Norm convergence of partial sums of $H^1$ functions

TL;DR

The paper proves that partial sums of $H^1$ functions do not converge in $H^1$ norm but do converge in the Bergman norm, extending the result to higher dimensions and providing a new proof of non-convergence.

Contribution

It establishes the convergence of partial sums in the Bergman norm for $H^1$ functions and extends classical results to complex Reinhardt domains.

Findings

Partial sums of $H^1$ functions do not converge in $H^1$ norm.

Partial sums always converge in the Bergman norm $A^1$.

Extension of results to Reinhardt domains in $ extbf{C}^n$.

Abstract

A classical observation of Riesz says that truncations of a general $\sum_{n=0}^\infty a_n z^n$ in the Hardy space $H^1$ do not converge in $H^1$. A substitute positive result is proved: these partial sums always converge in the Bergman norm $A^1$. The result is extended to complete Reinhardt domains in $\C^n$. A new proof of the failure of $H^1$ convergence is also given.

Equations

$$S_{N}f(z):=\sum _{k=0}^N{a}_k{z}^k⟶f(z)$$

$${\Vert f\Vert }_{A^p(U)}^p={\int }_U|f{|}^{p}dV<\infty ,$$

$${\Vert f\Vert }_{H^p(U)}^p=\underset{0\le r<1}{\sup }\frac{1}{2\pi }{\int }_0^{2\pi }{\left|f(r{e}^{i\theta })\right|}^{p}d\theta <\infty ,$$

$${\Vert S_{N}f-f\Vert }_{A^1(U)}⟶0\text{as }N\to \infty .$$

$$I_c(z)={\int }_0^{2\pi }\frac{1}{{\left|1-z{e}^{-i\theta }\right|}^{1+c}}d\theta .$$

$$I_c(z)\approx \frac{1}{(1-{\left|z\right|}^2{)}^c}\text{ if }c>0\text{ and }{I}_0(z)\approx \log \frac{1}{1-{\left|z\right|}^2},$$

$$f_a(z)=\sum _{k=0}^{\infty }(1-{\left|a\right|}^2)(k+1)(\overline{a}z{)}^k.$$

$$\begin{split}S_{N}f_{a}(z)&=\sum_{k=0}^{N}(1-\left|a\right|^{2})(k+1)(\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz)^{k}=(1-\left|a\right|^{2})\sum_{k=0}^{N}\frac{d}{dt}\left(t^{k+1}\right)_{|_{t=\bar{a}z}}\\ &=(1-\left|a\right|^{2})\bigg{[}\frac{1-(\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz)^{N+2}}{(1-\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz)^{2}}-\frac{(N+2)(\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz)^{N+1}}{1-\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz}\bigg{]}\\ &=T_{1}+T_{2}.\end{split}$$

$$\frac{1}{2\pi }{\int }_0^{2\pi }\frac{(1-{\left|a\right|}^2)\left|1-(\overline{a}{e}^{i\theta }{)}^{N+2}\right|}{{\left|1-\overline{a}{e}^{i\theta }\right|}^2}d\theta \le \frac{2}{2\pi }{\int }_0^{2\pi }\frac{(1-{\left|a\right|}^2)}{{\left|1-\overline{a}{e}^{i\theta }\right|}^2}d\theta <\infty$$

$$\frac{(1-{\left|a\right|}^2)}{2\pi }{\int }_0^{2\pi }\frac{(N+2){\left|\overline{a}{e}^{i\theta }\right|}^{N+1}}{\left|1-\overline{a}{e}^{i\theta }\right|}d\theta \gtrsim (1-{\left|a\right|}^2){\left|a\right|}^{N+1}(N+2)\log \frac{1}{1-\left|a\right|}$$

$${\Vert T_{k}x\Vert }_Y$$

$${\int }_{U_r}\left|\sum _{j=0}^N{a}_j{z}^j\right|dV$$

$$\displaystyle=\frac{1}{2\pi}\int_{U_{r}}\left|\int_{\partial U_{R}}f(\xi)\sum_{j=0}^{N}\frac{1}{\xi}\bigg{(}\frac{z}{\xi}\bigg{)}^{j}d\xi\right|dV(z)$$

$$\displaystyle=\frac{1}{2\pi}\int_{U_{r}}\left|\int_{\partial U_{R}}f(\xi)\frac{1-\big{(}\frac{z}{\xi}\big{)}^{N+1}}{\xi-z}d\xi\right|dV(z)$$

$$\displaystyle\lesssim\int_{U_{r}}\int_{\partial U_{R}}\left|f(\xi)\right|\left|\frac{1-\big{(}\frac{z}{\xi}\big{)}^{N+1}}{\xi-z}\right|d\left|\xi\right|dV(z)=I.$$

$$I\lesssim {\int }_{U_r}{\int }_{\partial U_R}\left|f(\xi )\right|\frac{1}{\left|\xi -z\right|}d\left|\xi \right|dV(z)={\int }_{\partial U_R}\left|f(\xi )\right|{\int }_{U_r}\frac{1}{\left|\xi -z\right|}dV(z)d\left|\xi \right|,$$

$${\int }_{U_r}\frac{1}{\left|\xi -z\right|}dV(z)<{\int }_{U_{(\xi ;2)}}\frac{1}{\left|\xi -z\right|}dV(z)={\int }_0^{2\pi }{\int }_0^2\frac{1}{s}sdsd\sigma \lesssim 1,$$

$${\int }_{U_r}\left|\sum _{j=0}^N{a}_j{z}^j\right|dV\lesssim {\int }_{\partial U_R}\left|f(\xi )\right|d\left|\xi \right|\lesssim {\Vert f\Vert }_{H^1(U)},$$

$${\Vert S_{N}f\Vert }_{A^1(U)}\lesssim {\Vert f\Vert }_{H^1(U)}.$$

$$S_{N}f(z)=:\sum _{{|\alpha |}_{\infty }\le N}{b}_{\alpha }{z}^{\alpha }.$$

$${\Vert f\Vert }_{H^p(\mathcal{R})}^p=:\underset{\mathbf{r}\in \mathcal{F}}{\sup }{\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{r}\cdot e^{i\mathbf{\theta }})\right|}^{p}d{\theta }_1\cdots d{\theta }_n<\infty ,$$

$${\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{r}\cdot e^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }\approx {\int }_{\mathbf{r}\cdot {\mathbb{T}}^n}\left|f(\xi )\right|d\left|{\xi }_1\right|\cdots d\left|{\xi }_n\right|,$$

$${\int }_0^{2\pi }{\left|h(r{e}^{i\theta })\right|}^{p}d\theta \le {\int }_0^{2\pi }{\left|h(R{e}^{i\theta })\right|}^{p}d\theta ,h\in \mathcal{O}(U),$$

$${\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{r}\cdot e^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }\le {\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{R}\cdot e^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }.$$

$${\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{r}\cdot e^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }$$

$$\le {\int }_{{\mathbb{T}}^{n-1}}\left({\int }_0^{2\pi }{\left|f(R_1{e}^{i{\theta }_1},r_2{e}^{i{\theta }_2},\dots ,r_n{e}^{i{\theta }_n})\right|}^{p}d{\theta }_1\right)d{\theta }_2\cdots d{\theta }_n$$

$$I(\mathbf{\sigma })={\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{\sigma }{e}^{i\mathbf{\theta }})-f(s\mathbf{\sigma }{e}^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }$$

$$\le 2^{p+1}{\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{\sigma }{e}^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }.$$

$${\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{\sigma }{e}^{i\mathbf{\theta }})-f_s(\mathbf{\sigma }{e}^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }<ϵ.$$

$$f_{\rho }(z)=\sum _{\alpha \in {\mathbb{N}}^n}\left(a_{\alpha }{\rho }^{|\alpha |}\right)z^{\alpha }=:\sum _{\alpha \in {\mathbb{N}}^n}{b}_{\alpha }(\rho )z^{\alpha }$$

$$\underset{z\in \mathcal{R}}{\sup }\left|f_{\rho }(z)-\sum _{|\alpha {|}_{\infty }\le M}{b}_{\alpha }(\rho )z^{\alpha }\right|<ϵ.$$

$${\int }_{{\mathbb{T}}^n}{\left|f(\mathbf{\sigma }{e}^{i\mathbf{\theta }})-Q(\mathbf{\sigma }{e}^{i\mathbf{\theta }})\right|}^{p}d\mathbf{\theta }$$

$$<2^p(ϵ+{ϵ}^p\cdot (2\pi {)}^n)$$

Full text

1. Introduction

Let $U\subset{\mathbb{C}}$ be the unit disc and ${\mathcal{O}}(U)$ denote the set of holomorphic functions on $U$. If $f\in{\mathcal{O}}(U)$, with power series $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$, then

[TABLE]

uniformly on compact subsets of $U$. If \big{(}X,\|\cdot\|_{X}\big{)}\subset{\mathcal{O}}(U) is a Banach space of functions, it is natural to ask whether $S_{N}f$ also converges to $f$ in the norm $\|\cdot\|_{X}$.

Two classically studied spaces, the Bergman and Hardy spaces, will be considered here. For $p>0$, the Bergman space $A^{p}(U)$ is the set of $f\in{\mathcal{O}}(U)$ such that

[TABLE]

$dV$ denoting Lebesgue measure on ${\mathbb{C}}$. The Hardy space $H^{p}(U)$ is the set of $f\in{\mathcal{O}}(U)$ such that

[TABLE]

$d\theta$ denoting Lebesgue measure on $[0,2\pi]$. For $1\leq p<\infty$, $A^{p}(U)$ and $H^{p}(U)$ are Banach spaces.

When $p=2$, norm convergence of $S_{N}f$ in either the Bergman or Hardy norm is elementary. For $f\in H^{2}(U)$, orthogonality of $\left\{e^{ik\theta}\right\}$ on $\partial U$ shows $\|f\|^{2}_{H^{2}(U)}=\sum_{k=0}^{\infty}|a_{k}|^{2}$. Orthogonality also shows $\left\|S_{N}f-f\right\|^{2}_{H^{2}(U)}=\sum_{k=N+1}^{\infty}\left|a_{k}\right|^{2}$, which tends to [math] as $N\to\infty$. Minor modifications of the argument hold when $A^{2}(U)$ replaces $H^{2}(U)$. When $1<p<\infty$ and $p\neq 2$, the result is the same as in the Hilbert space case but proving this is no longer elementary. Norm convergence of $S_{N}f$ in $H^{p}(U)$ for $1<p<\infty$ is considered classical; a proof is contained in [10] on pages 104–110. Convergence of $S_{N}f$ in $A^{p}(U)$ for the same range of $p$ is established by Zhu [19], utilizing the result on $H^{p}(U)$.

The focus in this paper is $p=1$. Our interest in this case stems from the widespread occurrence of $L^{1}$ holomorphic functions, not as an endpoint consideration. For $A^{1}(U)$ and $H^{1}(U)$, it is known that partial sum approximation fails; this is also addressed in [10] and [19]. For $A^{1}(U)$, [19] gives an explicit family of functions $g_{\alpha}\in A^{1}(U)$, $\alpha\in U$, such that $\left\|S_{N}g_{\alpha}\right\|_{A^{1}}$ is not bounded uniformly in $\alpha$ and $N$. The fact that $\left\|S_{N}f-f\right\|_{A^{1}}\not\to 0$ for all $f\in A^{1}(U)$ then follows from the uniform boundedness principle. For $H^{1}(U)$, the proofs in print are somewhat oblique. In [10], it is first shown that $S_{N}f$ approximating $f$ in $H^{1}$ is equivalent to $h^{1}$ boundedness of the harmonic conjugation operator, where $h^{1}(U)$ denotes the harmonic functions with the norm $\|\cdot\|_{H^{1}}$. Both properties are also shown to be equivalent to the $L^{1}$ boundedness of the Szegő projection on $U$. The fact that harmonic conjugation is not bounded on $h^{1}(U)$ – evidenced, e.g., by the Poisson kernel, as discussed in [10] and [3] – then implies that partial sum approximation on $H^{1}(U)$ fails.

The role of harmonic conjugation in this argument does not readily generalize to domains in ${\mathbb{C}}^{n}$ or to non-simply connected domains in the plane. Consequently a new proof of the failure of $H^{1}$-approximation of partial sums, in the spirit of [19], is given in the Section 2.1.

Using polar coordinates, it is easy to see $H^{p}(U)\subset A^{p}(U)$. The main purpose of this paper is a substitute positive result for the failure of $H^{1}$ partial sum approximation: partial sums of $f\in H^{1}(U)$ are norm convergent, but in the weaker norm $A^{1}(U)$.

Theorem 1.2**.**

If $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\in H^{1}(U)$ and $S_{N}f(z)=\sum_{k=0}^{N}a_{k}z^{k}$, then

[TABLE]

In the Section 3, Theorem 1.2 is extended to give an analogous result on complete bounded Reinhardt domains ${\mathcal{R}}\subset{\mathbb{C}}^{n}$.

There are other substitutes for the failure of $H^{1}(U)$ partial sum approximation in the literature: [16, 17, 2] on the $H^{1}(U)$ boundedness of the Cesàro operator, [18, 14] on certain Toeplitz and Hankel operators, [9, 12] on boundedness of the Hausdorff operator with particular choices of Borel measure, and [13] on boundedness of the Libera operator from $H^{1}(U)$ to $H^{p}(U)$ with $0<p<1$. Unlike Theorem 1.2, these results involve modifications of $S_{N}f$.

As notational shorthand, $\left|a\right|\lesssim\left|b\right|$ will mean there exists a constant $C>0$ such that $\left|a\right|\leq C\left|b\right|$, with $C$ independent of specified parameters. Let $\left|a\right|\approx\left|b\right|$ mean both $\left|a\right|\lesssim\left|b\right|$ and $\left|b\right|\lesssim\left|a\right|$ hold.

2. The result on $U$

For $h(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\in{\mathcal{O}}(U)$ and $N\in{\mathbb{Z}}^{+}$, let $S_{N}h(z)=\sum_{k=0}^{N}a_{k}z^{k}$ denote the $N$-th partial sum of the power series of $h$. Since power series are unique, call these polynomials partial sums of $h$ for short.

2.1. Failure of norm convergence in $H^{1}(U)$

A family of integral estimates is used in the proof of Theorem 2.2 below.

Lemma 2.1**.**

For $z\in U$ and $c$ real, define

[TABLE]

If $c<0$ then $I_{c}\in L^{\infty}(U)$. Furthermore,

[TABLE]

for constants independent of $z\in U$.

Proof.

See [15] Proposition 1.4.10, [8], or [20] for the standard proof involving asymptotics of the Gamma function. See [4, 5, 6] for alternate, elementary proofs that extend to other singular integrands. ∎

Theorem 2.2**.**

There exists $g\in H^{1}(U)$ such that $S_{N}g$ does not converge in $H^{1}(U)$.

Proof.

For $a\in U$, define $f_{a}(z)=\frac{1-\left|a\right|^{2}}{(1-\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz)^{2}}$. By Lemma 2.1, $\left\|f_{a}\right\|_{H^{1}(U)}\lesssim 1$ with constant independent of $a$. The power series of $f_{a}$ is

[TABLE]

Consider the partial sum

[TABLE]

$\|T_{1}\|_{H^{1}}$ is uniformly bounded in $N$ and $a$, since

[TABLE]

by Lemma 2.1. $\|T_{2}\|_{H^{1}}$ is estimated

[TABLE]

for a constant independent of $N$ and $a$, by Lemma 2.1. Let $a=\frac{N}{N+1}$. The lower bound on $\|T_{2}\|_{H^{1}}$ goes to infinity as $N\rightarrow+\infty$. Thus $\left\|S_{N}f_{a}\right\|_{H^{1}}$ is unbounded as a function of $N$ and $a$. The uniform boundedness principle in contrapositive form gives the stated conclusion. ∎

Remark 2.3*.*

Holomorphic polynomials are dense in $H^{1}(U)$ (see Proposition 3.6), but Theorem 2.2 says the sequence of natural polynomials $\left\{S_{N}h\right\}$ does not approximate a general $h\in H^{1}(U)$. Is there a best association of $h\in H^{1}(U)$ to a sequence of holomorphic polynomials $\left\{p_{n}\right\}$ such that $\|p_{n}-h\|_{H^{1}}\to 0$? Several interpretations of “best” are possible; the authors are unaware of results in this direction.

2.2. Convergence in $A^{1}(U)$

Convergence of $S_{N}f$ can be reduced to a bound on the operator norm of $S_{N}$. The following is a slight generalization of [19, Proposition 1].

Lemma 2.4**.**

Let $T_{k}$, $k=1,2,\dots$, be a sequence of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Suppose that there is a dense subset $D$ of $X$ such that for each $x\in D$, $T_{k}x\to 0$ in the norm of $Y$ as $k\to\infty$.

Then the following are equivalent

1. (i)

$\lim_{k\to\infty}\left\|T_{k}x\right\|_{Y}=0$* for each $x\in X$.* 2. (ii)

there is a $C>0$ such that for each $k$, we have $\left\|T_{k}\right\|_{\rm op}\leq C$.

$\left\|T_{k}\right\|_{\rm op}$* is the operator norm of $T_{k}:X\to Y$.*

Proof.

Assume (i). Then (ii) holds by the uniform boundedness principle.

Assume (ii). Fix $x\in X$ and $\epsilon>0$. Since $D$ is dense in $X$, there exists $p\in D$ such that $\left\|x-p\right\|_{X}<\frac{\epsilon}{2C}$. Therefore

[TABLE]

Choosing $k$ so large that $\left\|T_{k}p\right\|_{Y}<\frac{\epsilon}{2}$ yields (i). ∎

The key idea of the proof of Theorem 1.2 is to represent the coefficients in $S_{N}f$ as integrals, reducing the problem to estimates of geometric series.

Proof of Theorem 1.2.

For $b\in\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu$ and $\rho>0$, let $U(b;\rho)$ denote the disc centered at $b$ of radius $\rho$. Let $U_{r}=U(0;r)$. For each $U_{r}$, choose $U_{R}$ such that $0<r<R<1$. If $f(z)=\sum_{j=0}^{\infty}a_{j}z^{j}\in H^{1}(U)$, the Cauchy integral formula gives $a_{j}=\frac{1}{2\pi i}\int_{\partial U_{R}}\frac{f(\xi)}{\xi^{j+1}}d\xi$. Therefore

[TABLE]

Since $\xi\in\partial U_{R}$ and $z\in U_{r}$, $\left|\frac{z}{\xi}\right|<\frac{r}{R}<1$. Thus Fubini’s theorem implies

[TABLE]

with constant independent of $N$.

For any $\xi\in\partial U_{R}$ fixed, note $U\subset U(\xi;2)$. Letting $z=\xi+se^{i\sigma}$,

[TABLE]

with constant independent of $\xi$. Thus (2.5) implies

[TABLE]

with constant independent of $N$. Since $\lim_{r\rightarrow 1}\left\|S_{N}f\right\|_{A^{1}(U_{r})}=\left\|S_{N}f\right\|_{A^{1}(U)}$,

[TABLE]

Let $T_{N}=S_{N}-\text{id}$, $X=H^{1}(U)$, $Y=A^{1}(U)$, and $D=\{\text{holomorphic polynomials}\}$ in Lemma 2.4. Note that for any $p\in D$, $T_{N}p\equiv 0$ if $N\geq\text{deg}(p)$. Lemma 2.4 says that (2.6) implies $\left\|S_{N}f-f\right\|_{A^{1}(U)}\to 0$ as $N\to\infty$. ∎

3. Several variable extension

A domain $\Omega\subset{\mathbb{C}}^{n}$ is a complete Reinhardt domain if $(z_{1},\dots z_{n})\in\Omega$ implies $(\lambda_{1}z_{1},\dots,\lambda_{n}z_{n})\in\Omega$ for all $\lambda_{k}\in{\mathbb{C}}$ with $|\lambda_{k}|<1$, $k=1,\dots n$.

Let $\mathcal{R}$ be a bounded complete Reinhardt domain in ${\mathbb{C}}^{n}$ and $\mathcal{O}(\mathcal{R})$ denote the set of holomorphic functions on $\mathcal{R}$. Each $f\in\mathcal{O}(\mathcal{R})$ has a power series expansion $f(z)=\sum_{\alpha\in{\mathbb{N}}^{n}}a_{\alpha}z^{\alpha}$, using standard multi-index notation, converging uniformly on compact subsets of $\mathcal{R}$.

A choice of partial sum of $f\in\mathcal{O}(\mathcal{R})$ is required, since the index set is an $n$-dimensional lattice. Let $\left|\alpha\right|_{\infty}=\max\{\alpha_{j}:\alpha=(\alpha_{1},\cdots,\alpha_{n})\}$ if $\alpha\in{\mathbb{N}}^{n}$. For $f(z)=\sum_{\alpha}b_{\alpha}z^{\alpha}\in\mathcal{O}(\mathcal{R})$, define

[TABLE]

Call $S_{N}f$ the square partial sum of $f$.

Let ${\mathbb{T}}^{n}=\{z\in{\mathbb{C}}^{n}:\left|z_{j}\right|=1,j=1,\cdots,n\}$ and $U^{n}=\{z\in{\mathbb{C}}^{n}:\left|z_{j}\right|<1,j=1,\cdots,n\}$ denote the unit torus and polydisc, respectively. In the sequel, quantities depending on several variables are sometimes written in bold typeface, scalar quantities in regular, to avoid ambiguity. For $\bm{r}=(r_{1},\cdots,r_{n})\in\left({\mathbb{R}}^{n}\right)^{+}$, $\bm{\theta}=(\theta_{1},\cdots,\theta_{n})\in{\mathbb{R}}^{n}$, and $\bm{z}=(z_{1},\dots z_{n})\in{\mathbb{C}}^{n}$ let $\bm{r}\cdot e^{i\bm{\theta}}=(r_{1}e^{i\theta_{1}},\cdots,r_{n}e^{i\theta_{n}})$ and $\bm{r}\cdot\bm{z}=(r_{1}z_{1},\dots r_{n}z_{n})$. Dilations of ${\mathbb{T}}^{n}$ and $U^{n}$ are denoted $\bm{r}\cdot{\mathbb{T}}^{n}=\{\bm{r}\cdot e^{i\bm{\theta}}:e^{i\bm{\theta}}\in{\mathbb{T}}^{n}\}$ and $\bm{r}\cdot U^{n}=\{\bm{r}\cdot\bm{z}:\bm{z}\in U^{n}\}$.

3.1. Hardy spaces of Reinhardt domains

There is not a canonical definition of Hardy spaces on a general domain, especially in several variables. See [7, 11, 1] for a few of the definitions used. On a Reinhardt domain, the following is reasonable.

Definition 3.2**.**

Let $0<p<\infty$. Say $f\in H^{p}(\mathcal{R})$ if $f\in\mathcal{O}(\mathcal{R})$ and

[TABLE]

where $\mathcal{F}=\left\{\bm{r}:\bm{r}\cdot{\mathbb{T}}^{n}\subset\mathcal{R}\right\}$.

An alternate form of the integrals in Definition 3.2 is used in section 3.2. As shorthand, let $d\bm{\theta}=d\theta_{1}\cdots d\theta_{n}$. Then

[TABLE]

where $d\left|\xi_{k}\right|$ denotes arc length measure.

3.1.1. Density of polynomials in $H^{p}({\mathcal{R}})$

A fundamental fact about holomorphic functions on $U$ is

[TABLE]

if $r\leq R<1$. See [3, Theorem 1.5].

A version of this monotonicity holds on ${\mathcal{R}}$. For $\bm{r}=(r_{1},\dots,r_{n})$ and $\bm{R}=(R_{1},\dots,R_{n})$, write $\bm{r}\prec\bm{R}$ to denote $r_{k}\leq R_{k}$ for all $k=1,\dots,n$.

Lemma 3.4**.**

If $\bm{r},\bm{R}\in\mathcal{F}$, $\bm{r}\prec\bm{R}$, and $f\in{\mathcal{O}}({\mathcal{R}})$,

[TABLE]

Proof.

[TABLE]

by (3.3). Iteratively applying this to the integrals $d\theta_{2}\cdots d\theta_{n}$ gives (3.5). ∎

The density of holomorphic polynomials in $H^{p}(U)$ is well-known, see [3, Theorem 3.3]. This fact also holds on complete Reinhardt domains in ${\mathbb{C}}^{n}$:

Proposition 3.6**.**

If ${\mathcal{R}}\subset{\mathbb{C}}^{n}$ is a bounded complete Reinhardt domain and $0<p<\infty$, the set of holomorphic polynomials is dense in $H^{p}({\mathcal{R}})$.

Proof.

Let $f(z)=\sum_{\alpha\in{\mathbb{N}}^{n}}a_{\alpha}z^{\alpha}\in H^{p}({\mathcal{R}})$. For $0<s<1$, define $f_{s}(\bm{z})=f(s\bm{z})$.

For any $\bm{\sigma}\in{\mathcal{F}}$, consider $I(\bm{\sigma})=:\int_{{\mathbb{T}}^{n}}\left|f(\bm{\sigma}e^{i\bm{\theta}})-f_{s}(\bm{\sigma}e^{i\bm{\theta}})\right|^{p}\,d\bm{\theta}$. Lemma 3.4 implies

[TABLE]

In particular $I(\bm{\sigma})\lesssim\|f\|^{p}_{H^{p}}$.

Let $\epsilon>0$. Since \lim_{s\to 1}\big{(}f(\bm{\sigma}e^{i\bm{\theta}})-f(s\bm{\sigma}e^{i\bm{\theta}})\big{)}=0 for all $\bm{\theta}$, the dominated convergence theorem gives $\rho$ such that for all $\rho\leq s<1$

[TABLE]

Note $\rho$ depends on both $\bm{\sigma}$ and $\epsilon$.

The function $f_{\rho}\in{\mathcal{O}}\left(\frac{1}{\rho}{\mathcal{R}}\right)$ and $\bar{{\mathcal{R}}}\subset\subset\frac{1}{\rho}{\mathcal{R}}$. Since the power series

[TABLE]

converges uniformly on $\bar{\mathcal{R}}$, there exists $M=M(\rho,\epsilon)$ such that

[TABLE]

Let $Q(z)=\sum_{|\alpha|_{\infty}\leq M}b_{\alpha}(\rho)z^{\alpha}$. Then

[TABLE]

by (3.7) and (3.8). This holds for any $\bm{\sigma}\in{\mathcal{F}}$ and $\epsilon>0$ was arbitrary, so the claimed density holds.

∎

3.2. Partial sums in $H^{1}({\mathcal{R}})$ converge in $A^{1}({\mathcal{R}})$

The definition of Bergman spaces is canonical: $f\in A^{p}(\mathcal{R})$ if

[TABLE]

Theorem 1.2 generalizes to the pair $H^{1}(\mathcal{R}),A^{1}(\mathcal{R})$. As in one variable, the key fact is an estimate on the operator norm of $S_{N}$.

Proposition 3.9**.**

There exists a constant independent of $N\in{\mathbb{Z}}^{+}$ and $f\in H^{1}(\mathcal{R})$ such that

[TABLE]

where $S_{N}f$ is the square partial sum (3.1).

Proof.

Choose $\bm{r}=(r_{1},\cdots,r_{n})\in\mathcal{F},\bm{R}=(R_{1},\cdots,R_{n})\in\mathcal{F}$ such that $r_{j}<R_{j},j=1,\cdots,n$. For $f(z)=\sum_{\alpha}a_{\alpha}z^{\alpha}\in H^{1}(\mathcal{R})$, the Cauchy integral formula implies

[TABLE]

On $\bm{r}\cdot U^{n}$, the $A^{1}$ norm of $S_{N}f$ is

[TABLE]

Since $\xi\in\bm{R}\cdot{\mathbb{T}}^{n}$, $z\in\bm{r}\cdot U^{n}$, $\left|\frac{z_{j}}{\xi_{j}}\right|<1$ for each $j$. Therefore

[TABLE]

by Fubini’s theorem.

Since $\mathcal{R}$ is bounded, for $\bm{r}=(r_{1},\dots,r_{n})\in\mathcal{F}$, the argument that gave (2.2) shows that for each $r_{j}$, $\int_{r_{j}U}\frac{1}{\left|\xi_{j}-z_{j}\right|}dV(z_{j})\lesssim 1$ for a constant independent of $\xi\in\bm{R}\cdot{\mathbb{T}}^{n}$. Consequently

[TABLE]

Inserting this into (3.2) yields

[TABLE]

with all constants independent of $N$ and $f$.

Finally note that for any $f\in A^{1}(\mathcal{R})$ there are sequences $\{\bm{r}_{k}\}_{k=1}^{\infty}\subseteq\mathcal{F}$ such that $\left\|f\right\|_{A^{1}(\mathcal{R})}=\lim_{k\rightarrow\infty}\int_{\bm{r}_{k}\cdot U^{n}}\left|f\right|dV(z)$, by the dominated convergence theorem. Therefore

[TABLE]

as claimed. ∎

Convergence of $S_{N}f$ is $A^{1}({\mathcal{R}})$ now follows as in the conclusion to the proof of Theorem 1.2. Let $T_{N}=S_{N}-\text{id}$, $X=H^{1}({\mathcal{R}})$, $Y=A^{1}({\mathcal{R}})$, and $D=\{\text{holomorphic polynomials}\}$. Note $T_{N}p\equiv 0$ for any $p\in D$ if $N$ is large enough. The hypotheses of Lemma 2.4 are thus satisfied. Proposition 3.9 and this lemma yield

Corollary 3.11**.**

If $f\in H^{1}({\mathcal{R}})$ and $S_{N}f$ is given by (3.1), then

[TABLE]