Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.     

The best <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Close two-sided estimates are obtained for the best approximation in the space L _p (? ^m ), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L _p (? ^m ). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best.     

Differences of idempotents in <Emphasis Type="Italic">C</Emphasis>*-algebras

Suppose that P and Q are idempotents on a Hilbert space H, while Q = Q* and I is the identity operator in H. If U = P ? Q is an isometry then U = U* is unitary and Q = I ? P. We establish a double inequality for the infimum and the supremum of P and Q in H and P ? Q. Applications of this inequality are obtained to the characterization of a trace and ideal F-pseudonorms on a W*-algebra. Let φ be a trace on the unital C*-algebra A and let tripotents P and Q belong to A. If P ? Q belongs to the domain of definition of φ then φ(P ? Q) is a real number. The commutativity of some operators is established.     

Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form f(x) = P(f)+K(?f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.     

Quasiopen and <Emphasis Type="Italic">p</Emphasis>-Path Open Sets,and Characterizations of Quasicontinuity

In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.     

Different prime <Emphasis Type="Italic">N</Emphasis>-ideals and IFP <Emphasis Type="Italic">N</Emphasis>-Ideals

We provide some characterizations of completely prime (completely semiprime) and 3-prime (3-semiprime) N-groups. The relationship between a 3-prime (completely prime) N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N is investigated. Moreover, the notion of IFP N-ideal is defined. We prove that the concept of IFP N-ideal occurs naturally where N is a left permutable (left self distributive, subcommutative) near-ring and Γ a monogenic N-group. Also, we obtain some relationships between an IFP N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N.     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

Sobolev-type integral representations for functions defined on Carnot groups

We obtain some integral representations of the form f(x) = P(f) + K(?f) on the Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. These representations are employed to prove the weak Poincaré inequality.     

Weak <Emphasis Type="Italic">WT</Emphasis><Subscript>2</Subscript>-class of differential forms and weakly <Emphasis Type="Italic">A</Emphasis>-harmonic tensors

In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.     

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.     

Martin Boundary of a Fine Domain and a Fatou-Naïm-Doob Theorem for Finely Superharmonic Functions

Let Γ be a graph with the doubling property for the volume of balls and P a reversible random walk on Γ. We introduce H¹ Hardy spaces of functions and 1-forms adapted to P and prove various characterizations of these spaces. We also characterize the dual space of H¹ as a BMO-type space adapted to P. As an application, we establish H¹ and H¹- L¹ boundedness of the Riesz transform.     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

Some new scales of weight characterizations of the class <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We present an equivalence theorem, which includes all known characterizations of the class B _p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B _p ^* , B _∞ ^* and RB _p , and prove and apply a “gluing lemma” of independent interest.     

Some connections between frames of radical ideals and frames of <Emphasis Type="Italic">z</Emphasis>-ideals

For any semisimple f-ring A with bounded inversion, we show that the frame of radical ideals of A and the frame of z-ideals of A have isomorphic subfit coreflections. If we assume the Axiom of Choice, then the two coreflections are actually identical. If the f-ring has the property that the sum of two z-ideals is a z-ideal (as in the case of rings of continuous functions), then the epicompletion of the frame of z-ideals is shown to be a dense quotient of the epicompletion of the frame of radical ideals. Baer rings, exchange rings, and normal rings that lie in the class of semisimple f-rings with bounded inversion are shown to have characterizations in terms of frames of z-ideal which are similar to characterizations in terms of frames of radical ideals.     

On ordered-covering mappings and implicit differential inequalities

We define the set of ordered covering of a mapping that acts in partially ordered spaces; we suggest a method for finding the set of ordered covering of vector functions of several variables and the Nemytskii operator acting in Lebesgue spaces. We prove assertions on operator inequalities in arbitrary partially ordered spaces. We obtain conditions that use a set of ordered covering of the corresponding mapping and ensure that the existence of an element u such that f(u) ≥ y implies the solvability of the equation f(x) = y and the estimate x ≤ u for its solution. We study the problem on the existence of the minimal and least solutions. These results are used for the analysis of an implicit differential equation. For the Cauchy problem, we prove a theorem on an inequality of the Chaplygin type.     

On the semiclassical functional calculus for <Emphasis Type="Italic">h</Emphasis>-dependent functions

We study the functional calculus for operators of the form \(f_h(P(h))\) within the theory of semiclassical pseudodifferential operators, where \(\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty _c}({{\mathbb {R}}})\) denotes a family of h-dependent functions satisfying some regularity conditions, and P(h) is either an appropriate self-adjoint semiclassical pseudodifferential operator in \(\mathrm{L}^2({{\mathbb {R}}}^n)\) or a Schrödinger operator in \(\mathrm{L}^2(M), M\) being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P(h) in spectral windows of width of order \(h^\delta \), where \(0\le \delta <\frac{1}{2}\).     

Discriminator positively complete classes of ternary logic

The article addresses the operator of positive closure on the set P _k of functions of k-valued logic. For each k ? 3, k ≠ 4, the set H _k of all homogeneous functions from P _k is proved to form an atom in the lattice of the positively closed classes from P _k . Also, we find all 17 positively closed classes from P ₃ containing the class H ₃ (i.e., discriminator positively closed classes). Positively generating systems of these classes are defined.     

On the sharpness of Jackson’s inequality in the spaces L p on the half-line with power weight

In the space L _p , 1 ≤ p < 2, on the half-line with power weight, Jackson’s inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson’s inequality, we obtain a lower bound, which proves its sharpness.     

Two classes of <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P ₁ and P ₂ of τ-measurable operators and investigate their properties. The class P ₂ contains P ₁. If a τ-measurable operator T is hyponormal, then T lies in P ₁; if an operator T lies in P _k , then UTU* belongs to P _k for all isometries U from M and k = 1, 2; if an operator T from P ₁ admits the bounded inverse T ^?1, then T ^?1 lies in P ₁. We establish some new inequalities for rearrangements of operators from P ₁. If a τ-measurable operator T is hyponormal and T ⁿ is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P ₁ coincides with the set of all paranormal operators on H.