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Vol 211, No 12 (2020)
- Year: 2020
- Articles: 5
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7472
Necessary and sufficient conditions for extending a function to a Schur function
Abstract
A criterion for a function given by its values (with multiplicities) at a sequence of points in the disc $\mathbb D=\{|z|<1\}$ to extend to a holomorphic function in $\mathbb D$ with modulus at most $1$ is stated and proved. In the case when the function is defined by the values of its derivatives at $z=0$, this coincides with Schur's well-known criterion. Bibliography: 16 titles.
Matematicheskii Sbornik. 2020;211(12):3-48
3-48
On Weyl multipliers of the rearranged trigonometric system
Abstract
We prove that the condition $\sum_{n=1}^\infty1/(nw(n))<\infty$ is necessary for an increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property was known long ago for Haar, Walsh, Franklin and some other classical orthogonal systems. The proof of this result is based on a new sharp logarithmic lower bound on $L^2$ for the majorant operator related to the rearranged trigonometric system. Bibliography: 32 titles.
Matematicheskii Sbornik. 2020;211(12):49-82
49-82
Renormalized solutions of elliptic equations with variable exponents and general measure data
Abstract
A class of second-order elliptic equations with variable nonlinearity exponents and the right-hand side in the form of the general Radon measure with finite total variation is considered. The existence of a renormalized solution of the Dirichlet problem is proved as a consequence of stability with respect to the convergence of the right-hand side of the equation. Bibliography: 37 titles.
Matematicheskii Sbornik. 2020;211(12):83-122
83-122
Two purity theorems and the Grothendieck-Serre conjecture concerning principal $\mathbf G$-bundles
Abstract
The main results of the paper are two purity theorems for reductive group schemes over regular local rings containing a field. Using these two theorems a well-known Grothendieck-Serre conjecture on principal bundles is reduced to the simply-connected case. We point out that the mentioned reduction is one of the major steps in the proof of the conjecture that the author published in another work. Bibliography: 25 titles.
Matematicheskii Sbornik. 2020;211(12):123-142
123-142
Proof of a conjecture of Wiegold for nilpotent Lie algebras
Abstract
Let $\mathfrak{g}$ be a nilpotent Lie algebra. By the breadth $b(x)$ of an element $x$ of $\mathfrak{g}$ we mean the number $[\mathfrak{g}:C_{\mathfrak{g}}(x)]$. Vaughan-Lee showed that if the breadth of all elements of the Lie algebra $\mathfrak{g}$ is bounded by a number $n$, then the dimension of the commutator subalgebra of the Lie algebra does not exceed $n(n+1)/2$. We show that if $\dim \mathfrak{g'} > n(n+1)/2$ for some nonnegative $n$, then the Lie algebra $\mathfrak{g}$ is generated by the elements of breadth $>n$, and thus we prove a conjecture due to Wiegold (Question 4.69 in the Kourovka Notebook) in the case of nilpotent Lie algebras. Bibliography: 4 titles.
Matematicheskii Sbornik. 2020;211(12):143-148
143-148