Karen Avetisyan

Yerevan State University | YSU · Faculty of Mathematics and Mechanics

The paper studies families of two-parameter Bergman type operators Tβ,λ,Sβ,λ,Φβ,δ in mixed norm and Besov spaces on the unit ball of Rn. Motivated by a series of papers by Choe et al., we extend estimation theorems for harmonic reproducing kernels. This enables us to obtain boundedness of operators Tβ,λ,Sβ,λ,Φβ,δ on mixed norm and Besov spaces for...

The action of the fractional integration operator in weighted Lebesgue classes and mixed-norm spaces is studied in the unit ball from R n. Some results of Hardy, Littlewood, and Flett are refined and generalized.

Boundedness of some Poisson-Bergman type operators is stated over the unit ball in ℝⁿ. Forelli-Rudin type theorems are proved and bounded harmonic projections are found on Lipschitz and mixed norm spaces.

In a recent paper of ours, we proved that a special “harmonic conjugation” operator is not bounded in weighted Bergman spaces of quaternion‐valued functions in the 3D ball. In the present paper, we prove that, in contrast to the Bergman spaces case, the same operator is bounded in weighted Dirichlet spaces of quaternion‐valued functions in the 3D b...

The paper studies some new ℂⁿ -generalizations of Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We find the values of parameter β for which these operators are bounded on mixed norm spaces L(p, q, β) over the unit ball in ℂⁿ. Moreover, these operators are bounded projections as well, and t...

The paper considers Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We prove that there exist values of parameter ß for which these operators are bounded on mixed norm spaces L(p, q, ß) on the unit ball in Cⁿ.

In the paper, a new form of fractional derivative is introduced for functions defined in the unit ball in Rn . As an application, an integral representation for harmonic functions with finite mixed-norm in the ball is obtained.

The aim of the paper is to prove a monogenic version of classical M. Riesz theorem on harmonic conjugates in the framework of quaternionic analysis in \({\mathbb{R}^{4}}\) . Our proof is subharmonic and somewhat simpler than that for less general Riesz-Stein-Weiss systems of harmonic conjugate functions.

In continuation of recent studies, we discuss two constructive approaches for the generation of harmonic conjugates to find null solutions to the Riesz system in . This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. Our first algorithm for harmonic conjugates is based on special systems...

This note announces some results that will be presented in the forthcoming paper [10]. In continuation to these studies we discuss a constructive approach for the generation of harmonic conjugates to find nullsolutions to the Riesz system in R3. This class of solutions coincides with the subclass of monogenic functions with values in the reduced qu...

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ℝn are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogu...

We establish the sharpness and strictness of continuous inclusions in mixed-norm spaces of n-harmonic functions on the unit polydisc of ℂ. To this end, we modify a wellknown counterexample of Hardy and Littlewood and give a characterization of lacunary series with Hadamard gaps in mixed-norm and weighted Hardy spaces.

The paper establishes a necessary and sufficient condition under which a lacunary series belong to a mixed norm space of functions holomorphic in the unit disc. As a corollary, some sharp pointwise estimates are obtained for lacunary series. Key wordsLacunary series-Hadamard gaps-mixed norm spaces MSC2010 numbers30B10-30H05-30D55

In this paper we study the harmonic conjugation problem in weighted Bergman spaces of quaternion-valued functions on the unit ball. For a scalar-valued harmonic function belonging to a Bergman space, harmonic conjugates in the same Bergman space are found.

The main results of this note prove that the generalized Libera operator is bounded on the Besov mixed-norm space as well as on the spaces BMOA and VMOA on the unit disk. The compactness of the operator on is also studied.

The paper continues the investigation of holomorphic mixed norm spaces A<(p,q)((omega)over right arrow>) in the unit polydisc of C-n. We prove that a mixed norm is equivalent to a "derivative norm" for all 0 < p <= infinity, 0 < q < infinity and a large class of weights <(omega)over right arrow>. As an application, we prove that pluriharmonic conju...

Let Hp denote the Hardy space of holomorphic functions on the unit ball B. This note gives some sufficient and necessary conditions for the boundedness and compactness of the following extended Cesàro operatorswhere z∈B and g is a fixed holomorphic map on B, acting from the space Hp into the space Hq, for the case p<q. Our results extend and simpli...

We study anisotropic mixed norm spaces h(p,q,α) consisting of n-harmonic functions on the unit polydisc of \mathbb Cn{\mathbb C}^n by means of fractional integro-differentiation including small 0 < p < 1 and multi-indices α = (α 1,...,α n ) with non-positive α j  ≤ 0. As an application, two different Bloch spaces of n-harmonic functions are cha...

It is proved that the inequality delta(X) (epsilon) >= c epsilon(p) p >= 2, where delta(X) is the modulus of convexity of X, is sufficient and necessary for the inequality integral parallel to del f(z) parallel to(p) (1-vertical bar z vertical bar)(p-1) dA(z) <= C(parallel to f parallel to(p)(p.X) - parallel to f(0) parallel to(p)), where f is an X...

The paper generalizes the well-known Littlewood-Paley inequality and Hardy-Stein identity. As an application, some area inequalities and quasinorm representations in the space A ω p over the polydisc are obtained.

We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of . Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixe...

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.

In this paper we show that the following integrals ∫B |f(z)p(1 - |z|)αdV(z), ∫B |f(z)| p-q|∇ f(z)|q(1 - |z|)α+qdV(z), and ∫B |f(z)|p-q|R f(z)|q(1 -|z|) α+qdV(z), where p > 0, q ∈ [0, p], α ∈ (-1, ∞), and where f is a holomorphic function on the unit ball B in ℂn are comparable. This result confirms a conjecture proposed by the second author at seve...

The paper generalizes the well-known inequality of Littlewood-Paley in the poly-disc. We establish a family of inequalities which are analogues and extensions of Littlewood-Paley type inequalities proved by Sh. Yamashita and D. Luecking in the unit disk. Some other gener-alizations of the Littlewood-Paley inequality are stated in terms of anisotrop...

We introduce fractional -integro-differentiation for functions holomorphic in the upper half-plane. It gives us a tool to construct Cauchy-Bergman type kernels associated with the weights Some estimates of the kernels enable us to obtain reproducing integral formulas for Bergman spaces with general weights which may decrease to zero with arbitrary...

A new family of Littlewood–Paley type g-functions is defined and the related L p-inequalities are proved for n-harmonic and holomorphic functions on the unit polydisc of C n. The paper generalizes and improves the results of author's recent work, that gave a positive answer to Littlewood's question on extension of L p-inequalities to the case of se...

We study anisotropic mixed norm spaces of n-harmonic functions in the unit polydisc of Cn. Bergman type reproducing integral formulas are established by means of fractional derivatives and some continuous inclusions. It gives us a tool to construct corresponding projections and related operators and prove their boundedness on the mixed norm and Bes...

For n-harmonic functions on the unit polydisk in the space $$\mathbb{C}^n $$ we define g-functions of Littlewood--Paley type and establish Lp-inequalities related to them. In the present paper, the main theorems deal with the extension of results of Littlewood, Paley, and Flett to the polydisk and their generalizion to fractional derivatives of ar...

A Bergman type operator is constructed on the unit disc, that continuously projects a Besov space onto its harmonic subspace.

In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces h(p,q,�) on the half-space are established. We prove that mixed norm is equivalent to a "fractional derivative norm" and that harmonic conjugation is bounded in h(p,q,�) for the range 0 < p ≤ ∞, 0 < q ≤ ∞. As an application of t...

In this paper, we establish certain embedding theorems and integral representations in weighted classes of functions harmonic or holomorphic in the unit disk of the complex plane. In our representations, we use the Lipschitz classes of O. V. Besov on the unit circle.