Related papers: Elementary approach to the Hartogs extension theor…

We discuss several questions which remained open in our joint work with M. Sodin "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character--automorphic functions". In…

The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $\mathbb D\subset \mathbb C$ in the setting of Clifford algebras, based on a new convex combination identity.…

We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of…

We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use…

We prove, for Hermitian algebras, the multiplicative version of the Kowalski-S\l{}odkowski Theorem which identifies the characters among the collection of all complex valued functions on a Banach algebra $A$ in terms of a spectral…

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and H\"older-Zygmund spaces for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$, extending the…

Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction…

We prove a realization theorem for rational functions of several complex variables which extends the main theorem of M. Bessmertnyi, "On realizations of rational matrix functions of several complex variables," in Vol. 134 of Oper. Theory…

The Carath\'eodory extension theorem is a fundamental result in measure theory. Often we do not know what a general measurable subset looks like. The Carath\'eodory extension theorem states that to define a measure we only need to assign…

Examples by Poletsky and the author and by Zwonek show the existence nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions and contains…

The purpose of this paper is to develop some methods to study Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some…

Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that <a|b> is different from 1 if one of the points a, b is in B and…

Let $D, G\subset{\Bbb C}$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup(A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then…

Let M be a compact, connected, C^2-smooth and globally minimal hypersurface M in P_2(C) which divides the projective space into two connected parts U^{+} and U^{-}. We prove that there exists a side, U^- or U^+, such that every continuous…

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in $\mathbb{C}^n, \ n > 1$. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in…

The phenomenon of removable singularity is studied for overedetermined systems of differential equations. We show that the dimension of the characteristic variety plays a key role in the problem.

The results from the article [Strachan I.A.B., Szablikowski B.M., Stud. Appl. Math. 133 (2014), 84-117] are extended over consideration of central extensions allowing the introducing of additional independent variables. Algebraic conditions…

Let D be a bounded, finitely connected domain in the complex plane without isolated points in the boundary and let f be a continuous function on the boundary bD. Let F be a continuous extension of f to the closure of D. We prove that f…

We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of Horodecki Theorem, giving a necessary and sufficient criterion for…

A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…