One-element differential standard bases with respect to inverse lexicographical orderings

We give a simplified proof of the following fact: for all nonnegative integers n and d the monomial y _n ^d forms a differential standard basis of the ideal [y _n ^d ]. In contrast to Levi’s combinatorial proof, in this proof we use the Gröbner bases technique. Under some assumptions we prove the converse result: if an isobaric polynomial f forms a differential standard basis of [f], then f = y _n ^d .     

Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions

In this note we investigate the convexity of zero-balanced hypergeometric functions with respect to Hölder mean.     

Weighted bounded mean oscillation applied to backward stochastic differential equations

We deduce conditional $L_p$-estimates for the variation of a solution of a BSDE. Both quadratic and sub-quadratic types of BSDEs are considered, and using the theory of weighted bounded mean oscillation we deduce new tail estimates for the solution $(Y,Z)$ on subintervals of $[0,T]$. Some new results for the decoupling technique introduced in Geiss and Ylinen (2019) are obtained as well and some applications of the tail estimates are given.     

Some new characterizations on spaces of functions with bounded mean oscillation

Let X be a space of homogeneous type. The authors introduce some generalized approximations to the identity (for short, GAI) with optimal decay conditions in the sense that these conditions are the sufficient and necessary conditions for these GAI's to characterize BMO(X), the space of functions with bounded mean oscillation on X. The authors also obtain a new John‐Nirenberg‐type inequality associated with GAI's, which leads some new characterizations of BMO(X) in terms of rearrangement functions, and certain maximal functions related to GAI's. Some variants of these characterizations for BMO‐type spaces on X of Duong and Yan are also established. Moreover, the equivalence between BMO and the space of BMO type on Ahlfors n ‐regular quasimetric measure spaces is obtained, which confirms an assertion of Duong and Yan (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Representation of analytic functions by series with respect to solutions of differential equations

Translated from Matematicheskie Zametki, Vol. 47, No. 4, pp. 106–114, April, 1990.     

Constructive decomposition of functions of finite central mean oscillation

The space CMO of functions of finite central mean oscillation is an analogue of BMO where the condition that the sharp maximal function is bounded is replaced by the convergence of the sharp function at the origin. In this paper it is shown that each element of CMO is a singular integral image of an element of the Beurling space of functions whose Hardy-Littlewood maximal function converges at zero. This result is an analogue of Uchiyama's constructive decomposition of BMO in terms of singular integral images of bounded functions. The argument shows, in fact, that to each element of CMO one can construct a vector Calderón-Zygmund operator that maps that element into the proper subspace .

     

The Riemann problem and expansion of entire functions of two variables with respect to quasiexponential bases

It is shown with the aid of the two-dimensional Borel transform that the expansion of entire functions of two variables of a finite degree with respect to a system of primitive functions or with respect to the system of Ozhegov polynomials is possible only when a certain two-dimensional Riemann problem which is equivalent to the Wiener-Hopf problem for a bidomain of the form {¦z¦>r, ¦¦>} is solvable uniquely and unconditionally.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 473–483, April, 1977.