Zero distribution of solutions of complex linear differential equations determines growth of coefficients

It is shown that the exponent of convergence λ(f) of any solution f of with entire coefficients A₀(z), …, A_k?2(z), satisfies λ(f) ? λ ∈ [1, ∞) if and only if the coefficients A₀(z), …, A_k?2(z) are polynomials such that for j = 0, …, k ? 2. In the unit disc analogue of this result certain intersections of weighted Bergman spaces take the role of polynomials. The key idea in the proofs is W. J. Kim’s 1969 representation of coefficients in terms of ratios of linearly independent solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Carleson Measures for the Bloch Space

In this paper we study the positive Borel measures μ on the unit disc in for which the Bloch space is continuously included in , 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure μ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for μ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator , defined by , maps continuously into the Bergman space A ¹, . Furthermore, we prove that if p > 1, α >-1 and ω is a weight which satisfies the Bekollé-Bonami -condition, then the measure defined by is a p-Bloch-Carleson-measure. We also consider the Banach space of those functions f which are analytic in and satisfy , as . The Bloch space is contained in . We describe the p-Carleson measures for and study weighted composition operators and a class of integration operators acting in this space. We determine which of these operators map continuously to the weighted Bergman space and show that they are automatically compact. This research is partially supported by several grants from “the Ministerio de Educación y Ciencia, Spain” (MTM2005-07347, MTM2007-60854, MTM2006-26627-E, MTM2007-30904-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032); from “La Junta de Andalucía” (FQM210 and P06-FQM01504); from “the Academy of Finland” (210245) and from the European Networking Programme “HCAA” of the European Science Foundation.     

Straightforward Synthesis of the Brønsted Acid hfipOSO3H and its Application for the Synthesis of Protic Ionic Liquids

The easily accessible hexafluoroisopropoxysulfuric acid ( 1 , hfipOSO₃H ; hfip=C(H)(CF₃)₂) was synthesized by the reaction of hexafluoroisopropanol and chlorosulfonic acid on the kilogram scale and isolated in 98 % yield. The calculated gas‐phase acidity (GA) value of 1 is 58 kJ mol^?1 lower in ΔG° than that of sulfuric acid (GA value determined by a CCSD(T)‐MP2 compound method). Considering the gas‐phase dissociation constant as a measure for the intrinsic molecular acid strength, a hfipOSO₃H molecule is more than ten orders of magnitude more acidic than a H₂SO₄ molecule. The acid is a liquid at room temperature, distillable at reduced pressure, stable for more than one year in a closed vessel, reactive towards common solvents, and decomposes above 180 °C. It is a versatile compound for further applications, such as the synthesis of ammonium‐ and imidazolium‐based air‐ and moisture‐stable protic ionic liquids (pILs). Among the six synthesized ionic compounds, five are pILs with melting points below 100 °C and three of them are liquids at nearly room temperature. The conductivities and viscosities of two representative ILs were investigated in terms of Walden plots, and the pILs were found to be little associated ILs, comparable to conventional aprotic ILs.     

Univalent functions in Hardy, Bergman, Bloch and related spaces

The aim of this paper is to show that univalent functions in several classical function spaces can be characterized by integral conditions involving the maximum modulus function. For a suitable choice of parameters, the established condition or its appropriate variant reduces to a known characterization of univalent functions in the Hardy or weighted Bergman space and gives a new characterization of univalent functions in several M?bius invariant function spaces, such as BMOA, Q _p or the Bloch space. It is proved, for example, that univalent functions in the Dirichlet type space are the same as the univalent functions in H _α ^p and S _α ^p if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller M?bius invariant subspace of the Bloch space than Q _p still containing all univalent Bloch functions. This research has been supported in part by the MEC-Spain MTM2005-07347, the Spanish Thematic Network MTM2006-26627-E, and the Academy of Finland 210245.     

Lipschitz Continuous and Compact Composition Operators in Hyperbolic Classes

Natural metrics in the hyperbolic α-Bloch-, weighted Dirichlet- and Q _p -classes are introduced, and these classes are shown to be complete metric spaces with respect to the corresponding metrics. Then Lipschitz continuous and compact composition operators C _φ (f) = f ◦ φ acting from the hyperbolic α-Bloch-class to the hyperbolic weighted Dirichlet- or Q _p -class are characterized by conditions depending on the symbol φ only.     

Finiteness of φ-order of solutions of linear differential equations in the unit disc

If φ: [0, 1) → (0,∞) is a non-decreasing unbounded function, then the φ-order of a meromorphic function f in the unit disc is defined as $$ \sigma _\phi (f) = \mathop {\lim \sup }\limits_{r \to 1^ - } \frac{{\log ^ + T(r,f)}} {{\log \phi (r)}}, $$ where T(r, f) is the Nevanlinna characteristic of f. In particular, $ \sigma _{\tfrac{1} {{1 - r}}} $ f is the order of f, and $ \sigma _{\log \tfrac{1} {{1 - r}}} $ f is the logarithmic order of f. Several results on the finiteness of the φ-order of solutions of $$ f^{(k)} + A_{k - 1} (z)f^{(k - 1)} + \cdots + A_1 (z)f' + A_0 (z)f = 0 $$ are obtained in the case when the coefficients A ₀(z), ...,A _k?1(z) are analytic functions in the unit disc. This paper completes some earlier results by various authors.     

Zero sequences,factorization and sampling measures for weighted Bergman spaces

The zero sets of the Bergman space \(A^p_\omega \) induced by either a radial weight \(\omega \) admitting a certain doubling property or a non-radial Bekollé-Bonami type weight are characterized in the spirit of Luecking’s results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz’s factorization result from 1977 for functions in \(A^p_\omega \). The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space \(A^p_\omega \) induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.

     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.