§9.4 Hardy Spaces

9.4 Hardy Spaces

(The knowledge of measure spaces or measure theory will not be required for the understanding of this section, but rather for purely motivational purposes.)

The definition of a Lebesgue space or $𝐿^{𝑝}$ space ( $0 < 𝑝 \leq \infty$ ) in real analysis approximately defines the $𝐿^{𝑝} (𝑆, 𝜇)$ space for a given space (a measure space) $(𝑆, Σ, 𝜇)$ (for $1 \leq 𝑝 < \infty$ ), by

𝐿^{𝑝} (𝑆, 𝜇) = {𝑓 : 𝑆 \to ℂ measurable : {‖ 𝑓 ‖}_{𝐿^{𝑝} (𝑆, 𝜇)} < \infty},

where the $𝐿^{𝑝}$ norm of $𝑓$ is given by

{‖ 𝑓 ‖}_{𝐿^{𝑝} (𝑆, 𝜇)} = {(\int_{𝑆} {| 𝑓 |}^{𝑝} d 𝜇)}^{\frac{1}{𝑝}}

Moreover,

{‖ 𝑓 ‖}_{𝐿^{\infty} (𝑆, 𝜇)} = {\begin{cases} ess. sup | 𝑓 | & if 𝜇 (𝑆) = 0 \\ 0 & otherwise \end{cases}

(where $ess. sup$ is the essential supremum, or the infimum of supremums over subsets of $𝑆$ omitting a set of measure $0$ ). (There is also the notion of a normalized $𝐿^{𝑝}$ norm that has an additional factor $\frac{1}{𝜇 (𝑆)}$ in front of the integral. In the case where we have a probability space, where $𝜇 (𝑆) = 1$ , then the normalization gives the same as without.)

Note that usually we are usually concerned only with the case where $𝑝 \geq 1$ , as strictly speaking, ${‖ 𝑓 ‖}_{𝐿^{𝑝}}$ no longer satisfies the triangle inequality otherwise, although the same notation of a “norm” is still used for convenience.

In $𝔻$ , we provide the following (somewhat) analogous formulation:

Definition 9.4.1 (Hardy Space).

For a holomorphic function $𝑓 : 𝔻 \to ℂ$ , define the $𝐻^{𝑝}$ norm (for $0 < 𝑝 < \infty$ ) to be

{‖ 𝑓 ‖}_{𝐻^{𝑝}} = sup_{𝑟 \in (0, 1)} {(\frac{1}{2 π} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃)}^{\frac{1}{𝑝}},

and

{‖ 𝑓 ‖}_{𝐻^{\infty}} = sup_{𝑟 \in (0, 1)} sup_{𝜃 \in [0, 2 π]} | 𝑓 (𝑟 e^{i 𝜃}) | = sup_{𝑧 \in 𝔻} | 𝑓 (𝑧) | .

Then the space

𝐻^{𝑝}

is simply the set of such functions with a finite Hardy

𝑝

-norm. Moreover

𝐻^{\infty}

class is simply the space of bounded holomorphic functions on

𝔻

Immediately, we have

Theorem 9.4.2.

Any $𝑓 \in 𝐻^{\infty}$ with $𝑓 ≢ 0$ can be factorized by

𝑓 (𝑧) = 𝐹 (𝑧) 𝐵 (𝑧), (9.4 . 1)

where

𝐵

is a Blaschke Product (Theorem 4.4.4.5

Theorem 4.4.4.5 Theorem 4.4.4.5

with an additional

𝑧^{𝑚}

factor) constructed on the zeros of

𝑓

; and

𝐹

is a bounded, non-vanishing holomorphic function on

𝔻

such that

{‖ 𝑓 ‖}_{𝐻^{\infty}} = {‖ 𝐹 ‖}_{𝐻^{\infty}}

Proof.

The assertions follow immediately from an application of Lemma 4.4.4.4

Lemma 4.4.4.4 Lemma 4.4.4.4

and Corollary 4.4.4.5.1

Corollary 4.4.4.5.1 Corollary 4.4.4.5.1

. ⁠

□

In a series of papers published by Godfrey H. Hardy and Frigyes Riesz, it was found that many of the properties of $𝐻^{\infty}$ functions apply to the weaker ( $𝐻^{\infty} \subseteq 𝐻^{𝑝}$ ) class of $𝐻^{𝑝}$ for finite $𝑝$ , including that of Theorem 9.4.2 $Theorem 9.4.2 Theorem 9.4.2$ .

Proposition 9.4.3.

For

𝑝 \in ℝ_{> 0}

and

𝑓 \in 𝐻^{𝑝}

{| 𝑓 |}^{𝑝}

is subharmonic.

Proof.

For each point $𝑧 \in 𝔻 ∖ {𝑧 \in 𝔻 : 𝑓 (𝑧) = 0}$ , $log | 𝑓 (𝑧) |$ is locally harmonic as it is the real part of a locally holomorphic function. Therefore, $𝑧 \mapsto exp [𝑝 log | 𝑓 (𝑧) |]$ , the composition between a non-decreasing convex function and a harmonic function, is subharmonic on $𝔻 ∖ {𝑧 \in 𝔻 : 𝑓 (𝑧) = 0}$ by Theorem 7.3.8 $Theorem 7.3.8 Theorem 7.3.8$ . Then at each point at which ${| 𝑓 |}^{𝑝}$ does not vanish, ${| 𝑓 |}^{𝑝}$ satisfies the local sub-mean-value property.

Since ${| 𝑓 |}^{𝑝}$ vanishes at each point of ${𝑧 \in 𝔻 : 𝑓 (𝑧) = 0}$ , it satisfies the local sub-mean-value property at such points (since ${| 𝑓 |}^{𝑝} \geq 0$ everywhere).

By Theorem 7.3.5 $Theorem 7.3.5 Theorem 7.3.5$ , ${| 𝑓 |}^{𝑝}$ is subharmonic on $𝔻$ . ⁠ $□$

Lemma 9.4.4 (Hardy).

For any $𝑓 : 𝔻 \to ℂ$ holomorphic and any $𝑝 \in ℝ_{> 0}$ , for any two $0 < 𝑟_{1} < 𝑟_{2} < 1$ ,

\int_{0}^{2 π} {| 𝑓 (𝑟_{1} e^{i 𝜃}) |}^{𝑝} d 𝜃 \leq \int_{0}^{2 π} {| 𝑓 (𝑟_{2} e^{i 𝜃}) |}^{𝑝} d 𝜃 .

Proof.

By Theorem 7.4.2 $Theorem 7.4.2 Theorem 7.4.2$ , $\exists! 𝑢 \in 𝐶 (𝜕 𝐷 (0, 𝑟_{2}))$ such that $𝑢$ is harmonic in $𝐷 (0, 𝑟_{2})$ and that $𝑢 \equiv {| 𝑓 |}^{𝑝}$ on $𝜕 𝐷 (0, 𝑟_{2})$ . By the subharmonicity of ${| 𝑓 |}^{𝑝}$ , $𝑢 \geq {| 𝑓 |}^{𝑝}$ on $\overline{𝐷 (0, 𝑟_{2})}$ , and hence on $\overline{𝐷 (0, 𝑟_{1})}$ .

Therefore,

\begin{aligned} \int_{0}^{2 π} {| 𝑓 (𝑟_{2} e^{i 𝜃}) |}^{𝑝} d 𝜃 & = \int_{0}^{2 π} 𝑢 (𝑟_{2} e^{i 𝜃}) d 𝜃 \\ = 2 π 𝑢 (0) = \int_{0}^{2 π} 𝑢 (𝑟_{1} e^{i 𝜃}) d 𝜃 (mean-value property) \\ \geq \int_{0}^{2 π} {| 𝑓 (𝑟_{1} e^{i 𝜃}) |}^{𝑝} d 𝜃 . □ \end{aligned}

Theorem 9.4.5.

For $0 < 𝑝_{1} < 𝑝_{2} \leq \infty$ , $𝐻^{𝑝_{2}}$ is a proper subclass of $𝐻^{𝑝_{1}}$ :

𝐻^{𝑝_{2}} ⊊ 𝐻^{𝑝_{2}} .

Proof.

We aim to prove the existence of some function $𝑓$ which lies in $𝐻^{𝑝_{1}}$ but not $𝐻^{𝑝_{2}}$ . The function we concern ourselves with is

𝑓 : 𝑧 \mapsto {(1 - 𝑧)}^{- \frac{1}{𝑝}} = exp [- \frac{Log (1 - 𝑧)}{𝑝}],

for $𝑧 \in 𝔻$ , where $Log$ is the principal branch logarithm and $𝑝 \in (𝑝_{1}, 𝑝_{2})$ is fixed. The use of the principal branch is justified, as $𝑧 \mapsto 1 - 𝑧$ maps $𝔻$ to $𝐷 (1, 1) \subset ℂ ∖ 𝑅_{\leq 0}$ .

Since for any $𝑟 \in (0, 1)$ ,

\begin{aligned} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{1}} d 𝜃 & = \int_{0}^{2 π} {| 1 - 𝑟 e^{i 𝜃} |}^{- \frac{𝑝_{1}}{𝑝}} d 𝜃 \\ \leq \int_{0}^{2 π} {| ℑ𝔪 (1 - 𝑟 e^{i 𝜃}) |}^{- \frac{𝑝_{1}}{𝑝}} d 𝜃 \\ \leq \int_{0}^{2 π} {(𝑟 | sin 𝜃 |)}^{- \frac{𝑝_{1}}{𝑝}} d 𝜃 \\ = 4 𝑟^{- \frac{𝑝_{1}}{𝑝}} \int_{0}^{\frac{π}{2}} {(sin 𝜃)}^{- \frac{𝑝_{1}}{𝑝}} d 𝜃, \end{aligned}

and since $sin 𝜃 \geq \frac{2 𝜃}{π}$ for $𝜃 \in [0, \frac{π}{2}]$ ,

\begin{aligned} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{1}} d 𝜃 & = 4 {(\frac{2}{π})}^{- \frac{𝑝_{1}}{𝑝}} 𝑟^{- \frac{𝑝_{1}}{𝑝}} \int_{0}^{\frac{π}{2}} 𝜃^{- \frac{𝑝_{1}}{𝑝}} d 𝜃 \\ = 4 {(\frac{2}{π})}^{- \frac{𝑝_{1}}{𝑝}} 𝑟^{- \frac{𝑝_{1}}{𝑝}} {𝜃^{1 - \frac{𝑝_{1}}{𝑝}} |}_{0}^{\frac{π}{2}} \\ = 𝑐 𝑟^{- \frac{𝑝_{1}}{𝑝}} since 1 - \frac{𝑝_{1}}{𝑝} > 0 . \end{aligned}

Moreover, since $| 𝑓 |$ is bounded on $\overline{𝐷 (0, \frac{1}{2})}$ by some constant $𝑀$ ,

\begin{aligned} | \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{1}} d 𝜃 | & \leq max {2 π 𝑀^{𝑝_{1}}, sup_{𝑟 \in (\frac{1}{2}, 1)} 𝑐 𝑟^{- \frac{𝑝_{1}}{𝑝}}} \\ \leq max {2 π 𝑀^{𝑝_{1}}, 𝑐 2^{\frac{𝑝_{1}}{𝑝}}} < \infty, \end{aligned}

we then have $𝑓 \in 𝐻^{𝑝_{1}}$ . Moreover, observe that for all $𝑟 \in (0, 1)$ , if $𝑝_{2} \neq \infty$ ,

\begin{aligned} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{2}} d 𝜃 & = \int_{0}^{2 π} {| 1 - 𝑟 e^{i 𝜃} |}^{- \frac{𝑝_{2}}{𝑝}} d 𝜃 \\ \geq \int_{0}^{2 π} {(| ℑ𝔪 (1 - 𝑟 e^{i 𝜃}) | + | ℜ𝔢 (1 - 𝑟 e^{i 𝜃}) |)}^{- \frac{𝑝_{2}}{𝑝}} d 𝜃 \\ \geq 2 \int_{0}^{π} {[𝑟 (sin 𝜃 - cos 𝜃) + 1]}^{- \frac{𝑝_{2}}{𝑝}} d 𝜃 \\ = 2 \int_{0}^{π} {[𝑟 (sin 𝜃 - cos 𝜃) + 1]}^{- \frac{𝑝_{2}}{𝑝}} d 𝜃 . \end{aligned}

Since for any $𝑟 \in (0, 1)$ , $𝜃 \in [0, π]$ ,

𝑟 (sin 𝜃 - cos 𝜃) + 1 \leq 1 - 𝑟 + 2 𝜃,

it follows that

\begin{aligned} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{2}} d 𝜃 & = 2 \int_{0}^{π} {(1 - 𝑟 + 2 𝜃)}^{- \frac{𝑝_{2}}{𝑝}} d 𝜃 \\ = {\frac{{(1 - 𝑟 + 2 𝜃)}^{1 - \frac{𝑝_{2}}{𝑝}}}{1 - \frac{𝑝_{2}}{𝑝}} |}_{0}^{π} \\ = \frac{{(1 - 𝑟 + 2 π)}^{1 - \frac{𝑝_{2}}{𝑝}} - {(1 - 𝑟)}^{1 - \frac{𝑝_{2}}{𝑝}}}{1 - \frac{𝑝_{2}}{𝑝}} . \end{aligned}

Taking the limit superior as $𝑟 \to 1^{-}$ ,

\underset{𝑟 \to 1^{-}}{lim sup} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{2}} d 𝜃 \geq \frac{{(1 - 𝑟 + 2 π)}^{1 - \frac{𝑝_{2}}{𝑝}} - {(1 - 𝑟)}^{1 - \frac{𝑝_{2}}{𝑝}}}{1 - \frac{𝑝_{2}}{𝑝}} = \infty

since $1 - \frac{𝑝_{2}}{𝑝} < 0$ (if $𝑝_{2} = \infty$ then this is trivial as $𝑓$ is not bounded). Hence

(𝐻^{𝑝_{1}} ∋) 𝑓 \notin 𝐻^{𝑝_{2}} .

It suffices to show $𝐻^{𝑝_{2}} ⊊ 𝐻^{𝑝_{1}}$ (the inclusion). This is already known for $𝑝_{2} = \infty$ , so assume $𝑝_{2} \neq \infty$ .

Assume an arbitrary $𝑓 \in 𝐻^{𝑝_{2}}$ . Then there exists $𝑀 > 0$ (independent of $𝑟$ ) such that $\forall 𝑟 \in (0, 1)$ ,

\int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{2}} d 𝜃 \leq 𝑀 .

Let $𝜙 : ℝ_{\geq 0} \to ℝ$ be given by $𝑥 \mapsto 𝑥^{\frac{𝑝_{2}}{𝑝_{1}}}$ . Let $𝑔 (𝜃) = {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{1}}$ .

Then by holomorphy, $𝑔 \in 𝐶 ([0, 2 π])$ and $𝜙$ is convex since $\frac{𝑝_{2}}{𝑝_{1}}$ . Then

\begin{aligned} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{1}} d 𝜃 & = 2 π {[{(\frac{1}{2 π} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{1}} d 𝜃)}^{\frac{𝑝_{2}}{𝑝_{1}}}]}^{\frac{𝑝_{1}}{𝑝_{2}}} \\ = 2 π {[𝜙 \circ (average of 𝑔 in [0, 2 π])]}^{\frac{𝑝_{1}}{𝑝_{2}}} \\ \leq 2 π {[average of 𝜙 \circ 𝑔 in [0, 2 π]]}^{\frac{𝑝_{1}}{𝑝_{2}}} \\ = 2 π {[\frac{1}{2 π} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝_{2}} d 𝜃]}^{\frac{𝑝_{1}}{𝑝_{2}}} \leq 𝑀 {(2 π)}^{1 - \frac{𝑝_{1}}{𝑝_{2}}} . \end{aligned}

by application of Jensen’s Inequality (Theorem 7.3.7 $Theorem 7.3.7 Theorem 7.3.7$ ). This proves the inclusion. ⁠ $□$

Theorem 9.4.6.

Fix

𝑝 \in ℝ_{> 0} \cup {\infty}

. For any

𝑓 \in 𝐻^{𝑝}

such that

{𝑎_{𝑛}}_{𝑛 \in ℕ} \subset 𝔻^{*}

are its zeros in order of increasing modulus, counting multiplicities (

𝑓 (0) \neq 0

);

𝑓

satisfies the Blaschke condition (refer to Theorem 4.4.4.5

Theorem 4.4.4.5 Theorem 4.4.4.5

Proof.

If $𝑓$ has only finitely many nonzero zeros, the assertion is immediate. We may assume $𝑝 \neq \infty$ as we have already proved this case in Lemma 4.4.4.4 $Lemma 4.4.4.4 Lemma 4.4.4.4$ .

By Jensen’s Formula (Theorem 4.4.4.2 $Theorem 4.4.4.2 Theorem 4.4.4.2$ ), for each $𝑘 \in ℕ$ , for any $𝑟 \in (| 𝑎_{𝑘} |, 1)$ , $𝑘 \leq 𝑛 (𝑟, 0, 𝑓)$ , giving that

\begin{aligned} log | 𝑓 (0) | & = \frac{1}{2 π} \int_{0}^{2 π} log | 𝑓 (𝑟 e^{i 𝜃}) | d 𝜃 + \sum_{𝑗 = 1}^{𝑛 (𝑟, 0, 𝑓)} log | \frac{𝑎_{𝑗}}{𝑟} | \\ \leq \frac{1}{2 π} \int_{0}^{2 π} log | 𝑓 (𝑟 e^{i 𝜃}) | d 𝜃 + \sum_{𝑗 = 1}^{𝑘} log | \frac{𝑎_{𝑗}}{𝑟} |, (9.4 . 2) \end{aligned}

where $𝑛 (𝑟, 0, 𝑓)$ counts the zeros of $𝑓$ in $\overline{𝐷 (0, 𝑟)}$ . Since $\frac{1}{𝑡} \leq 𝑡^{𝑝 - 1}$ for $𝑡 \geq 1$ , integrating from $1$ to $𝑥 \geq 1$ , we have

\int_{1}^{𝑥} \frac{d 𝑡}{𝑡} \leq \int_{1}^{𝑥} 𝑡^{𝑝 - 1} ⟹ log 𝑥 \leq \frac{1}{𝑝} 𝑥^{𝑝} - \frac{1}{𝑝} < \frac{1}{𝑝} 𝑥^{𝑝} .

For $𝑥 \in (0, 1)$ , the inequality continues to hold as the left-hand side is negative. Therefore, applying this to (9.4.2) $(9.4.2) (9.4.2)$ , we have

\begin{aligned} log | 𝑓 (0) | & \leq lim_{𝑟 \to 1^{-}} \frac{1}{2 π 𝑝} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃 + \sum_{𝑗 = 1}^{𝑘} log | \frac{𝑎_{𝑗}}{𝑟} | \\ \leq \frac{1}{𝑝} {‖ 𝑓 ‖}_{𝐻^{𝑝}}^{𝑝} + \sum_{𝑗 = 1}^{𝑘} log | 𝑎_{𝑗} | . \end{aligned}

Since $𝑘$ was arbitrary, taking $𝑘 \to \infty$ ,

\sum_{𝑗 = 1}^{\infty} log \frac{1}{| 𝑎_{𝑗} |} \leq \frac{1}{𝑝} {‖ 𝑓 ‖}_{𝐻^{𝑝}}^{𝑝} - log | 𝑓 (0) | . (9.4 . 3)

For $𝑡 \geq 1$ , $- 1 \leq - \frac{1}{𝑡}$ , therefore for $𝑥 \geq 1$

\int_{1}^{𝑥} - d 𝑡 \leq \int_{1}^{𝑥} - \frac{d 𝑡}{𝑡} ⟹ 1 - 𝑥 \leq log \frac{1}{𝑥} .

Similarly, for $𝑥, 𝑡 < 1$ , we have $- 1 > - \frac{1}{𝑡}$ . Thus,

\int_{𝑥}^{1} - d 𝑡 > \int_{𝑥}^{1} - \frac{d 𝑡}{𝑡} ⟹ 𝑥 - 1 > log 𝑥 ⟹ 1 - 𝑥 \leq log \frac{1}{𝑥},

which holds everywhere in $ℝ_{> 0}$ . Then (9.4.3) $(9.4.3) (9.4.3)$ yields

\sum_{𝑗 = 1}^{\infty} (1 - | 𝑎_{𝑗} |) \leq \frac{1}{𝑝} {‖ 𝑓 ‖}_{𝐻^{𝑝}}^{𝑝} - log | 𝑓 (0) | < \infty . □

Theorem 9.4.7.

Suppose $𝑝 > 0$ or $𝑝 = \infty$ and let $𝑓 \in 𝐻^{𝑝}$ such that $𝑓$ is not identically $0$ . Then the Blaschke product $𝐵 (𝑧)$ constructed on the zero set of $𝑓$ (which may contain a $𝑧^{𝑚}$ term if $𝑓 (0) = 0$ ) locally uniformly converges on $𝔻$ , and furthermore, $𝑓$ can be factorized by

𝑓 (𝑧) = 𝐹 (𝑧) 𝐵 (𝑧),

such that

𝐹 \in 𝐻^{𝑝}

and does not vanish in

𝔻

and

{‖ 𝐹 ‖}_{𝐻^{𝑝}} = {‖ 𝑓 ‖}_{𝐻^{𝑝}}

Proof.

Let ${𝑎_{𝑛}}_{𝑛 \in ℕ}$ be the zero set of $𝑓$ in $𝔻^{*}$ in order of increasing modulus, and let $𝑚 \geq 0$ be the multiplicity of the zero ( $𝑚 = 0$ if there isn’t one) at $0$ . Let

𝐵 (𝑧) = 𝑧^{𝑚} \prod_{𝑛 = 1}^{\infty} [- \frac{| 𝑎_{𝑛} |}{𝑎_{𝑛}} \frac{𝑧 - 𝑎_{𝑛}}{1 - \overline{𝑎_{𝑛}} 𝑧}], 𝐹 (𝑧) = \frac{𝑓 (𝑧)}{𝐵 (𝑧)} .

Then $𝐹$ extends holomorphically to $𝔻$ and does not vanish therein. It remains to show the final assertion. If $𝑝 = \infty$ , the assertion follows from Theorem 9.4.2 $Theorem 9.4.2 Theorem 9.4.2$ , so assume $0 < 𝑝 < \infty$ .

If $𝑓$ has only finitely many nonzero zeros, then $𝐵$ is a finite Blaschke product, and the boundary argument below, applied directly to $𝐵$ and $𝐹$ , proves the norm equality. Thus suppose ${𝑎_{𝑛}}$ is infinite.

The use of this infinite product is not actually of great utility as it is not necessarily continuous on $\overline{𝔻}$ . Thus we use the partial products

𝐵_{𝑁} (𝑧) = 𝑧^{𝑚} \prod_{𝑛 = 1}^{𝑁} [- \frac{| 𝑎_{𝑛} |}{𝑎_{𝑛}} \frac{𝑧 - 𝑎_{𝑛}}{1 - \overline{𝑎_{𝑛}} 𝑧}], 𝐹_{𝑁} (𝑧) = \frac{𝑓 (𝑧)}{𝐵_{𝑁} (𝑧)}

for $𝑁 \in ℕ$ . Each $𝐹_{𝑁}$ extends holomorphically to $𝔻$ . Because $𝐵_{𝑁}$ is a finite product, $| 𝐵_{𝑁} (𝑧) | \leq 1$ on $𝔻$ and $| 𝐵_{𝑁} (e^{i 𝜃}) | = 1$ . If

𝑚_{𝑁} (𝑟) = min_{𝜃 \in [0, 2 π]} | 𝐵_{𝑁} (𝑟 e^{i 𝜃}) |,

then $𝑚_{𝑁} (𝑟) \to 1$ as $𝑟 \to 1^{-}$ . For all $𝑟$ sufficiently close to $1$ ,

\begin{aligned} \frac{1}{2 π} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃 & \leq \frac{1}{2 π} \int_{0}^{2 π} {| 𝐹_{𝑁} (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃 \\ \leq 𝑚_{𝑁} {(𝑟)}^{- 𝑝} \frac{1}{2 π} \int_{0}^{2 π} {| 𝑓 (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃 . \end{aligned}

Taking $𝑟 \to 1^{-}$ , all inequalities become equalities, and using Lemma 9.4.4 $Lemma 9.4.4 Lemma 9.4.4$ gives

{‖ 𝐹_{𝑁} ‖}_{𝐻^{𝑝}} = {‖ 𝑓 ‖}_{𝐻^{𝑝}} . (9.4 . 4)

Moreover,

𝐹_{𝑁} (𝑧) = 𝐹 (𝑧) \frac{𝐵 (𝑧)}{𝐵_{𝑁} (𝑧)} = 𝐹 (𝑧) \prod_{𝑛 = 𝑁 + 1}^{\infty} [- \frac{| 𝑎_{𝑛} |}{𝑎_{𝑛}} \frac{𝑧 - 𝑎_{𝑛}}{1 - \overline{𝑎_{𝑛}} 𝑧}],

so $𝐹_{𝑁} ⇉ 𝐹$ locally uniformly on $𝔻$ . Hence, for every $0 < 𝑟 < 1$ ,

\begin{aligned} \frac{1}{2 π} \int_{0}^{2 π} {| 𝐹 (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃 & = lim_{𝑁 \to \infty} \frac{1}{2 π} \int_{0}^{2 π} {| 𝐹_{𝑁} (𝑟 e^{i 𝜃}) |}^{𝑝} d 𝜃 \\ \leq {‖ 𝑓 ‖}_{𝐻^{𝑝}}^{𝑝} . (by (9.4.4) (9.4.4) (9.4.4)) \end{aligned}

Thus $𝐹 \in 𝐻^{𝑝}$ and ${‖ 𝐹 ‖}_{𝐻^{𝑝}} \leq {‖ 𝑓 ‖}_{𝐻^{𝑝}}$ . The reverse inequality follows from $𝑓 = 𝐵 𝐹$ and $| 𝐵 | \leq 1$ on $𝔻$ , proving equality. ⁠ $□$

Remark.

We would also like to state that two Hardy spaces $𝐻^{𝑝_{1}}, 𝐻^{𝑝_{2}}$ (which may be $\infty$ ) share their zero sets: the sets

𝑆_{1} = {{𝑧 \in 𝔻 : 𝑓 (𝑧) = 0} : 𝑓 \in 𝐻^{𝑝_{1}}}

and

𝑆_{2} = {{𝑧 \in 𝔻 : 𝑓 (𝑧) = 0} : 𝑓 \in 𝐻^{𝑝_{2}}}

are equal. Indeed, any

𝑓 \in 𝐻^{𝑝_{1}}

has a zero set satisfying the Blaschke condition by Theorem 9.4.6

Theorem 9.4.6 Theorem 9.4.6

(the negligence of multiplicities is not substantial, as the series consists solely of positive terms). Then a Blaschke product

𝐵

may be constructed on this set, which lies in

𝐻^{𝑝_{2}}

. Similarly, a function in

𝐻^{𝑝_{1}}

may be constructed for any function in

𝐻^{𝑝_{2}}