Lacunary Walsh series in rearrangement invariant spaces

We prove that the classical results by Rodin and Semenov and by Lindenstrauss and Tzafriri on the subspace generated by the Rademacher system in rearrangement invariant spaces also hold for lacunary Walsh series.     

Lacunary series in weighted spaces of analytic functions

We improve a recent result of Yang and Xu (Arch. Math. 96 (2011), 151–160) by proving that if ψ is a normal function on [1, ∞) and \({f(z)=\sum_{n=0}^\infty a_n z^{k_n}}\) (|z| < 1) is an analytic function with Hadamard gaps, then

$\frac 1C \sup_{n\ge 0} \frac{|a_n|}{\psi(k_n)} \le \sup_{0 < r < 1} \frac{|f(r\zeta)|}{\psi(1/(1-r))} \le C\sup_{n\ge 0}\frac{|a_n|}{\psi(k_n)}, \quad |\zeta|=1,$

where C is a constant independent of ζ and {a _n }.

     

Korenblum maximum principle in mixed norm spaces

Archiv der Mathematik - It is well known that the Korenblum maximum principle holds in Bergman spaces $$mathrm {A}^{p}$$ if and only if $$pge 1$$ . In this note, we improve this result by proving...     

Lacunary series and sharp estimates in weighted spaces of holomorphic functions

The well-known Paley inequalities for lacunary series are applied in investigation of weighted spaces H(p, α) and H(p, log(α)) of functions holomorphic in the unit disc of the complex plane. These are spaces which are similar to the Bloch and Hardy spaces and naturally arise as the images of some fractional operators.     

Boundedness for iterated commutators on the mixed norm spaces

This paper studies the iterated commutators on mixed norm spaces L²(?) characterizing the conjugate holomorphic symbols for which the corresponding iterated commutators are bounded by using the Bergman geometry, properties of holomorphic functions and related analysis.     

Zeros of functions in weighted spaces with mixed norm

In the spaces of analytic functions f in the unit disk with mixed norm and measure satisfying the Δ₂-condition, sharp necessary conditions on subsequences of zeros $\{ z_{n_k } (f)\} $ of the function f are obtained in terms of subsequences of numbers {n _k }. These conditions can be used to define, in the spaces with mixed norm, subsets of functions with certain extremal properties; these subsets provide answers to a number of questions about the zero sets of the spaces under consideration and, in particular, about weighted Bergman spaces.     

Boundedness of the Bergman type operators on mixed norm spaces

Conditions sufficient for boundedness of the Bergman type operators on certain mixed norm spaces of functions on the unit ball of are given, and this is used to solve Gleason's problem for the mixed norm spaces .

     

Calderón--Lozanovskii; construction for mixed norm spaces

We show that the Calderón--Lozanovskii; construction φ(.) commutes with arbitrary mixed norm spaces, that is, φ(E₀[F₀], E₁[F₁]) = φ(E₀, E₁) [φ(F₀, F₁)] if and only if φ is equivalent to a power function. This result we obtain by giving characterizations of the corresponding embeddings of φ(E₀[F₀], E₁[F₁]) into φ₀ (E₀, E₁)[φ₁ (F₀, F₁)] and vice versa in terms of the functions φ, φ₀, φ₁. As a particular case, we get embeddings of an Orlicz space with mixed norms into an Orlicz space on a product of measure spaces. Applications to classical operators between mixed norm Orlicz spaces are also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Boundedness of the Cesàro operator on mixed norm spaces

In this note, the boundedness of the Cesàro operator on mixed norm space , , is proved.

     

Lacunary series and pseudocontinuations

The main goal of this paper is to prove the following statement. Let be a function holomorphic and of bounded characteristic in the unit disk , where E is a Λ(1)-subset of ℤ₊. Assume that f has a pseudocontinuation of bounded characteristic to the annulus {zɛℂ:1<|z|<R}. Then f admits analytic continuation to the disk R . In particular, f is a polynomial if R=+∞. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 16–32. Translated by A. B. Aleksandrov.     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

Lacunary series and Tauberian-type hypotheses

In this article, we introduce some Tauberian-type conditions and prove variants of Hardy-Littlewood’s high indices theorem for certain classes of series.     

A class of integral operators on mixed norm spaces in the unit ball

This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball. The author is supported in part by the NNSF China (No. 10671115), grants from Specialized Research Fund for the doctoral program of Higher Education (No. 20060560002) and NSF of Guangdong Province (No. 06105468).     

Counting functions of zeros of analytic functions in spaces with mixed norm

We study the behavior of counting functions of zeros of analytic in a disk functions in spaces with mixed norm, in particular, the Bergman-Dzhrbashyan spaces with standard weights. We obtain corollaries that strengthen the known results on zero sets of spaces with mixed norm.     

The Gleason''''s problem on mixed norm spaces in convex domains

Let Ω be a bounded convex domain with C2 boundary in C2 and for given 0 < p, q ≤∞ and normal weight function (r) let Hp,q, be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω, a, Hp,q,) is solvable for any fixed point a ∈ Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on Hp,q,.     

Lacunary sets based on Lorentz spaces

A new lacunary set for compact abelian groups is introduced; this is called a set. This set is defined in terms of the Lorentz spaces and is shown to be a generalization of sets and Sidon sets. A number of functional-analytic statements about sets are established by making use of the structural similarities between spaces and Lorentz spaces. These statements are analogous to several well-known properties of a set which are equivalent to the definition of a set. Some general set-theoretic and arithmetic properties of sets are also developed; these properties extend known results on the structure of sets. Open problems and directions for further research are outlined.