we have

$$\begin{aligned} C(z,w)= & {} \frac{1}{(1-\langle z,w\rangle )^n} = \sum _{j=0}^\infty \left( {\begin{array}{c}j+n-1\\ n-1\end{array}}\right) (\langle z, w\rangle )^j \\= & {} \sum _{j=0}^\infty \left( {\begin{array}{c}j+n-1\\ n-1\end{array}}\right) \sum _{|\omega |=j} \left( {\begin{array}{c}j\\ \omega _1, \omega _2,\ldots , \omega _n\end{array}}\right) z^\omega \overline{w}^\omega \\= & {} \sum _{\omega } \left( {\begin{array}{c}|\omega |+n-1\\ n-1\end{array}}\right) \left( {\begin{array}{c}|\omega |\\ \omega _1, \omega _2,\ldots , \omega _n\end{array}}\right) z^\omega \overline{w}^\omega \\= & {} \sum _{\omega }c_{\omega }^{-1} z^\omega \overline{w}^\omega . \end{aligned}$$

The rearranging of the sum above is justified as it absolutely converges. Indeed, using the Cauchy-Schwarz inequality we have

$$\begin{aligned}{} & {} \sum _{j=0}^\infty \left( {\begin{array}{c}j+n-1\\ n-1\end{array}}\right) \sum _{|\omega |=j} \left( {\begin{array}{c}j\\ \omega _1, \omega _2,\ldots , \omega _n\end{array}}\right) |z^\omega \overline{w}^\omega |\nonumber \\{} & {} \quad = \sum _{j=0}^\infty \left( {\begin{array}{c}j+n-1\\ n-1\end{array}}\right) \left( \sum _{k=1}^n |z_k||w_k|\right) ^j \le \sum _{j=0}^\infty \left( {\begin{array}{c}j+n-1\\ n-1\end{array}}\right) (|z||w|)^j\nonumber \\{} & {} \quad = \frac{1}{(1-|z||w|)^n}, \end{aligned}$$

(6)

as $|z||w|<1$. Thus, again using absolute convergence, we have for $z\in B$ and $\zeta \in S$

$$\begin{aligned} P(z,\zeta )=\frac{C(z,\zeta )C(\zeta ,z)}{C(z,z)}=C(z,z)^{-1}\sum _{\upsilon }\sum _{\omega }c_{\omega }^{-1}c_{\upsilon }^{-1}z^\omega \overline{\zeta }^\omega \zeta ^\upsilon \overline{z}^\upsilon , \end{aligned}$$

(7)

where each sum is taken over all multi-indices $\omega$ and $\upsilon$. Furthermore, using a similar computation as in Inequality (6), and using the assumption that $f\in L^p(S,\sigma )\subseteq L^1(S,\sigma )$ we have

$$\begin{aligned} \int _{S} \sum _{\upsilon }\sum _{\omega }c_{\omega }^{-1}c_{\upsilon }^{-1}\left| z^\omega \overline{\zeta }^\omega \zeta ^\upsilon \overline{z}^\upsilon f(\zeta )\right| d\sigma (\zeta )\le & {} \int _{S} \frac{1}{(1-|z||\zeta |)^{2n}}|f(\zeta )|d\sigma (\zeta )\\\le & {} \frac{1}{(1-|z|)^{2n}} \int _{S}|f(\zeta )|d\sigma (\zeta ) <\infty . \end{aligned}$$

Thus, by Equation (7) and Fubini’s Theorem we have

$$\begin{aligned} P[f](z) = C(z,z)^{-1}\sum _{\upsilon }\sum _{\omega }c_{\omega }^{-1}c_{\upsilon }^{-1}z^\omega \overline{z}^\upsilon \int _{S}\zeta ^\upsilon \overline{\zeta }^\omega f(\zeta )d\sigma (\zeta ). \end{aligned}$$

(8)

By Equation (3), the integrals of $\int _{S}\zeta ^\upsilon \overline{\zeta }^\omega f(\zeta )d\sigma (\zeta )$ vanish except when $\upsilon _j\le \omega _j$ for each $1\le j\le k$. That is, the integrals vanish except when $\omega =\upsilon +\lambda$ for some multi-index $\lambda$. Thus, Equation (8) becomes

$$\begin{aligned} P[f](z)= & {} C(z,z)^{-1}\sum _{\upsilon } \sum _{\lambda }c_{\upsilon }^{-1}z^{\upsilon +\lambda }\overline{z}^\upsilon \left( c_{\upsilon +\lambda }^{-1}\int _{S}\zeta ^\upsilon \overline{\zeta }^{\upsilon +\lambda } f(\zeta )d\sigma (\zeta )\right) . \end{aligned}$$

Using Equation (4) in the summation above yields

$$\begin{aligned} P[f](z)= & {} C(z,z)^{-1}\sum _{\upsilon }\sum _{\lambda }c_{\upsilon }^{-1}z^{\upsilon +\lambda }\overline{z}^\upsilon \left( c_{\lambda }^{-1}\int _{S}\overline{\zeta }^{\lambda } f(\zeta )d\sigma (\zeta )\right) \\= & {} C(z,z)^{-1}\left( \sum _{\upsilon }c_{\upsilon }^{-1}z^{\upsilon }\overline{z}^\upsilon \right) \left( \int _{S}\sum _{\lambda }c_{\lambda }^{-1}z^{\lambda } \overline{\zeta }^{\lambda } f(\zeta )d\sigma (\zeta ) \right) \\= & {} C(z,z)^{-1}C(z,z)\int _{S} C(z,\zeta )f(\zeta )d\sigma (\zeta ) = C[f](z), \end{aligned}$$

where we can use the Dominated Convergence Theorem with dominating function $\frac{1}{(1-|z||\zeta |)^n}|f(\zeta )|$ to justify swapping the sum and the integral in the second-to-last line, and used Equation (5) in the last line, completing the proof. $\square$

4 Closing Remarks: Can Theorem 4 be Generalized?

In the case of one complex variable, Hardy spaces can be defined on a variety of different subsets of $\mathbb {C}$. In particular, in the case that $\Omega \subseteq \mathbb {C}$ is a simply-connected open set with some minimal smoothness assumptions on its boundary $\partial \Omega$, then the Hardy spaces $H^p(\Omega )$ and $H^p(\partial \Omega )$ have several equivalent definitions (see [6]). In this general setting, the statement of Theorem 2 still holds with $\partial \Omega$ replacing $S^1$ (see [3, Thm. 10.4]). So while one might hope that Theorem 4 could carry over to Hardy spaces of a wide class of open subsets of $\mathbb {C}^n$, there are potential difficulties in proving it. For example, the proof of the generalization of Theorem 2 in [3] uses the Riemann Mapping Theorem and the Riemann Mapping Theorem does not hold in $\mathbb {C}^n$ if $n>1$ (see the Corollary to [1, Thm. 2.2.4]). Also, the technical machinery needed for Hardy spaces on more general subsets of $\mathbb {C}^n$ for $n>1$ is gnarly (see [7] or [8, Ch. 8]). So even if Theorem 4 is true in a more general setting, it will likely not be a short proof like we have presented here. In any case, the main point of this note is that boundary traces of holomorphic functions form a very special club, a fact that is almost certainly true for more exotic domains in $\mathbb {C}^n$ than just the ball B.