Harmonic Hardy classes on the unit disc

Assume countable choice. Identify the torus $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ with the unit circle as in The one-dimensional torus and its normalized Haar integral, and write $m$ for its normalized Haar measure, a probability measure. For a function $u:\mathbb{D}\to \mathbb{C}$ and $0\le r<1$ write $u_r:\mathbb{T}\to \mathbb{C},u_r(\zeta ):=u(r\zeta )\mathrm{.}$

A complex-valued function $u=U+iV$ on $\mathbb{D}$ is called harmonic when both components $U$ and $V$ are real-valued plane harmonic functions in the sense of Plane harmonic functions; this is the componentwise convention of Complex Lp classes and Euclidean test-function conventions. Every such $u$ is continuous, because a $C^2$ real function is continuous and both components are of class $C^2$.

The classes ${\mathbf{h}}^p(\mathbb{D})$. For $1\le p<\mathrm{\infty }$ let $h^p(\mathbb{D}):=\{u:\mathbb{D}\to \mathbb{C}\text{ harmonic}:{\sup }_{0\le r<1}\mathrm{\parallel }{u}_r{\mathrm{\parallel }}_{L^p(\mathbb{T},m)}<+\mathrm{\infty }\},$ where $u_r$ is regarded as an element of the quotient space $L^p(\mathbb{T},m;\mathbb{C})$ of Complex Lp classes and Euclidean test-function conventions through its continuous representative, and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L^p(\mathbb{T},m)}$ is the norm of Complex Holder, Minkowski, and the quotient norm. For $p=\mathrm{\infty }$ set $h^{\mathrm{\infty }}(\mathbb{D}):=\{u:\mathbb{D}\to \mathbb{C}\text{ harmonic}:{\sup }_{0\le r<1}{\sup }_{\zeta \in \mathbb{T}}\mathrm{\mid }u(r\zeta )\mathrm{\mid }<+\mathrm{\infty }\}\mathrm{.}$ Here the inner supremum may equivalently be read as the essential supremum of $u_r$ with respect to $m$ (The essential supremum of a measurable function with respect to a measure): a continuous function has the same supremum and essential supremum, because a nonempty open subset of $\mathbb{T}$ contains the image $q((a,b))$ of an open interval with $0<b-a<1$ (the map $q$ is open and its images of rational-endpoint intervals form a base, as proved in The one-dimensional torus and its normalized Haar integral), and such a set has $m$-measure $b-a>0$; hence a continuous function bounded by $M$ almost everywhere is bounded by $M$ everywhere. In particular ${\sup }_{0\le r<1}{\sup }_{\zeta \in \mathbb{T}}\mathrm{\mid }u(r\zeta )\mathrm{\mid }={\sup }_{z\in \mathbb{D}}\mathrm{\mid }u(z)\mathrm{\mid },$ because every $z\in \mathbb{D}$ has the form $r\zeta$ with $r=\mathrm{\mid }z\mathrm{\mid }$ and $\zeta \in \mathbb{T}$.

The Hardy norms. For $u\in h^p(\mathbb{D})$ put $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^p}:={\sup }_{0\le r<1}\mathrm{\parallel }{u}_r{\mathrm{\parallel }}_{L^p(\mathbb{T},m)}(1\le p<\mathrm{\infty }),\mathrm{\parallel }u{\mathrm{\parallel }}_{h^{\mathrm{\infty }}}:={\sup }_{z\in \mathbb{D}}\mathrm{\mid }u(z)\mathrm{\mid }\mathrm{.}$ Then $h^p(\mathbb{D})$ is a complex vector space: harmonicity and finiteness of the suprema are preserved by finite linear combinations, and $\mathrm{\parallel }u+v{\mathrm{\parallel }}_{h^p}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{h^p}+\mathrm{\parallel }v{\mathrm{\parallel }}_{h^p}$, $\mathrm{\parallel }\lambda u{\mathrm{\parallel }}_{h^p}=\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\parallel }u{\mathrm{\parallel }}_{h^p}$ for $\lambda \in \mathbb{C}$, by the corresponding statements at each radius in Complex Holder, Minkowski, and the quotient norm. The assignment is definite: if $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^p}=0$, then $\mathrm{\parallel }{u}_{1\mathrm{/}2}{\mathrm{\parallel }}_{L^p}=0$, so $u_{1\mathrm{/}2}=0$ almost everywhere, hence $u_{1\mathrm{/}2}=0$ everywhere by continuity and the preceding paragraph; the Poisson representation formula A harmonic function is recovered from its values on any containing circle by the Poisson formula then gives $u=0$ on the disc $\mathrm{\mid }z\mathrm{\mid }<\frac{1}{2}$, and the identity principle A plane harmonic function that vanishes on a nonempty open set vanishes everywhere on the domain, applied to the two components on the domain $\mathbb{D}$, gives $u=0$. Thus $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{h^p}$ is a norm on $h^p(\mathbb{D})$ for every $1\le p\le \mathrm{\infty }$. No containment among the classes $h^r(\mathbb{D})\subseteq h^p(\mathbb{D})$ for $p<r$ is asserted here; only the containment $h^{\mathrm{\infty }}(\mathbb{D})\subseteq h^1(\mathbb{D})$ will be used, and it is proved where it is needed.