Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces \(T^{p,q}_s(X)\), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z -spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

Towards strong Banach property (T) for SL(3,ℝ)

We prove that SL(3, ?) has Strong Banach property (T) in Lafforgue’s sense with respect to the Banach spaces that are θ > 0 interpolation spaces (for the complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, every action of SL(3, ?) or its lattices by affine isometries on such a Banach space X has a fixed point, and the expanders contructed from SL(3, ?) do not admit a coarse embedding into X. We also prove a quantitative decay of matrix coefficients (Howe-Moore property) for representations with small exponential growth of SL(3, ?) on X.     

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m ₀: Σ → X ₀ and m ₁: Σ → X ₁ be positive vector measures with values in the Banach Köthe function spaces X ₀ and X ₁. If 0 < α < 1, we define a new vector measure [m ₀, m ₁] _α with values in the Calderón lattice interpolation space X ₀ ^1?ga X ₁ ^α and we analyze the space of integrable functions with respect to measure [m ₀, m ₁] _α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L ^p -spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.     

Cardinal interpolation with general multiquadrics: convergence rates

This article pertains to interpolation of Sobolev functions at shrinking lattices \(h\mathbb {Z}^{d}\) from L _p shift-invariant spaces associated with cardinal functions related to general multiquadrics, ? _{α, c} (x) := (|x|² + c ²) ^α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, L _p error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h ^k ) for functions with derivatives of order up to k in L _p , \(1<p<\infty \)). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.     

On Symmetric Spaces With Convergence in Measure on Reflexive Subspaces

A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L₁. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.     

Real Interpolation of Sobolev Spaces Associated to a Weight

We hereby study the interpolation property of Sobolev spaces of order 1 denoted by \(W^{1}_{p,V}\), arising from Schrödinger operators with positive potential. We show that for 1?≤?p ₁?p?p ₂?q ₀ with p?>?s ₀, \(W^{1}_{p,V}\) is a real interpolation space between \(W_{p_1,V}^{1}\) and \(W_{p_2,V}^{1}\) on some classes of manifolds and Lie groups. The constants s ₀, q ₀ depend on our hypotheses.     

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.     

Off-Shell Supersymmetry and Filtered Clifford Supermodules

An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1|N}\), correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1|p,q}\), correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.     

On one transformation of parameter-spaces of real interpolation method

In connection with estimation of interpolation orbits and coorbits we introduce a new transformation acting in the class of all parameter-space of real interpolation K-method. We “calculate” the result of the transformation of classical parameters. It is revealed that the transformation of weighted L₁-spaces leads to Orlicz spaces.     

A note on the perturbations of compact quantum metric spaces

In this short note,we consider the perturbation of compact quantum metric spaces.We first show that for two compact quantum metric spaces(A,P) and(B,Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively,the quantum Gromov–Hausdorff distance between(A,P) and(B,Q) is small under certain conditions.Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.     

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving <Emphasis Type="Italic">k</Emphasis>-Modulus of Smoothness

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H ^σ X(IR ⁿ ) with order of smoothness σ?∈?(0, n), modelled upon rearrangement invariant Banach function spaces X(IR ⁿ ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR ⁿ ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W ^k?+?1 L ^n/k(logL) ^α (IR ⁿ ) and W ^k L ^n/k(logL) ^α (IR ⁿ ) into generalized Hölder spaces.     

Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

New algorithm for computing the Hermite interpolation polynomial

Let x ₀, x ₁,? , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,? , n, and k = 0 ,1 ,? , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N ? 1}(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n _i . P _N?1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,? , n, k = 0,1,? , n _i }. The polynomial p _N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.     

On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle

Given a three-dimensional dynamical system on the interval t ₀ < t < +∞, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighbor-hood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue λ = ? + iβ, with β ? ? > 0 and a real eigenvalue with δ < 0 with |δ| ? ?. On the arbitrary interval [t ₀, +∞), an approximate solution is sought as a polynomial P _N (?) in powers of the small parameter with coefficients from Hölder function spaces. It is proved that there exist ? _N and C _N depending on the initial data such that, for 0 < ? < ? _N , the difference between the exact and approximate solutions does not exceed C _N ? ^N+1.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C ₁,…,C _N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥?∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.     

On Measure Contraction Property without Ricci Curvature Lower Bound

Measure contraction properties M C P (K, N) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N, then M C P (K, N) is equivalent to Ricci curvature bounded below by K. On the other hand, it was observed in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013) that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy M C P (0,5). In this paper, we give sufficient conditions for a 2n+1 dimensional weakly Sasakian manifold to satisfy M C P (0, 2n + 3). This extends the above mentioned result on the Heisenberg group in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013).     

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.