Том 106, № 1-2 (2019)

We establish nontrivial solvability conditions for the homogeneous double integral equation

\(S(x, y)=\int_{0}^{\infty} \int_{0}^{\infty} K\left(x-x^{\prime}, y-y^{\prime}\right) S\left(x^{\prime}, y^{\prime}\right) d x^{\prime} d y^{\prime}, \quad(x, y) \in \mathbb{R}_{+} \times \mathbb{R}_{+},\)

\(0 \leq K \in L_{1}, \quad \iint_{\mathbb{R}^{2}} K(x, y) \ d x\ d y=1\)

In this paper, we prove the boundedness of matrix Hausdorff operators and rough Hausdorff operators in the two weighted Herz-type Hardy spaces associated with both power weights and Muckenhoupt weights. By applying the fact that the standard infinite atomic decomposition norm on two weighted Herz-type Hardy spaces is equivalent to the finite atomic norm on some dense subspaces of them, we generalize some previous known results due to Chen et al. [7] and Ruan, Fan [35].

Estimates of the rate of convergence in the Birkhoff ergodic theorem which hold almost everywhere are considered. For the action of an ergodic automorphism, the existence of such estimates is proved, their structure is studied, and unimprovability questions are considered.

This paper deals with one-parameter families of integer translates of functions. It is shown that, as the scaling multiplier tends to infinity, the nodal interpolation functions converge to the sample function and the ratio of the upper and lower Riesz constants tends to 2. The assertion about convergence in the limit to the sample function is also proved for functions obtained by orthogonalization of the system of translates of the Gauss function and for the tight Gabor window functions as the ratio of the parameters of the time-frequency window tends to infinity.

A new estimate of the Kloosterman sum with primes modulo a prime number q is obtained, in which the number of summands can be of order q^0.5+^ε. This estimate refines results obtained earlier by J. Bourgain (2005) and R. Baker (2012).

In this paper, we compute the degree of the Kodiyalam polynomials of an ideal in the case where its Rees ring is Cohen—Macaulay and its fiber ring is a domain. We apply this result to some classes of polymatroidal ideals.

On the space of a principal bundle, a Lorentzian metric and a time orientation are given that are invariant with respect to the action of the structure group. These objects form a fibered space-time and, in the case of spacelike fibers, induce the same structures on the base. The following causality conditions are discussed: chronology, causality, stable and strong causality, and global hyperbolicity. It is proved that if the base space-time satisfies one of the above conditions, then so does the fibered space-time.

Necessary and sufficient conditions are found under which a symmetric space X on [0,1] of type 2 has the following property, which was first proved for the spaces L_p, p > 2, by Kadets and Pełczyński: if \(\left\{ {{u_n}} \right\}_{n = 1}^\infty \) is an unconditional basic sequence in X such that

\({\left\| {{u_n}} \right\|_X}\;\asymp\;{\left\| {{u_n}} \right\|_{{L_1}}},\;\;\;\;\;\;\;\;n\; \in \;\mathbb{N},\)

then the norms of the spaces X and L₁ are equivalent on the closed linear span [u_n] in X. For sequences of martingale differences, this implication holds in any symmetric space of type 2.

For the classes of functions

\({W^r}\left( {{\omega _{\cal M}},\;{\rm{\Phi }}} \right)\;: = \left\{ {f\; \in \;L_2^r\left(\mathbb{R}\right)\;:\;{\omega _{\cal M}}\left( {{f^{\left( r \right)}},\;t} \right)\; \le \;{\rm{\Phi }}\left( t \right)\;\forall \;t\; \in \;\left( {0,\;\infty } \right)} \right\},\)

where Φ is a majorant and r ∈ ℤ₊, lower and upper bounds for the Bernstein, Kolmogorov, and linear mean ν-widths in the space L₂(ℝ) are obtained. A condition on the majorant Φ under which the exact values of these widths can be calculated is indicated. Several examples illustrating the results are given.

This paper deals with several problems related to functions of the class \({\mathcal C}{\mathcal M}\) of completely monotone functions and functions of the class Φ(E) of positive definite functions on a real linear space E. Theorem 1 verifies some conjectures of Moak related to the complete monotonicity of the function x^−μ (x²+ 1)^−ν. Theorem 2 states that if f ∈ C^∞(0, + ∞) and δ ∈ ℝ, then

\(f\left( x \right) - {a^\delta }f\left( {ax} \right)\; \in \;{\mathcal C}{\mathcal M}\;\;\;\;\;\;{\rm{for}}\;{\rm{all}}\;\;\;\;a > 1\)

if and only if \( - \delta f\left( x \right) - xf\prime \left( x \right)\; \in \;{\cal C}{\cal M}\). A similar result for functions in Φ(E) is obtained in Theorem 9: if ε ∈ ℝ and a function h:[0, + ∞) → ℝ is continuous on [0, +œ) and differentiable on the interval (0, + œ) and satisfies the condition xh′ (x) → 0 as x → +0, then

\(h\left( {\rho \left( u \right)} \right) - {a^{ - \varepsilon }}h\left( {a\rho \left( u \right)} \right)\; \in \;{\rm{\Phi }}\left( E \right)\;\;\;\;\;\;{\rm{for}}\;{\rm{all}}\;\;\;\;a > 1\)

if and only if ψ_ε(p(u)) ∈ Φ(E), where ip_ε(x):= εh(x) − xh^′(x) for x > 0 and ψ_ε(0): = εh(0). Here p is a nonnegative homogeneous function on E and p(u) ≢ 0. It is proved (Example 6) that: (1) e^−α∥u∥ (1 − β∥u∥) ∈ Φ(ℝ^m) if and only if −α ≤ β ≤ a/m;(2) e^{−α∥u∥2} (1 − β∥u∥²) ∈ Φ(ℝ^m) if and only if 0 ≤ β ≤ 2α/m. Here ∥u∥ is the Euclidean norm on ℝ^m. Theorem 11 deals with the case of radial positive definite functions h_μ,ν.

Existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of x-dependent coefficients at u and u_x in a nondivergent parabolic equation from integral observation are obtained. Estimates of the maximum absolute values of these coefficients with constants explicitly expressed via the input data of the problem are given. An example of an inverse problem to which the proved theorems apply is presented.

The problem of sums of products in \(\mathbb{F}_p \times \mathbb{F}_p\) is considered. An estimate for sums of products improving Bourgain’s result of 2005 is obtained. This estimate is applied to the problem of estimating polynomial exponential sums over multiplicative subgroups in \(\mathbb{F}_p^*\).

The problem of the *-weak decomposability into ergodic components of a topological ℕ₀-dynamical system (Ω, φ), where φ is a continuous endomorphism of a compact metric space Ω, is considered in terms of the associated enveloping semigroups. It is shown that, in the tame case (where the Ellis semigroup E(Ω, φ) consists of endomorphisms of Ω of the first Baire class), such a decomposition exists for an appropriately chosen generalized sequential averaging method. A relationship between the statistical properties of (Ω, φ) and the mutual structure of minimal sets and ergodic measures is discussed.

Let G be a finite non-Abelian p-group, where p is an odd prime, such that G/Z(G) is metacyclic. We prove that all commuting automorphisms of G form a subgroup of Aut(G) if and only if G is of nilpotence class 2.

Let G be a finite non-Abelian p-group, where p is a prime. An automorphism α of G is called an IA-automorphism if x⁻¹α(x) ∈ G′ for all x ∈ G. An automorphism α of G is called an absolute central automorphism if, for all x ∈ G, x⁻¹α(x) ∈ L(G), where L(G) is the absolute center of G. Let C_IA(G)(Z(G)) and C_Var(_G)(Z(G)) denote, respectively, the group of all IA-automorphisms and the group of all absolute central automorphisms of G fixing the center Z(G) of G elementwise. We give necessary and sufficient conditions on a finite p-group G under which C_IA(G)(Z(G)) = C_Var(G)(Z(G)).