Volume deficiency of Banach-space-valued meromorphic functions beyond admissible compact-projection maps and the vanishing counting hypothesis, and sharpness of the Ricci-index finiteness bounds

Pujari (2018) introduced the volume deficiency $\delta_V(a)=1-\limsup_{r\to\infty}V(r,a)/T(r,f)$ for meromorphic maps from the complex plane into an infinite-dimensional complex Banach space $E$ with a Schauder basis, where the volume function $V(r,a)$ is the new term generated by the Laplacian of $\log\|f-a\|$, which has no counterpart in scalar Nevanlinna theory. Her finiteness bound (at most $(4-2\delta_G)/\eta$ completely multiple values with $\delta_V(a)\ge\eta$ for admissible maps of compact projection, and at most $(2-2\delta_G)/\eta$ in the entire case, where $\delta_G$ is the Ricci index) and her countability theorem for the deficiency set $E(\delta_V)$ (conditional on countability of the auxiliary set $E(\Delta_{\overline N})$) rest on the structural hypotheses of admissibility, compact projection, and a Schauder basis on $E$, while her deficiency-sum inequalities hold for arbitrary $E$-valued meromorphic maps under the vanishing counting hypothesis $N(r,a)/T(r,f)\to 0$ for every $a$. The open problem is to determine which of these finiteness, countability, and sum phenomena survive without their hypotheses or under weaker conditions on the Banach space or the map, to characterize when the auxiliary deficiency set $E(\Delta_{\overline N})$ is countable, and to decide whether the Ricci-index-dependent constants are sharp.

Background

Classical Nevanlinna theory describes the value distribution of a scalar meromorphic function $f$ on the complex plane through its characteristic function $T(r,f)=m(r,f)+N(r,f)$, where $m(r,f)=\frac{1}{2\pi}\int_0^{2\pi}\log^+|f(re^{i\varphi})|\,d\varphi$ measures the size of $f$ on circles and $N(r,f)$ counts poles (an integrated counting function: for a discrete set with counting function $n(t)$, the associated $N$ is $\int_0^r n(t)\,dt/t$). The second fundamental theorem implies the deficiency relation $\sum_a \delta(a)\le 2$ for the Nevanlinna deficiency $\delta(a)=1-\limsup_{r\to\infty}N(r,a)/T(r,f)$, from which it follows that the set of deficient values is at most countable and that for each $\eta>0$ only finitely many values (at most $2/\eta$) can have $\delta(a)\ge\eta$; a value $a$ is called completely multiple if every solution of $f(z)=a$ has multiplicity at least $2$, and ramification versions of the second fundamental theorem bound the number of such values.

This theory has been extended to vector-valued maps. Ziegler (1982) developed Nevanlinna theory for meromorphic maps into $\mathbb C^n$; Hu and collaborators (the book of Hu and Yang, 1992, and Hu and Hu, 2006) extended the first and second fundamental theorems to maps into an infinite-dimensional complex Banach space $E$ equipped with a Schauder basis, i.e. a sequence $(e_j)$ such that every $x\in E$ has a unique norm-convergent expansion $x=\sum_j x_j e_j$. A map $f:\mathbb C\to E$ is called $E$-valued meromorphic if each component function $f_j$ (the coefficient of $e_j$) is meromorphic. The characteristic is defined as in the scalar case with the norm in place of the modulus, $m(r,f)=\frac{1}{2\pi}\int_0^{2\pi}\log^+\|f(re^{i\varphi})\|\,d\varphi$, and for $a\in E$ one has the counting function $N(r,a)$ of $a$-points (with multiplicity) and its reduced version $\overline N(r,a)$ (ignoring multiplicity).

The extension produces one genuinely new object. For scalar $f$, $\log|f-a|$ is harmonic away from the $a$-points, so its Laplacian is a purely point measure carried by those points; for a norm, $\log\|f(z)-a\|$ is merely subharmonic and its Laplacian carries a diffuse positive mass spread over the plane. The integrated Laplacian

$$v(t,a)=\frac{1}{2\pi}\int_{|\xi|<t}\Delta\log\|f(\xi)-a\|\,d\sigma\,d\tau, \qquad V(r,a)=\int_0^r \frac{v(t,a)}{t}\,dt,$$

defines the volume function $V(r,a)$, which enters the $E$-valued first and second fundamental theorems alongside the counting terms (the same $V$ already appears in the $E$-valued Borel-exceptional-value theory of Bhoosnurmath and Pujari, 2010). The volume deficiency is $\delta_V(a)=1-\limsup_{r\to\infty}V(r,a)/T(r,f)$, and one also considers $\delta(a)=1-\limsup_{r\to\infty}(V(r,a)+N(r,a))/T(r,f)$, the counting deficiencies $\delta_N(a)=\liminf N(r,a)/T(r,f)$ and $\Delta_N(a)=\limsup N(r,a)/T(r,f)$ (similarly $\delta_{\overline N},\Delta_{\overline N}$ with $\overline N$), and $\Theta(a)=1-\limsup_{r\to\infty}(V(r,a)+\overline N(r,a))/T(r,f)$.

The infinite-dimensional theory depends on two structural hypotheses. First, $f$ is of compact projection if the coordinate projections $P_n f$ ($P_n$ projects onto $\operatorname{span}(e_1,\dots,e_n)$) converge to $f$ uniformly on every compact subset of the domain; compact projection is what lets the finite-dimensional second fundamental theorem be passed to the limit. Second, $f$ is admissible (in the convention of this line of work) if along some sequence $r_\gamma\to\infty$ the error term $S(r_\gamma)$ of the second fundamental theorem satisfies $S(r_\gamma)/T(r_\gamma,f)\to 0$, the standard device for neutralizing that error term. Finally, the curvature function $G(r,f)=\int_0^r\frac{dt}{2\pi t}\int_{|\xi|<t}\Delta\log\|f'(\xi)\|\,d\sigma\,d\tau$ plays the role of counting the spherical derivative, and its Ricci index $\delta_G=\liminf_{r\to\infty}G(r,f)/T(r,f)$ enters the $E$-valued deficiency relations in place of part of the classical ramification bookkeeping.

Within this framework, deficiency relations for $E$-valued maps were obtained by Wu and Xuan (2012, 2013) and Wu and Chen (2013), all for finite-order admissible maps of compact projection. Pujari (2018) then initiated the theory of the volume deficiency itself: she proved that $v(r,a)$ and $V(r,a)$ are non-negative and increasing and that $V$ is convex in $\log r$; that $V(\cdot,a)\equiv 0$ exactly when $f$ has an essentially scalar image, $f(z)=a+\lambda(z)b$ with $\lambda$ scalar meromorphic and $b\in E$ fixed; the two-sided comparisons $\delta_N(a)+\delta(a)\le\delta_V(a)\le\Delta_N(a)+\delta(a)$ and $\delta_{\overline N}(a)+\Theta(a)\le\delta_V(a)\le\Delta_{\overline N}(a)+\Theta(a)$; the finiteness bounds stated in the abstract; the countability theorem that, for admissible maps of compact projection, $E(\delta_V)$ is countable whenever the auxiliary set $E(\Delta_{\overline N})=\{a:\Delta_{\overline N}(a)>0\}$ is countable, and conversely countability of $E(\delta_V)$ forces countability of $E(\delta_{\overline N})$; and deficiency-sum inequalities such as $\sum_{a\in E}\delta_V(a)+\delta_G\le 1+\limsup_{r\to\infty}\overline N(r,f)/T(r,f)$, which hold for arbitrary $E$-valued meromorphic maps under the vanishing counting hypothesis $N(r,a)/T(r,f)\to 0$ for every $a$. What remains unsettled is precisely the robustness and optimality of this edifice: the finiteness and countability conclusions lean on admissibility, compact projection, and the Schauder basis, the sum inequalities lean on the vanishing counting hypothesis, and nothing is known about the sharpness of the numerical constants.

Problem Statement

Let $E$ be an infinite-dimensional complex Banach space with a Schauder basis (a sequence $(e_j)$ in $E$ such that every $x\in E$ has a unique norm-convergent expansion $x=\sum_j x_j e_j$) and let $f:\mathbb C\to E$ be a non-constant $E$-valued meromorphic function (each component function $f_j$, the coefficient of $e_j$, meromorphic; the theory extends verbatim to maps from a disc $|z|<R\le\infty$, and the deficiency questions below concern the plane case $R=\infty$). Using the characteristic $T(r,f)=m(r,f)+N(r,f)$ with $m(r,f)=\frac{1}{2\pi}\int_0^{2\pi}\log^+\|f(re^{i\varphi})\|\,d\varphi$, the counting functions $N(r,a)$ (with multiplicity) and $\overline N(r,a)$ (reduced) of $a$-points for $a\in E\cup\{\infty\}$, the volume function $V(r,a)=\int_0^r\frac{v(t,a)}{t}\,dt$ with $v(t,a)=\frac{1}{2\pi}\int_{|\xi|<t}\Delta\log\|f(\xi)-a\|\,d\sigma\,d\tau$, the volume deficiency $\delta_V(a)=1-\limsup_{r\to\infty}V(r,a)/T(r,f)$, the curvature function $G(r,f)=\int_0^r\frac{dt}{2\pi t}\int_{|\xi|<t}\Delta\log\|f'(\xi)\|\,d\sigma\,d\tau$ with Ricci index $\delta_G=\liminf_{r\to\infty}G(r,f)/T(r,f)$, the further deficiencies $\delta(a)=1-\limsup_{r\to\infty}(V(r,a)+N(r,a))/T(r,f)$, $\Theta(a)=1-\limsup_{r\to\infty}(V(r,a)+\overline N(r,a))/T(r,f)$, $\delta_N(a)=\liminf_{r\to\infty}N(r,a)/T(r,f)$, $\Delta_N(a)=\limsup_{r\to\infty}N(r,a)/T(r,f)$, $\delta_{\overline N}(a)=\liminf_{r\to\infty}\overline N(r,a)/T(r,f)$, and $\Delta_{\overline N}(a)=\limsup_{r\to\infty}\overline N(r,a)/T(r,f)$, the deficiency sets $E(\delta_V)=\{a:\delta_V(a)>0\}$, $E(\Delta_{\overline N})=\{a:\Delta_{\overline N}(a)>0\}$, and $E(\delta_{\overline N})=\{a:\delta_{\overline N}(a)>0\}$ (similarly for the other functionals), and calling $a$ completely multiple when every $a$-point has multiplicity at least $2$ (the multiplicity of an $a$-point $z_0$ is the order of vanishing of $f-a$ at $z_0$, i.e. the least integer $m\ge 1$ such that $(f-a)^{(m)}(z_0)\neq 0$; for $a=\infty$ it is the order of the pole), Pujari (2018) proved, for admissible $f$ of compact projection (admissibility: $S(r_\gamma)/T(r_\gamma,f)\to 0$ along some $r_\gamma\to\infty$ for the second-fundamental-theorem error term $S$; compact projection: the coordinate projections $P_n f$, with $P_n$ the projection of $E$ onto $\operatorname{span}(e_1,\dots,e_n)$, converge to $f$ uniformly on compact subsets of the domain):

(F) for every $\eta$ with $0<\eta\le 1$ there are at most $(4-2\delta_G)/\eta$ distinct points $a\in E\cup\{\infty\}$ that are completely multiple for $f$ and satisfy $\delta_V(a)\ge\eta$; if $f$ is entire, the bound improves to $(2-2\delta_G)/\eta$;

(C) if $E(\Delta_{\overline N})$ is countable then $E(\delta_V)$ is countable, and conversely, if $E(\delta_V)$ is countable then $E(\delta_{\overline N})$ is countable;

and, separately, for an arbitrary $E$-valued meromorphic $f$ (no admissibility or compact projection assumed) under the vanishing counting hypothesis $N(r,a)/T(r,f)\to 0$ for every $a\in E\cup\{\infty\}$:

(S) the sum inequalities $\sum_{a\in E}\delta_V(a)+\delta_G\le 1+\limsup_{r\to\infty}\overline N(r,f)/T(r,f)$ (and the analogous inequalities for $\sum\delta(a)$ and $\sum\Theta(a)$) hold, and in this regime $\delta(a)=\delta_V(a)=\Theta(a)$ with $E(\delta_V)$ countable.

The common scientific objective of the problem is to determine the robustness and optimality of this volume-deficiency theory: decide, for each of the result families (F), (C), and (S), whether it is a genuine infinite-dimensional phenomenon or an artifact of its auxiliary structural hypotheses, and settle the optimality of the numerical constants. Concretely:

(Q1) Hypothesis necessity and extension. For an $E$-valued meromorphic $f$ (with $E$ as above) that fails admissibility or fails compact projection: does (F), or the countability conclusion of (C), fail, or can these conclusions (possibly with modified constants) be proven for all $E$-valued meromorphic maps, or under explicitly weaker hypotheses on $f$ or on $E$? For (S), which is already available for arbitrary $E$-valued meromorphic maps, the corresponding open regime is whether its vanishing counting hypothesis can be weakened or dropped. In particular, characterize the maps or spaces for which the auxiliary set $E(\Delta_{\overline N})$ is countable - the condition on which the source countability theorem leans - including whether an $E$-valued meromorphic map with uncountable $E(\Delta_{\overline N})$ or uncountable $E(\delta_V)$ exists.

(Q2) Sharpness. Are the constants $(4-2\delta_G)/\eta$ and $(2-2\delta_G)/\eta$ in (F) best possible - that is, can $4$ or $2$ be replaced by a smaller constant while the corresponding bound remains valid for every admissible compact-projection map (respectively, for every such entire map)? In particular, already for admissible compact-projection maps with $\delta_G=0$: for each $\eta$ with $0<\eta\le 1$, does there exist such a map (entire, for the second constant; the map may depend on $\eta$) with at least $\lfloor 4/\eta\rfloor$ (respectively $\lfloor 2/\eta\rfloor$) completely multiple values of volume deficiency at least $\eta$, so that the constants $4$ and $2$ cannot be decreased - or can the bounds be improved, with or without additional geometric hypotheses on the norm of $E$?

The problem therefore has two parts, (Q1) and (Q2), which are posed jointly.

References

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