Vol 102, No 5-6 (2017)

Some new extensions and refinements of Hermite–Hadamard and Fejér type inequalities for functions which are N-quasiconvex are derived and discussed.

We establish criteria for the validity of modular inequalities for the Hardy operator on the cone Ω of nonnegative decreasing functions from weighted Orlicz spaces with general weight. The result is based on the theorem on the reduction of modular inequalities for positively homogeneous operators on the cone Ω, which enables passing to modular inequalities for modified operators on the cone of all nonnegative functions from an Orlicz space. It is shown that, for the Hardy operator, the modified operator is a generalized Hardy operator. This enables us to establish explicit criteria for the validity of modular inequalities.

In L₂(ℝ₃;ℂ₃), we consider a self-adjoint operator ℒ_ε, ε > 0, generated by the differential expression curl η(x/ε)⁻¹ curl−∇ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τℒ_ε^1/2) and ℒ_ε^−1/2 sin(τℒ_ε^1/2) for τ ∈ ℝ and small ε. It is shown that these operators converge to cos(τ(ℒ⁰)^1/2) and (ℒ₀)^−1/2 sin(τ(ℒ₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H^s (with a suitable s) to ℒ₂. Here ℒ⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ∂_τ²v_ε = −ℒ_εv_ε, div v_ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).

We give conditions on the exponent function p( · ) that imply the existence of embeddings between the grand, small, and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent results by the second author, Rakotoson and Sbordone.

Let Γ be a simply connected unbounded C²-hypersurface in ℝⁿ such that Γ divides ℝⁿ into two unbounded domains D^±. We consider the essential spectrum of Schrödinger operators on ℝⁿ with surface δ_Γ-interactions which can be written formally as

\({H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}\)

\({H_\Gamma } = - \Delta + Won{\mathbb{R}^n}\backslash \Gamma \)

In this paper, we study the boundedness of the fractional integral operator I_α on Carnot group G in the generalized Morrey spaces M_{p, φ}(G). We shall give a characterization for the strong and weak type boundedness of I_α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.

Asymptotic formulas are obtained for a class of integrals that are Fourier transforms of rapidly oscillating functions. These formulas contain special functions and generalize the well-known method of stationary phase.

We study composition operators on Lorentz spaces. In particular, we obtain necessary and sufficient conditions under which a measurable mapping induces a bounded composition operator.

Using geometrical and algebraic ideas, we study tunnel eigenvalue asymptotics and tunnel bilocalization of eigenstates for certain class of operators (quantum Hamiltonians) including the case of Penning traps, well known in physical literature. For general hyperbolic traps with geometric asymmetry, we study resonance regimes which produce hyperbolic type algebras of integrals of motion. Such algebras have polynomial (non-Lie) commutation relations with creation-annihilation structure. Over this algebra, the trap asymmetry (higher-order anharmonic terms near the equilibrium) determines a pendulum-like Hamiltonian in action-angle coordinates. The symmetry breaking term generates a tunneling pseudoparticle (closed instanton). We study the instanton action and the corresponding spectral splitting.

It is proved that every elementary carpet of nonzero additive subgroups which is associated with a Chevalley group of a Lie rank exceeding one over a locally finite field coincides, up to conjugation by a diagonal element, with a carpetwhose additive subgroups are equal to some chosen subfield of the ground field. A similar result is obtained for a full matrix carpet (a full net).

A chain bicomplex for A_∞-algebras, which generalizes the Tsygan chain bicomplex in the theory of cyclic homology of associative algebras, is constructed by using the techniques of differential modules with ∞-simplicial faces and D_∞-differential modules. For homotopy unital A_∞-algebras, an exact sequence generalizing the Connes–Tsygan exact sequence for unital associative algebras is obtained.

In a multidimensional projective space, a distribution of planes is considered. Under the assumption that there is a relative invariant scoped by a subobject of a fundamental object of the first order, an internal composite equipment of the distribution is made, which is an analog of the Cartan equipment and Norder normalization of the second kind. It is proved that the composition equipment induces six bunches of group connections in the associated principal bundle which are intrinsically determined by the distribution itself. In every bundle, a unique intrinsic connection is distinguished. Analytic and geometric conditions for the coincidence of different types of connections are found. In the paper, the Cartan–Laptev method is used. All considerations are of local nature.

For solutions of linear, weakly nonlinear, and quasilinear first-order differential equations, we obtain theorems on the unique unbounded continuation of germs along continuous curves contained in the integral submanifolds of the distribution induced by the major part of the equation.