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Cauchy representation of an $H^{1}$ function from its boundary values

Statement

Assume countable choice, as in the Hardy and circle conventions. Let $f \in H^{1} (D)$ and let $μ$ be the unique finite complex Borel measure on $T$ with $f = P [μ]$ . Then $μ ≪ m$ and $μ = f^{*} m$ for the boundary function $f^{*}$ of the Fatou theorem, and for every $z \in D$ $f (z) = P [f^{*}] (z) = \int_{T} P (z, ζ) f^{*} (ζ) d m (ζ) = \frac{1}{2 π i} \oint_{T} \frac{f^{*} (ζ)}{ζ - z} d ζ .$ Moreover $∥ f ∥_{H^{1}} = ∥ f^{*} ∥_{1} = ∣ μ ∣ (T)$ and $∥ f_{r} - f^{*} ∥_{1} \to 0$ as $r ↑ 1$ .

Facts & Assumptions

Given: Countable choice and $f \in H^{1} (D)$ .

[F1]

The CC analytic Hardy boundary theorem gives $f^{*} \in L^{1}$ , $f = P [f^{*}]$ , $∥ f_{r} - f^{*} ∥_{1} \to 0$ and $∥ f^{*} ∥_{1} = ∥ f ∥_{H^{1}}$ . Its analytic H1 representing measure exists and is unique under CC. (Fatou's boundary theorem for analytic Hardy spaces, Analytic Hardy spaces on the unit disc, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F2]

The density measure $f^{*} m$ is countably additive by dominated convergence; its finite positive variation is supplied by the local CC circle-variation lemma, and the direct density formula gives total variation integral $\int ∣ f^{*} ∣ d m$ , and finite complex circle measures with equal Poisson integrals are equal under CC. (A complex L^1 density defines a complex measure whose total variation is |h| dmu, Complex circle measures have finite regular total variation under countable choice, Finite complex circle measures are determined by Fourier coefficients and Poisson integrals)

[F3]

Cauchy's integral formula applies on every radius-r circle with |z|<r<1. The torus parametrization is $ζ = e^{2 π i t}$ with $d m = d t$ . (Cauchy's integral formula on a circle compactly contained in a disc of holomorphy, The one-dimensional torus and its normalized Haar integral)

[F4]

The Poisson integral is the kernel integral against the boundary datum. L1 convergence and uniform convergence of bounded weights imply convergence of their integrals, by the estimate $∣ \int (v_{r} w_{r} - v w) d m ∣ \leq ∥ v_{r} - v ∥_{1} ∥ w_{r} ∥_{\infty} + ∥ v ∥_{1} ∥ w_{r} - w ∥_{\infty}$ . (The Poisson integral of a finite complex boundary measure)

Proof

1.1F1F2givenconstructalgebra

Apply [F1] and set $μ = f^{*} m$ . By [F2] it is a finite complex representing measure for f, and its total variation is $∥ f^{*} ∥_{1} = ∥ f ∥_{H^{1}}$ . Every other finite complex representing measure has the same Poisson integral, so [F2] proves it equals mu. Thus the unique measure in the Statement is exactly this absolutely continuous density measure, including mu zero when f zero.

2.1step 1.1F1F4algebra

By [F1] and [F4], $f = P [f^{*}] = \int P (z, ζ) f^{*} (ζ) d m$ , and $∥ f_{r} - f^{*} ∥_{1} \to 0$ . Together with step 1.1 this gives the full Poisson, absolute-continuity, uniqueness and norm assertions under CC.

3.1step 2.1F3F4algebra

The Cauchy formula. Fix $z \in D$ and $r \in (∣ z ∣, 1)$ . The function $f$ is holomorphic on a neighbourhood of the closed disc of radius $r$ , so [F3] gives $f (z) = \frac{1}{2 π i} \oint_{∣ ζ ∣ = r} \frac{f (ζ)}{ζ - z} d ζ$ . Writing the circle integral in the torus parametrization, this equals $\int_{T} \frac{r ζ}{r ζ - z} f (r ζ) d m (ζ)$ (the factor $r ζ$ is $\frac{d ζ}{2 π i d m}$ ). As $r ↑ 1$ , the weights $\frac{r ζ}{r ζ - z}$ converge uniformly on $T$ to $\frac{ζ}{ζ - z}$ and are uniformly bounded for $r \geq (1 + ∣ z ∣) / 2$ , because $∣ r ζ - z ∣ \geq r - ∣ z ∣ \geq (1 - ∣ z ∣) / 2 > 0$ ; together with $∥ f_{r} - f^{*} ∥_{1} \to 0$ from step 2.1, [F4] gives $f (z) = \int_{T} \frac{ζ}{ζ - z} f^{*} (ζ) d m (ζ) = \frac{1}{2 π i} \oint_{T} \frac{f^{*} (ζ)}{ζ - z} d ζ,$ the last equality being the same parametrization at $r = 1$ .

4.1step 1.1step 2.1step 3.1∎

Assembly. Step 1.1 gives $μ = f^{*} m$ with the norm identity, step 2.1 gives $f = P [f^{*}] = \int P (z, ζ) f^{*} d m$ and the $L^{1}$ convergence of the radial functions, and step 3.1 gives the Cauchy representation of $f$ by its boundary values.

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Sources

R. K. Srivastava, Lecture Notes on Hardy Spaces (MA650, IIT Guwahati), §5.7 and §5.11 (standard reference, not scraped)
L. Ryzhik, Stanford Math 215 Course Notes, Chapter 5 (standard reference, not scraped)

Cauchy representation of an H1H1 function from its boundary values