Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

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.     

Reconstruction from Line Integrals in Spaces of Constant Curvature

New reconstruction formula for the line integral transformation in Euclidean spaces is found. The general k-plane integral transform in Euclidean space is related to a totally geodesic integral transform for an arbitrary Riemannian space of constant curvature by means of a factorization property. Duality theorems for the totally geodesic transforms are stated.     

On Monotonic Integrable Solutions for Quadratic Functional Integral Equations

We study the solvability of functional quadratic integral equations in the space of integrable functions on the interval I = [0, 1]. We concentrate on a.e. monotonic solutions for considered problems. The existence result is obtained under the assumption that the functions involved in the investigated equation satisfy Carathéodory conditions. As a solution space we consider both L ₁(I) and L _p (I) spaces for p > 1.     

«Assoluta continuità» e «Quasi sub-additività rispetto a una coppia» in spazi di misura

A well-known theorem due to J. C. Burkill and some existence theorems for Cesari's integral obtained in [1] are generalized here in measure spaces. Moreover, the last result, where is used the concept of quasi sub-additivity with respect to two families \((\mathfrak{D},\mathfrak{D}'),\mathfrak{D}' \subset \mathfrak{D}\) , and to the mesh for a set function ψ:I?ψ(I),I ∈ {I}, introduced in [1], is an extension of a theorem obtained by L. Cesari in [5].     

Maximal element theorems in product FC-spaces and generalized games

Let I be a finite or infinite index set, X be a topological space and (Yi,{φNi})_i∈I be a family of finitely continuous topological spaces (in short, FC-space). For each i∈I, let be a set-valued mapping. Some existence theorems of maximal elements for the family {Ai}_i∈I are established under noncompact setting of FC-spaces. As applications, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in noncompact FC-spaces. These theorems improve, unify and generalize many important results in recent literature.     

Integral operators on analytic Morrey spaces

In this note, we characterize the boundedness of the Volterra type operator T _g and its related integral operator I _g on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given. As a corollary, we get the compactness of those operators.     

Doubly stochastic operators and rearrangement theorems

This paper characterizes doubly stochastic operators between two L¹ spaces of random variables in terms of convex functions on the real line. This characterization is then applied to proving some Hardy-Littlewood-Pólya-type rearrangement theorems. The conditional form of Jensen's inequality is also derived and a condition for equality obtained. Moreover, some known results concerning doubly stochastic operators are also generalized.     

The Cauchy Transform of Continuous Linear Functionals and Projections on the Weighted Spaces of Analytic Functions

Under some general assumptions on weight, we give a complete characterization of the Cauchy transform of continuous linear functionals on the weighted spaces of holomorphic functions in a ball. We construct an integral projection that sends the weighted spaces of measurable and n-harmonic functions in the ball onto the corresponding spaces of holomorphic functions.     

On an Inverse Problem for a Parabolic Equation

In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in u in which the coefficients of u _t and u depend on u _(x,t) , with overdetermination given by the integral of the solution over time. The Fredholm property for this problem and the existence and uniqueness theorems in Sobolev spaces are established.     

Inner divisors and composition operators

This characterization is stated and proved in much greater generality, beginning with subspaces of arbitrary L^∞ spaces and using a notion of inner divisors. Among the consequences are a variant of Wermer's theorem on embedding disks in maximal ideal spaces and a result on linear isometrics of H^∞. The ranges of composition operators C_I when I is not necessarily inner are characterized. In particular, relatively closed sets E ?cΔ of zero logarithmic and zero analytic capacity are characterized in terms of the algebras of bounded analytic functions on A invariant under corresponding Fuchsian groups. The paper concludes with an example of a uniformly closed subalgebra of H^∞ which contains the constants and the inner factors of its members, but is not of the form H ∞ I.     

On convergence in mixed integer programming

Let ${P \subseteq {\mathbb R}^{m+n}}$ be a rational polyhedron, and let P _I be the convex hull of ${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$ . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in ${{\mathbb R}^m}$ . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P _I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvátal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P _I .     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Analysis of a problem of Raikov with applications to barreled and bornological spaces

Raikov’s conjecture states that semi-abelian categories are quasi-abelian. A first counterexample is contained in a paper of Bonet and Dierolf who considered the category of bornological locally convex spaces. We prove that every semi-abelian category I admits a left essential embedding into a quasi-abelian category K_l(I) such that I can be recovered from K_l(I) by localization. Conversely, it is shown that left essential full subcategories I of a quasi-abelian category are semi-abelian, and a criterion for I to be quasi-abelian is given. Applied to categories of locally convex spaces, the criterion shows that barreled or bornological spaces are natural counterexamples to Raikov’s conjecture. Using a dual argument, the criterion leads to a simplification of Bonet and Dierolf’s example.     

Sharp interpolation theorems in couples of l p spaces for generalized Lions-Peetre spaces of means

Sharp interpolation theorems for linear operators acting on arbitrary couples of L _p spaces are found in the family of generalized Lions-Peetre spaces of means. This family includes the Lorentz spaces with functional quasi-concave parameters, the Orlicz spaces, and spaces similar to them.     

Integral Representations and Transforms of N-Functions. I

The author has proposed a new approach to extrapolation of operators from the scale of Lebesgue spaces to the Orlicz spaces beyond this scale. In this article comprising two parts we develop some mathematical method that enables us to prove extrapolation theorems for arbitrary behavior of an operator in the Lebesgue scale (i.e., in the case when the norm of the operator is an arbitrary function of p) and also in the case when the basic scale is an interval of the Lebesgue scale with exponents separated from 1 or +∞. In this event, we face ill-posed problems of inversion of the classical Mellin and Laplace type integral transforms over nonanalytic functions in terms of their asymptotic behavior on the real axis and also the question about the properties of convolution type integral transforms on classes of N-functions. In the first part of the article we study integral representations for N-functions by expansions in power functions with a positive weight and the behavior of convolution type integral transforms on classes of N-functions.     

Interpolation of Lorentz martingale spaces

In this paper,we apply function parameters,introduced by Persson,to real interpolation of Lorentz martingale spaces.Some new interpolation theorems concerning Lorentz martingale spaces are formulated.The results that we obtain generalize some fundamental interpolation theorems in classical martingale Hp theory.     

Characterization of radon integrals as linear functionals

The problem of characterization of integrals as linear functionals is considered in this paper. It has its origin in the well-known result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and is directly connected with the famous theorem of J. Radon (1913) on integral representation of bounded linear functionals by Lebesgue integrals on a compact in ? ⁿ . After the works of J. Radon, M. Fréchet, and F. Hausdorff, the problem of characterization of integrals as linear functionals has been concretized as the problem of extension of Radon??s theorem from ? ⁿ to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and abundant history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such eminent mathematicians as S. Banach (1937?C38), S. Saks (1937?C38), S. Kakutani (1941), P. Halmos (1950), E. Hewitt (1952), R. E. Edwards (1953), Yu. V. Prokhorov (1956), N. Bourbaki (1969), H. K¨onig (1995), V. K. Zakharov and A. V. Mikhalev (1997), et al. Essential ideas and technical tools were worked out by A. D. Alexandrov (1940?C43), M. N. Stone (1948?C49), D. H. Fremlin (1974), et al. The article is devoted to the modern stage of solving this problem connected with the works of the authors (1997?C2009). The solution of the problem is presented in the form of the parametric theorems on characterization of integrals. These theorems immediately imply characterization theorems of the above-mentioned authors.     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

Nagata-like theorems for almost Prüfer v-multiplication domains

Let D be an integral domain and X an indeterminate over D . We show that if S is an almost splitting set of an integral domain D , then D is an APVMD if and only if both DS and DN(S) are APVMDs. We also prove that if {Dα}α∈I is a collection of quotient rings of D such that D=∩α∈IDα has finite character (that is, each nonzero d∈D is a unit in almost all Dα) and each of Dα is an APVMD, then D is an APVMD. Using these results, we give several Nagata-like theorems for APVMDs.     

Converse and Smoothness Theorems for Erdo&#x030B;s Weights inL_(0<p?∞)

We prove converse and smoothness theorems of polynomial approximation in weightedL_pspaces with norm ‖fW‖_Lp()(0<p?∞) for Erdo&#x030B;s weights on the real line. In particular we prove characterization theorems involving realization functionals and thereby establish some interesting properties of our weighted modulus of continuity.