On functions with zero spherical means on complex hyperbolic spaces

The class of functions with zero integrals over all balls of one fixed radius on the complex hyperbolic space is described. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 504–512, October, 2000.     

The removability problem for functions with zero spherical means

We study the functions on the punctured n-dimensional sphere having zero integrals over all admissible “hemispheres.” We find a condition for the point to be a removable set for this class of functions and show that the condition cannot be dropped or substantially improved.     

The extension problem for functions with zero weighted spherical means

We study functions on a sphere with a pricked point having zero integrals with a given weight over all admissible “hemispheres”. We find a condition under which the point is a removable set for such a class of functions. We show that this condition cannot be dropped or substantially weakened.     

Analyticity of functions analytic on circles

Let Δ be the open unit disc in, let pbΔ, and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p. Then f is holomorphic on Δ.     

A note on means of entire functions

A result of Singh and Sreenivasulu is proved under less restrictive hypothesis. It is also shown that if a condition on the coefficients is not satisfied, the theorem will not hold.     

A polynomial decomposition of Boolean functions

Translated from Matematicheskie Zametki, Vol. 53, No. 2, pp. 25–29, February, 1993.     

Acharacterization of Newtonian functions with zero boundary values

We give an intrinsic characterization of the property that the zero extension of a Newtonian function, defined on an open set in a doubling metric measure space supporting a strong relative isoperimetric inequality, belongs to the Newtonian space on the entire metric space. The theory of functions of bounded variation is used extensively in the argument and we also provide a structure theorem for sets of finite perimeter under the assumption of a strong relative isoperimetric inequality. The characterization is used to prove a strong version of quasicontinuity of Newtonian functions.     

Ratios of harmonic functions with the same zero set

We study the ratio of harmonic functions u,v which have the same zero set Z in the unit ball \({B\subset \mathbb{R}^n}\). The ratio \({f=u/v}\) can be extended to a real analytic nowhere vanishing function in B. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set \({K\subset B}\) we show that \({\sup_K|f|\le C_1\inf_K|f|}\) and \({\sup_K\left|\nabla f\right|\le C_2 \inf_K|f|}\), where C ₁ and C ₂ depend on K and Z only. In dimension two we specify the dependence of the constants on Z in these inequalities by showing that only the number of nodal domains of u, i.e. the number of connected components of \({B\setminus Z}\), plays a role.     

On functions with given spherical means on symmetric spaces

The inversion problem for a local Pompeiu transformation of rank one on sym- metric spaces X of the noncompact type is studied. The reconstruction of a function defined in the ball B _R ⊂ X by its averages on balls of two fixed radii lying in B _R is obtained.     

On entire functions with zero as a deficient value

We consider entire functions of finite order for which zero is a Nevanlinna deficient value. Estimates are determined for the ratio of the integrated moduli of the logarithmic derivative of the function on a circle of radius r to the Nevanlinna characteristic of the function at r.     

Random summation of independent identically distributed random vectors with zero means

Translated from:Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, 1989, pp. 20–26.