Albert Petrosyan's Publications (Full profile)

We introduce the Banach spaces $h_{\infty}(\varphi)$, $h_{0}(\varphi)$ and $h^{1}(\psi)$ functions harmonic in the unit ball $\mathbb{B}\subset\mathbb{R}^n$. These spaces depend on weight functions $\varphi$, $\psi$. We prove that if $\varphi$ and $\psi$ form a normal pair, then $h^{1}(\psi)^*\sim h_{\infty}(\varphi)$ and $h_{0}(\varphi)^*\sim h^{1}(\psi)$.

The paper studies the Banach spaces $h_{\infty}(\varphi)$, $h_{0}(\varphi)$, and $h^{1}(\eta)$ of harmonic functions over the unit ball in $\mathbb{R}^n$ . These spaces depend on a weight function $\varphi$ and a weight measure $\eta$. For a given function $\varphi$ from a sufficiently broad class of functions, we solve the duality problem. that is, we construct measures $\eta$ such that $h^{1}(\eta)^*\sim h_{\infty}(\varphi)$ and $h_{0}(\varphi)^*\sim h^{1}(\eta)$.

$\mathbb C^n$-generalizations of Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions are studied. We find the values of parameter $\beta$ for which these operators are bounded on mixed norm spaces $L(p, q,\beta)$ over the unit ball in $\mathbb C^n$. Moreover, these operators are also bounded projectors, and the images of $L(p, q,\beta)$ under the projections are found.

We introduce the Banach spaces $h_{\infty}(\varphi)$, $h_{0}(\varphi)$, and $h^{1}(\eta)$ of functions harmonic in the unit ball in $\mathbb{R}^n$, depending on weight function $\varphi$ and weighting measure $\eta$. The paper studies the following question: for which $\varphi$ and $\eta$ we have $h^{1}(\eta)^*\sim h_{\infty}(\varphi)$ and $h_{0}(\varphi)^*\sim h^{1}(\eta)$. We prove that the necessary and sufficient condition for this is that certain linear operator, which projects $L^\infty (d\eta\, d\sigma )$ onto the subspace $\varphi h_\infty (\varphi)$, is bounded.

The paper studies some bounded operators in the Banach spaces $L^{\infty}(B)$ and $L^{1}(B)$ over the unit ball $B$ of $ \mathbb{C}^n$, the range of which are the corresponding holomorphic subspaces $A_{\infty}(\varphi)$ and $A^{1}(\psi)$ depending on a normal pair of weight-functions $\{\varphi, \psi \}$.

In the present paper the class of semi-analytical functions in the polydisk $U^n\subset \mathbb C^n$ is introduced. This class is an extension of the set of holomorphic functions. For $n =1$ the concept of semi-analyticity coincides with analyticity. The Dirichlet problem with values given on the distinguished boundary of the polydisk always has a solution in the set of real parts of semi-analytical functions. Therefore, to investigate semi-analytical functions one can apply the potential theory methods, like one does it for the one-dimensional case. In the present paper the Schwarz type integral representation for the above-mentioned functions is obtained.

In the paper the parametric representation for Hilbert spaces $b^2_\alpha$ of functions harmonic in the unit ball $B\subset \mathbb R^n$ is obtained.

In the Banach space $H^\infty(U)$ of bounded, holomorphic functions in the unit disc $U$ of the complex plane, the paper considers the subset $H^\infty(U;K)$ of functions that take values in a compact $K$. A description of the extreme points of $H^\infty(U;K)$ is given under rather general conditions on $K$.

It is proved, that any compact subset $K$ of an interpolation $C^{m,\alpha}$-smooth manifold on the distinguished boundary of unit polydisc $\mathbb{U}^n$ is a peak set for the algebra of smooth functions $A^{m-1,\alpha}(\mathbb{U}^n)$. Besides the following interpolation result is obtained: given $f\in \mathcal{C}^m(\mathbb{T}^n)$ there is $F\in A^{m-1,\alpha}(\mathbb{U}^n)$ with $F|_K=f|_K$. If given $f$ belongs to $\mathcal{C}^{m-1,\alpha}(\mathbb{T}^n)$, then the interpolating function $F$ can be chosen from $A_n^{m-1,\alpha}(\mathbb{U}^n)$.

It is proved that any function holomorphic in a real, non-degenerate Weil polyhedron $G$ and continuous in its closure $\overline{G}$ can be uniformly approximated by functions holomorphic in a neighborhood of $\overline{G}$. Besides, it is proved that such functions can be approximated by polynomials if $G$ is a polynomial polyhedron.

This paper relates to arbitrarily large $A^p_\omega$ spaces in the unit ball of $\mathbb C^n$. The introduced M.M.Djrbashian multidimensional kernel and the used technics allow to obtain the similarities of the representations proved in the early works of M.M.Djrbashian (1945--1948), which in essence gave rise to the theory of $A^p_\alpha$ (or initially $H^p(\alpha)$) spaces in the unit disc of the complex plane, and hence to extend to $\mathbb C^n$ several results of the one-dimensional theory. Nonetheless, the paper gives only the representation connected with a natural isometry between $A^2_\omega$ and the ordinary Hardy space $H^2$ in the ball, which has an explicit form of integral operator along with its inversion.

In the paper the formulas for derivatives of Cauchy type integral $K[f]$ of smooth function $f$ on the distinguished boundary of polydisc are given. These formulas express the derivatives of the order $m$ from $K[f]$ through the derivatives of lower order. We use them for the estimates of smoothness of derivatives of Cauchy type integral in terms of the Holder's ordinary scale and in terms of anisotropic Holder's classes.

В работе анонсируются основные результаты теории пространств $A^p_\omega$ в единичном шаре из $\mathbb C^n$.

В работе получено интегральное представление типа Коши-Грина для функций, гладких в замыкании полидиска $D^n$.

In this article we obtain a complete description of peak sets for the algebras $A^m$ of holomorphic functions on a polydisk and on the future tube in the case of nonintegers $m>2$. In the case of integers $m$ we obtain a sufficient condition that is very close to being necessary. It is proved that every peak set for $A^m$ is also an interpolation set for $A^m$ for nonintegers $m>2$.

Let $D$ be a real nondegenerate Weil polyhedron in two-dimensional complex space; $f$ be a $\overline\partial$-closed differential form of the (0,1) type, which is continuous in $\overline D.$ It is proved, that the $\overline\partial u=f$ equation has the solution which satisfies the $\|u\|\leqslant\gamma\|f\|$ uniforme estimate.

The paper is devoted to study of functions, semianalytical in sense of Bergman in the unit bidisc of the space $\mathbb C^2$. For these functions the two-dimensional analog of the of the Schwarz integral representation and the theorem of Herglotz are proved.

Let $D$ be a nondegenerate Weil polyhedron in two-dimensional complex space. It is proved, that every holomorphic in $D$ and continuous in $\overline D$ function may be uniformly approximated by functions, holomorphic in the neighbourhood of $\overline D$.

The paper studies some new $\mathbb C^n $-generalizations of Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We find the values of parameter $\beta$ for which these operators are bounded on mixed norm spaces $ L(p,q,\beta)$ over the unit ball in $\mathbb C^n $. Moreover, these operators are bounded projections as well, and the images of $ L(p,q,\beta)$ under the projections are found.