ON SOME FAMILIES OF ANALYTIC FUNCTIONS

Theorem 1. Any analytic function f in a Riemann surface R belongs to the Smirnov class S(R) if and only if the subharmonic function log ⁺ | / | has a quasibounded harmonic majorant in R. Using a version of Garding and Hόrmander's theorem [7] as a lemma, we shall prove: THEORE ^M 2. Any analytic function f in a Riemann surface R belongs to the Hardy class H^P {R) (for p >0 ) if and only if the subharmonic function \f\^p has a quasi-bounded harmonic majorant in R. Seeing the above characterizations for the two classes, we are tempted to say the following:

Theorem 3. Let Ψ(r) be a continuous extended real-valued function defined for r > 0 satisfying the condition that for any finite positive real number c, the set of r such that the inequality Ψ(r) ^ c holds is bounded {from above). Let R be a Riemann surface, E be a closed polar set lying in R and f be an anaylytic function defined in R — E such that the composite function Ψ{\f\) has a quasi-bounded harmonic majorant in R — E. Then there exists an analytic function f defined in R such that the composite function 3^(1/1) has a quasi-bounded harmonic majorant in R and the restriction of f to R — E coincides with the function f. As corollaries we have an extension of Tumarkin-Havinson's theorem and a new proof of Parreau's. At the end, we shall give an example for the classification theory of open Riemann surfaces, which admits a non-constant analytic Lindelόfian function [9] and no non-constant analytic function in the Smirnov class. 2. Let R be a Riemann surface, HP^r {R) be the family of all the har monic functions u in R such that the subharmonic function \u\ has a harmonic majorant in R. It is well-known (see for example, [3]) that HP'(R) forms a vector lattice under the lattice operations: $$u \lor v =$$ (the least harmonic majorant of max $(u, v)$ ); $u \land v = -(-u) \lor (-v)$ for u,v in HP^f (R). For u in HP'(R) we define Λ as follows: $$Mu = u \vee 0 - u \wedge 0.$$ We know that MM — u \J (— u) and M(Mu) = MM . is, by definition, quasi-bounded if A function M in HP^f (R) $$Mu = \lim_{n \to +\infty} (Mu) \wedge n$$ , or equivalently , $$\lim_{n \to +\infty} (Mu - n) \vee 0 = 0,$$ where n are positive numbers which can be considered as elements in HP'(R) and the limit is taken in the sense of the lattice operation, namely, (Mu) Λ n (resp. (Mu — n) V 0) tends to Mu (resp. 0) non-decreasingly (resp. non increasingly) in R. A function u in HP^f (R) is called singular if $$\lim_{n\to +\infty} (Mu) \wedge n = 0.$$ It is shown by Parreau [13] that any u in HP'(R) can be decomposed uniquely as: $$u=u_B+u_S,$$ where u^B is quasi-bounded and u^s is singular. The operator u ->u^B (resp. u->u^s ) from HP^f (R) into itself is linear, positive, i.e., M> 0 implies uB>§ (resp. ^ ^s > 0 ) and idempotent, i.e., {u^B )B — u^B (resp. (M^S )S = ^s) Of course, u is quasi-bounded (resp. singular) if and only if u^s = 0 (resp. zί = 0). In the remainder of this paper we shall assume that the Riemann surface R is hyperbolic since the situation is obvious in the parabolic case. A subharmonic function v in R having a harmonic majorant in R can be decomposed uniquely as: $$v = v^{-} - p,$$ where v is the least harmonic majorant of v and \p>0 is a Green's potential in R* (F. Riesz's decomposition). We shall say that a subharmonic function v in R is quasi-bounded if v in the above decomposition is in HP^r {R) and quasi-bounded. A subharmonic function υ having a quasi-bounded harmonic majorant u and a quasi bounded harmonic minorant w simultaneously is quasi-bounded for 0 = w^s ^(v )^s ^Us = 0. Especially, a non-negative subharmonic function is quasi-bounded if and only if it has a quasi-bounded harmonic majorant. Let {Rⁿ y^L1 be a normal exhaustion of R in Pfluger's sense, dRⁿ = Γⁿ be the boundary of Rⁿ (consisting of a finite number of piecewise analytic closed Jordan curves), z⁰ be a fixed point in R and ωntZQ be the harmonic measure of Γⁿ with respect to the domain Rⁿ measured at the point z⁰ (for n = 1,2, . . .). Then obviously we have: $$v^{\wedge}(z_0) = \lim_{n \to +\infty} \int_{\Gamma_n} v(z) d\omega_{n,z_0}(z).$$ An extended real-valued function f(z) defined for points z in R is said to be uniformly absolutely integrable with respect to the system {.(Γⁿ , ωntZ0)}^lι (we shall say simply "U.A.L for z⁰ and {7?ⁿ }") if the followings are satisfied: (a) $$\sup_{n} \int_{\Gamma_n} |f(z)| d\omega_{n,z_0}(z) < \infty,$$ and (b) for any ε > 0, there exists a δ > 0 such that $$\left| \int_{A_n} f(z) d\omega_{n,z_0}(z) \right| < \varepsilon$$ uniformly for n = 1, 2, . . . , if only Λⁿ c Γⁿ and ωΛ>Z0(i4 ) < δ. According to de la Vallee Poussin [18] and Doob [4], [6], a function /(z) in R is U.A.I, for z⁰ and -C^J if and only if there exists a non-negative monotone non-decreasing convex function Φ[r) defined for r>0 satisfying the conditions: (i) $$\lim_{r \to +\infty} \Phi(r) / r = +\infty$$ and (ii) $$\sup_{n} \int_{\Gamma_{n}} \Phi(|f(z)|) d\omega_{n,z_{0}}(z) < \infty.$$ We shall call this de la Vallee Poussin-Doob's lemma. In particular, if a subharmonic function v{z)>0 in R is U.A.I, for z⁰ and {Rⁿ }, then the condition (ii) above can be read as: (ii)' The subharmonic function Φ(v) has a harmonic majorant in R. We state some lemmas which will be used later.

Lemma 1. Let υ be a quasi-bounded subharmonic function in a Riemann surface R. Then v is U.A.I, for arbitrary point z⁰ in R and arbitrary exhaustion {.Rv}> Zo in R\ Conversely assume that a subharmonic function v in R is U.A.I. for at least one point z⁰ and at least one exhaustion -[Rⁿ y > z^ in R . Then v is a quasi-bounded subharmonic function in R. Proof We know that any harmonic function belongs to HP^f {R) and is quasi-bounded if and only if it is U.A.I, for one point z^Q and for one ex haustion -[Rn}, z⁰ in i?i (and consequently for all) (see [4]). It is easy to check that Green's potential p> 0 is always U.A.I, for z⁰ and {Rⁿ } since $$\int_{\Gamma_n} p(z) d\omega_{n,z_0}(z) \to 0 \text{ as } n \to +\infty.$$ Using the above two facts, we have immediately the assertions.

Lemma 2. A subharmonic function v is quasi-bounded if and only if there exists a non-negative monotone non-decreasing convex function Φ{r) defined for r >: 0 satisfying the conditions (i) and (ii). Proof This is a consequence of de la Vallee Poussin-Doob's lemma and Lemma 1. 3. Here we remark the relations between some families of analytic functions defined in a Riemann surface R. We define the families ΛB(R) and AL(R) of analytic functions in R by the following: / is in AB{R) if and only if \f\ is bounded in R; f is in AL(R) if and only if the subharmonic function log ⁺ |/| has a harmonic majorant in R. Then the following inclusion relations: $$AB(R) \subset H_p(R) \subset S(R) \subset AL(R)$$ (for $p > 0$ ) are proved by the inequalities: $$\log^+(|f|/\mu) \le |f|^p/(p \cdot \mu^p)$$ and $$\log^+ |f| \le \log^+ (|f| / \mu) + \log^+ \mu$$ . REMARK. The functions / in the class AL{R) are Lindelofian analytic functions in the sense of Heins [9] and in the special case where R is the unit open disc, are analytic functions of bounded type in Nevanlinna's sense [12]. The Smirnov class S(R) was first investigated by V.I. Smirnov [16]. Now we give Proof of Theorem 1. Let μ>l. Then we obtain $$\log^+(|f|/\mu) = \max(\log^+|f| - \log \mu, 0).$$ Consequently we have $$\begin{split} (\log^+(|f|/\mu))^{\wedge} &= (\max(\log^+|f| - \log \mu, 0))^{\wedge} \\ &= (\max((\log^+|f|)^{\wedge} - \log \mu, 0))^{\wedge} \\ &= ((\log^+|f|)^{\wedge} - n) \vee 0 \,, \end{split}$$ where n = log μ and ψ' is the least harmonic majorant of φ (see §1). Hence the condition that $$(\log^+(|f|/\mu))^{\wedge} (z_0) \rightarrow 0 \text{ as } \mu \rightarrow +\infty$$ is equivalent to the condition that $$\lim_{n \to +\infty} ((\log^+|f|)^{\hat{}} - n) \vee 0 = 0$$ by Harnack's theorem, or (log⁺ |/|)^A , the least harmonic majorant of log⁺ |/|, is quasi-bounded. Q.E.D. REMARK. It is easy to show that log⁺ |/| has a quasi-bounded harmo nic majorant in R if and only if log |/ | has a quasi-bounded harmonic majorant in R. By Lemma 1 with υ = log⁺ |/| and by Theorem 1 we have

Corollary 1. {An extended form of Theorem 1 in [17]) Any analytic function f is in the Smirnov class S{R) if and only if the subharmonic function log ⁺ | / | is U.A.I, for arbitrary fixed point z⁰ in R and arbitrary exhaustion {Rⁿ }, z⁰ in R^x .

Corollary 2. (An extended form of Theorem 2 in [17]) Any analytic function f is in the Smirnov class S(R) if and only if the subharmonic function $\log^+|f|$ has a harmonic majorant which is U.A.I. for arbitrary fixed point $z_0$ in R and arbitrary exhaustion $\{R_n\}$ , $z_0$ in $R_1$ . The following corollary shows that Gehring's class $N^*$ in [8] is a special case of the Smirnov class S(R) where R is the unit open disc.

Corollary 3. Any analytic function f is in the class S(R) if and only if there exists a non-negative monotone non-decreasing convex function $\Phi(r)$ satisfying the condition (i) in §2 and the subharmonic function $\Phi(\log^+|f|)$ has a harmonic majorant in R.

Lemma 3. Let v be a subharmonic function defined in R. Let φ{r) be a non-negative monotone non-decreasing convex function defined for — oo < r < + co satisfying the condition (A) $$\lim_{r \to +\infty} \varphi(r) / r = +\infty$$ and assume that (B) the subharmonic function φ{v) has a harmonic majorant in R, where we set φ(— oo) = lim φ(r). r -> — oo Then - (C) the least harmonic majorant v of v exists and is in HF{R), - (D) the singular measure dμ^s on j x corresponding to the singular part {v )^s of v\ is non-positive,* ² ) E.D. Solomentsev proved partly the same results as Garding and Hδrmander's in his paper: Izv. Akad. Nauk SSSR (1938), pp. 571-582. (E) the least harmonic majorant $(\varphi(v))^{\wedge}$ of the subharmonic function $\varphi(v)$ exists and is quasi-bounded, and (F) $$(\varphi(v))^{\wedge}(z) = \int_{A_1} K(z,\zeta) \varphi(v^*(\zeta)) d\omega(\zeta) .$$

Corollary 1. {An extension of Tumarkin-HavinsorC s theorem [17]) Let E be a closed polar set lying in a Riemann surface R. If a function f is in the Smirnov class S(R — E), then there exists an analytic function f in the Smirnov class S(R) such that the restriction of f to R — E coincides with f. Proof This is a consequence of Theorem 1 and Theorem 3 with ψ{r) = logV.

Corollary 2. (Parreau [13], Theorem 20) Let E be a closed polar set lying in a Riemann surface R. If a function f is in the class H^P {R — E) for p > 0, then there exists f in the class H^P [R) such that the restriction of f to R — E coincides with f. Proof This is a consequence of Theorem 2 and Theorem 3 with Ψ(r) = r p . REMARK. Parreau's theorem can be proved, using Corollary 1 above, if we assume the fact that the polar set E is removable for non-negative superharmonic functions ([1], [2]). W. Rudin ([15], at p. 49) pointed out that the analogous assertion for the class AL is false. 6. As usual we shall denote by O^x the totality of open Riemann sur faces R (including parabolic types) on which the given family X(R) of functions consists only of constants. Then we have $$O_{AL} \subset O_S \subset O_{H_p} \subset O_{AB}$$ (for $p > 0$ ). Parreau ([13], p. 192) proved that the inclusion relation OAL C OH^P (for p > 0) is proper, using P.J. Myrberg's example in [10]. Using the fact that one point is removable for the Smirnov class S and the inequality: log⁺ | a — β \² ^ 2(log⁺ \a\ + log⁺ \βI + log 2), for complex numbers a and β, we can prove that the inclusion relation OAL C O^S is proper by the same method as in [10].