The Smirnov class on the disc

Assume countable choice. The Smirnov class $N^{+}(\mathbb{D})$ contains $0$ and consists otherwise of the nonzero $f\in N(\mathbb{D})$ such that, with $f^{\ast }$ the boundary function of Boundary values and log-integrability of Nevanlinna-class functions (so that $\log \mathrm{\mid }{f}^{\ast }\mathrm{\mid }\in L^1(\mathbb{T},m)$, extended by $-\mathrm{\infty }$ where $f^{\ast }=0$), $\log \mathrm{\mid }f(z)\mathrm{\mid }\le P[\log \mathrm{\mid }{f}^{\ast }\mathrm{\mid }](z)\text{for every }z\in \mathbb{D}\mathrm{.}$ For the zero function, the boundary function is zero; we interpret both sides of the displayed inequality as $-\mathrm{\infty }$. Its boundary logarithm is not an $L^1$ datum, and the finite-measure Poisson construction is used only for nonzero functions. The Poisson integral for the nonzero case is the one of The Poisson integral of a finite complex boundary measure.

By the Poisson-Jensen inequality of Poisson-Jensen inequality for Hardy functions, every $f\in H^p(\mathbb{D})$, $0<p\le \mathrm{\infty }$, lies in $N^{+}(\mathbb{D})$ by the displayed inequality; hence $H^p(\mathbb{D})\subseteq N^{+}(\mathbb{D})\subseteq N(\mathbb{D})(0<p\le \mathrm{\infty }),$ the second inclusion being part of the definition.

Equivalently, $N^{+}(\mathbb{D})$ is the class of quotients $g\mathrm{/}h$ with $g,h\in H^{\mathrm{\infty }}(\mathbb{D})$ and $h$ outer (and then $h$ may be normalized by $h(0)>0$); this equivalence is proved in The Smirnov class is the class of quotients by outer bounded functions ↗. The class $N^{+}$ is a complex vector space, as certified using the quotient characterization just proved by The Smirnov class is the class of quotients by outer bounded functions ↗, its justified_by supplier. Indeed, for nonzero $f_j=g_j\mathrm{/}{h}_j$ with bounded holomorphic numerators and bounded outer denominators, $h_1{h}_2$ is bounded and outer: its boundary logarithm is the sum of the two integrable boundary logarithms, and its interior log-modulus is their Poisson integral by linearity, in the outer convention of Inner, singular inner and outer functions. Thus $f_1+f_2=(g_1{h}_2+g_2{h}_1)\mathrm{/}(h_1{h}_2)$ has bounded numerator and bounded outer denominator. If the numerator is zero, the sum is the already included zero function; otherwise the characterization gives membership in $N^{+}$. Scalar multiplication uses the same denominator and numerator $c{g}_j$, including $c=0$. This certification is not used in the proof of the quotient characterization, which depends only on the displayed defining inequality.