Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous  , 1≤p≤∞$1\le p\le \infty$, when every sequence of pairwise disjoint normalized elements in E   has a subsequence equivalent to the unit vector basis of _?p${?}_p$. Employing methods from interpolation theory, we clarify which r.i. spaces on [0,1]$[0,1]$ are p  -disjointly homogeneous. In particular, for every 1<p<∞$1<p<\infty$ and any increasing concave function φ   on [0,1]$[0,1]$, which is not equivalent to neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function φ  . Moreover, it is shown that given 1<p<∞$1<p<\infty$ and an increasing concave function φ with non-trivial dilation indices, there is a unique p-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with φ.     

Closed subgroups as Ditkin sets

We prove that every closed subgroup of a locally compact group is locally p  -Ditkin for 1<p<∞$1<p<\infty$.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).     

Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

For a general subcritical second-order elliptic operator P   in a domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ (or noncompact manifold), we construct Hardy-weight W which is optimal   in the following sense. The operator P−λW$P-\lambda W$ is subcritical in Ω   for all λ<1$\lambda <1$, null-critical in Ω   for λ=1$\lambda =1$, and supercritical near any neighborhood of infinity in Ω   for any λ>1$\lambda >1$. Moreover, if P   is symmetric and W>0$W>0$, then the spectrum and the essential spectrum of W⁻¹P$W^{-1}P$ are equal to [1,∞)$[1,\infty )$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu=0$Pu=0$, the existence of which depends on the subcriticality of P in Ω.     

Chaotic dynamics of the heat semigroup on Riemannian symmetric spaces

We show that the heat semigroup generated by certain perturbations of the Laplace–Beltrami operator on the Riemannian symmetric spaces of noncompact type is chaotic   on their L^p$L^p$-spaces when 2<p<∞$2<p<\infty$. Both the range of p and the range of chaos-inducing perturbation are sharp. This extends a result of Ji and Weber [17] where it was shown that under identical conditions the heat operator is subspace-chaotic on these spaces.     

On everywhere divergence of the strong Φ-means of Walsh–Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems was established by Rodin in 1990. We prove, that if the growth of a function Φ(t):[0,∞)→[0,∞)$\Phi (t):[0,\infty )\to [0,\infty )$ is bigger than the exponent, then the strong Φ-summability of a Walsh–Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the authors of this paper.     

Pointwise tensor products of function spaces

By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as _L∞$L_{\infty }$-modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if 0→Y→X→Z→0$0\to Y\to X\to Z\to 0$ is an exact sequence of _L∞$L_{\infty }$-modules and homomorphisms, where Y and Z are function spaces, and V   is another function space, then the tensorized sequence 0→Y⊗V→X⊗V→Z⊗V→0$0\to Y\otimes V\to X\otimes V\to Z\otimes V\to 0$ is exact too.     

Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

On strictly singular nonlinear centralizers

We show that _c0$c_0$ is the only Banach space with unconditional basis that satisfies the equation Ext(X,X)=0$Ext(X,X)=0$. This partially improves an old result by Kalton and Peck. We prove that the Kalton–Peck maps are strictly singular on a number of sequence spaces, including _?p${?}_p$ for 0<p<∞$0<p<\infty$, Tsirelson and Schlumprecht spaces and their duals, as well as certain super-reflexive variations of these spaces. In the last section, we give estimates of the projection constants of certain finite-dimensional twisted sums of Kalton–Peck type.     

When is every linear transformation a sum of two commuting invertible ones?

A well-known result of Wolfson [7] and Zelinsky [8] says that every linear transformation of a vector space V over a division ring D   is a sum of two invertible linear transformations except when dim(V)=1$\dim (V)=1$ and D=_F2$D={\mathbb{F}}_2$. Indeed, many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. The goal of this note is to prove that every linear transformation of a vector space V over a division ring D   is a sum of two commuting invertible ones if and only if |D|?3$|D|?3$ and dim(V)<∞$\dim (V)<\infty$. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms.     

Benjamin–Ono periodic bifurcating water waves in presence of an essential spectrum

The mathematical study of travelling waves in the potential flow of two superposed layers of perfect fluid can be set as an ill-posed evolutionary problem, in which the horizontal unbounded space variable plays the role of “time”. In this paper we consider two problems for which the bottom layer of fluid is infinitely deep: for the first problem, the upper layer is bounded by a rigid top and there is no surface tension at the interface; for the second problem, there is a free surface with a large enough surface tension. In both problems, the linearized operator Lε$L_{\epsilon }$ (where ε is a combination of the physical parameters) around 0 possesses an essential spectrum filling the entire real line  , with in addition a simple eigenvalue in 0. Moreover, for ε<0$\epsilon <0$, there is a pair of imaginary eigenvalues which meet in 0 when ε=0$\epsilon =0$ and which disappear in the essential spectrum for ε>0$\epsilon >0$. For ε>0$\epsilon >0$ small enough, we prove in this paper the existence of a two parameter family of periodic travelling waves (corresponding to periodic solutions of the dynamical system). These solutions are obtained in showing that the full system can be seen as a perturbation of the Benjamin–Ono equation. The periods of these solutions run on an interval (T₀,∞)$(T_0,\infty )$ possibly except a discrete set of isolated points.     

Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the n$n$th generation are unlikely to be closely related if n$n$ is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {_pj}$\left\{p_j\right\}$ such that _p0=0$p_0=0$ and ψ(x)=_{∑jpjI{j≥x}}$\psi \left(x\right)={\sum }_j{p}_j{I}_{\{j\ge x\}}$ is asymptotic to x^−αL(x)$x^{-\alpha }L\left(x\right)$ as x→∞$x\to \infty$ where L(⋅)$L(\cdot )$ is slowly varying at ∞$\infty$ and 0<α<1$0<\alpha <1$ (and hence the mean m=∑j_pj=∞$m=\sum j{p}_j=\infty$) it is shown that if _Xn$X_n$ is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the n$n$th generation then n−_Xn$n-X_n$ converges in distribution to a proper distribution supported by N={1,2,3,…}$\mathbb{N}=\{1,2,3,\dots \}$. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean m$m$ satisfies 1<m≡∑j_pj<∞$1<m\equiv \sum j{p}_j<\infty$ and _p0=0$p_0=0$ then coalescence time _Xn$X_n$ does converge to a proper distribution as n→∞$n\to \infty$, i.e., coalescence does take place in the remote past.