Nevanlinna Class Research Articles

On the Ideal Structure of the Nevanlinna Class

On the Ideal Structure of the Nevanlinna Class

A conjecture of L. Carleson and applications.

Let $T_\alpha$ be the class of functions meromorphic in the unit disk ${\bf D}$ such that $$\int_D{|f'(z)|^2\over (1+|f(z)|^2)^2}(1-|z|)^{1-\alpha}dx\,dy<\infty,\quad 0\leq\alpha<1.$$ It is known that $T_\alpha\subset N$, where $N$ denotes the Nevanlinna class of functions meromorphic in $\bold D$ and of bounded characteristic. Because every $f\in N$ is a quotient of two bounded functions, analytic in ${\bf D}$, it is natural to ask whether any $f\in T_\alpha$ may be represented as a quotient of two bounded analytic functions of the same class. This problem was posed for the first time by Carleson in his thesis [``On a class of meromorphic functions and its associated exceptional sets'', Univ. Uppsala, Uppsala, 1950; MR0033354 (11,427c)]. Carleson proved that any $f\in T_\alpha$ is a quotient of bounded functions $u,v\in T_\beta$, analytic in ${\bf D}$, for all $\beta<\alpha$. The considerable success of the author in the paper under review is in proving the following result. Theorem: For $0<\alpha\leq 1$, every function in $T_\alpha$ is a quotient of two bounded analytic functions belonging to $T_\alpha$. (Less)

Entropy of inner functions

In this paper, we show that an inner functionf has finite entropy if and only if its derivativef′ lies in the Nevanlinna class. We prove also that the entropy off is given by the average of the logarithm of |f′|. The proof is based on the fact that, evenf being highly discontinuous on the circle, the action off −n on Borel subsets is smooth.

A note on radial variation of analytic functions

Let F denote a family of analytic functions in the unit disk Δ. Suppose that one has a “sharp” estimate on the almost everywhere radial variation of functions in the class Δ. We prove that if Δ is contained in the Nevanlinna class N then the estimate will be “sharp” in the algebra A of functions analytic in Δ and continuous in Δ.

Degenerate interpolation problems for Nevanlinna pairs

In this paper we study interpolation problems for pairs of matrix functions of the extended Nevanlinna class using two different approaches, namely via the Krei˛n-Langer theory of extensions of symmetric operators and via an adaptation of Dym's method to solve interpolation problems by means of the de Branges theory of Hilbert spaces of analytic functions. We give a description of all solutions when the corresponding Pick matrix is nonnegative and has a nontrivial kernel.

On normalized Bezout fractions of distributed LTI systems

The Hardy space H/sub infinity / and Bezout fractions are introduced. Some properties of transfer function matrices with entries in the Nevanlinna class are discussed. Shift-invariant subspaces in the Hardy space H/sub 2//sup k /are introduced, and the existence of normalized right Bezout fractions is proven. The existence of normalized left Bezout fractions and normalized Bezout fractions of discrete linear time-invariant (LTI) systems are considered. It is shown that if a transfer matrix with entries in the Nevanlinna class has a Bezout fraction, then it has a normalized one. This means that the full power of the theories developed by using normalized Bezout fractions can be applied to the transfer matrices with entries in the Nevanlinna class.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nevanlinna and Smirnov classes on the upper half plane

Nevanlinna and Smirnov classes on the upper half plane

Functions of bounded characteristic in n-connected domains

In this paper we define a metric d on the Nevanlinna class N(G) of an n-connected domain G. We show that ( N, d) is a complete translation invariant metric space in which multiplication by a constant is not continuous. We also show that N(G) is disconnected. Our main theorem characterizes the component of the origin of N(G), extending earlier work of J. Shapiro and A. Shields and of J. Roberts for the unit disk. We use the factorization theorem of R. Coifman and G. Weiss to show that the component of the origin of N(G) is equal to the set of all functions f in N(G) for which the positive part of the singular measure of the least harmonic majorant of log ¦ f¦ is continuous.

On Multipliers into Bergman Spaces and Nevanlinna Class

AbstractWe use the general factorisation theorems of Grothendieck, Nikishin and Maurey to characterise coefficient multipliers between Bergman spaces and into the Nevanlinna class.

Characteristic properties of the Nevanlinna class N and the Hardy classes H^p and H_h^p

Characteristic properties of the Nevanlinna class N and the Hardy classes H^p and H_h^p

Subsets of Hardy-class zero sets in the ball

We consider the problem of whether the union of complex hyperplanes can be a subset of a zero variety for the Hardy classes of the ball. A sufficient condition is found, consisting in a strong geometric separatedness requirement, together with a quantitative requirement slightly stronger than the necessary condition for Nevanlinna class zero varieties.

Boundary convergence of functions in the meromorphic Nevanlinna class

Boundary convergence of functions in the meromorphic Nevanlinna class

A nullset for normal functions in several variables

Suppose that Ω \Omega is a domain in C n , E ⊂ Ω {C^n},E \subset \Omega is closed in Ω \Omega , and f : Ω ∖ E → C ∗ f:\Omega \backslash E \to {C^ * } is a meromorphic function. We show that if f f is normal and E E is an analytic subvariety or, more generally, of locally finite ( 2 n − 2 ) (2n - 2) -dimensional Hausdorff measure in Ω \Omega satisfying a certain geometric condition, then f f can be extended to a meromorphic function (= holomorphic mapping) f ∗ : Ω → C ∗ {f^ * }:\Omega \to {C^ * } . In the case of a subvariety sufficient, but not necessary, for the geometric condition is that the singularities of E E are normal crossings. As a digression, we give a new proof for the following result, due to Parreau in the case n = 1 n = 1 : if f f is in the Nevanlinna class and E E is polar (in R 2 n {R^{2n}} ), then f f has a meromorphic extension f ∗ {f^ * } to Ω \Omega .

Prescribing Zeros of Functions in the Nevanlinna Class on Weakly Pseudo-Convex Domains in C 2

Prescribing Zeros of Functions in the Nevanlinna Class on Weakly Pseudo-Convex Domains in C 2

Some Classes of Functions with Exponential Decay in the Unit Ball of $\boldsymbol C^n$

One of the characterizing properties of the functions of bounded mean oscillation (BMO) is that their distribution functions have an exponential decay effect, i. e. the famous John-Nirenberg's theorema:i. In 1980, Baernstein® proved that the distribution functions of the non-tangential maximal functions decrease exponentially for a bounded subset of the Nevanlinna class in the unit disk, and as corollaries, he obtained an analytic form of John-Nirenberg's theorem with a weaker integrability assumption and pointed out that in the analytic category BMO is equivalent to BMO of logarithmic type. Long Ruilin and Yang LeC3] obtained similar results to Baernstein's theorem in the ^-dimensional real and complex ball by showing that BMO(10g)* = BMO for spaces of homogeneous type. In this paper, we try to generalize a series of the famous Baernstein's results for the unit disk to the unit ball with respect to different topological structures applying Rudin's function theory in the unit ball of CM. In order to lead to the discussion, we define a class of point sets in the ball in §2, where we point out that there is a useful geometric property of the intersections of this class of sets and the admissible domains Da (Q defined by Koranyi rs:i. The key part of this paper will be found in §3, where the decay characterizations will be studied for a bounded subset of a function space larger than the Nevanlinna class, which shall be referred to as H9 class in the present article. Hence the maximal function in the admissible domain

Prescribing zeros of functions in the Nevanlinna class on weakly pseudo-convex domains in 𝐶²

Let D D be a bounded weakly pseudo-convex domain in C 2 {{\mathbf {C}}^2} of uniform strict type. For any positive divisor M M of D D with finite area, there exists a holomorphic function f f in the Nevanlinna class such that M M is the zero set of f f . The proof is to study the solutions of ∂ ¯ \bar \partial with L 1 ( ∂ D ) {L^1}(\partial D) boundary values.

Estimates for the ∂̄-Neumann problem for pseudoconvex domains in C 2 of finite type

We outline results obtained for the partial differential-Neumann problem for an arbitrary pseudoconvex domain in C(2) of finite type. We obtain an approximation to the Neumann operator. A number of sharp estimates for the solution of partial differentialu = f are a consequence; one of these is an extension of the L(1) estimate of Henkin and Skoda used to characterize the zero sets of functions of the Nevanlinna class.

Invariant subspaces of ℋpfor multiply connected regions

A closed linear subspace oi^p(G) is said to be invariant if zf(z) is in Jt for all f(z) € Jt. It is said to be fully invariant if r(z)/(z) is in */# for all / € ^ and all rational functions r(z) with poles in the complement of G. This paper investigates those invariant subspaces of *3ΓP(G), for a multiply connected G, which are invariant but not fully invariant. We show that an invariant subspace ^# fails to be fully invariant if and only if there is one bounded component G, of the complement of G such that the ratio of any two functions in ^# has a pseudo-continu ation to a meromorphic function in the Nevanlinna class of G, . This allows us to give a complete characterization of those invariant subspaces of βfp(G) which contain the constants. 0. Introduction. Let G be a finitely connected bounded domain in C with smooth boundary contours. It is the purpose of this paper to study the closed linear subspaces of β?p(G) which are invariant under multiplication by z, that is, those subspaces Jt such that zf(z) is in Jt whenever f(z) is. The study of such spaces was initiated by Beurling [1] who gave a complete characterization of the invariant subspaces of ^ 2 (Δ), where Δ is the unit disk. Shortly thereafter Helson and Lowdenslager also investigated further problems of invariant subspaces using Beurling's methods. Beurling's characterization is not difficult to extend to general simply connected domains, but the problem of characterizin g the invariant subspaces for multiply connected domains is more complicated. A subspace Jt of JΓp(G) is said to be fully invariant if it is invariant under multiplication by rational functions whose poles are in the complement of G. For simply connected domains all invariant subspaces are fully invariant, but this is no longer true for multiply connected G. It is possible to give a characterizati on similar to Beurling's for the fully invariant subspaces of &P(G). This was carried out for the annulus by Sarason [12] and for more general domains by Hasumi [5]

Values in the interior or the $L^2$-minimal solutions of the $\partial\bar\partial$-equation in the unit ball of $\Bbb C^n$

We derive formulas for the values in the interior of the L2-minimal solutions of the ??-equation in the unit ball of Cn. These formulas generalize previously known formulas for the boundary values of the same solutions. We estimate the solution and obtain a (known) result concerning weighted Nevanlinna classes.

Removable Singularities in the Nevanlinna Class and in the Hardy Classes

Removable Singularities in the Nevanlinna Class and in the Hardy Classes