Plurisubharmonic functions in calibrated geometry and q-convexity

Let (M, ω) be a Kähler manifold. An integrable function ${\varphi}Let (M, ω) be a K?hler manifold. An integrable function j{\varphi} on M is called ω ^q -plurisubharmonic if the current dd^cjùw^q-1{dd^c\varphi\wedge \omega^{q-1}} is positive. We prove that j{\varphi} is ω ^q -plurisubharmonic if and only if j{\varphi} is subharmonic on all q-dimensional complex subvarieties. We prove that a ω ^q -plurisubharmonic function is q-convex, and admits a local approximation by smooth, ω ^q -plurisubharmonic functions. For any closed subvariety Z ì M{Z\subset M} , dim_\mathbbC Z ￡ q-1{\dim_\mathbb{C} Z\leq q-1} , there exists a strictly ω ^q -plurisubharmonic function in a neighbourhood of Z (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony’s lemma on integrability of positive closed (p, p)-forms which are integrable outside of a complex subvariety of codimension ≥  p + 1.     

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J, K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics. We prove a quaternionic analogue of A. D. Aleksandrov and ChernLevine-Nirenberg theorems.     

Simple global minimization algorithm for one-variable rational functions

Sturm's chain technique for evaluation of a number of real roots of polynomials is applied to construct a simple algorithm for global optimization of polynomials or generally for rational functions of finite global minimal value. The method can be applied both to find the global minimum in an interval or without any constraints. It is shown how to use the method to minimize globally a truncated Fourier series. The results of numerical tests are presented and discussed. The cost of the method scales as the square of the degree of the polynomial.     

Plurisubharmonic and holomorphic functions relative to the plurifine topology

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if it is locally bounded from above in the plurifine topology and f°h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.     

Classes characterized by best-possible approximation by spline functions

Certain classes of functions are introduced which are characterized by their best-possible approximation by splines.Translated from Matematicheskie Zametki, Vol. 8, No. 2, 137–148, August, 1970.     

A sharp estimate for extremal functions

We prove a sharp pointwise estimate for extremal functions of invariant subspaces of some weighted Bergman spaces on the unit disk. The allowed weights include standard radial weights and logarithmically subharmonic weights.

     

Plurisubharmonic domination

For a large class of separable Banach spaces we prove the following. Given a pseudoconvex open and that is locally bounded above, there is a plurisubharmonic such that . We also discuss applications of this result.

     

An extremal property of outer functions

Our main result is the following: iff (z) is in the space H², and F(z) is its outer part, then F⁽ⁿ⁾_H2F⁽ⁿ⁾_H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971.     

On extremal points of quasiconvex functions

In this paper we examine the linear sectionwise relative minimums of a quasiconvex function and give a sufficient condition for quasiconvex functions to have a strict global minimum on an open convex set.