Composition operators on the polydisc

In this paper we study the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc ${\mathbb{D}}^n$. Studying the volume of sublevel sets we show for which n the necessary conditions obtained by Bayart are sufficient. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.     

On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the $L_{\infty }$ norm. We consider algorithms that use standard information ${\mathrm{\Lambda }}^{\mathrm{std}}$ consisting of function values or general linear information ${\mathrm{\Lambda }}^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for ${\mathrm{\Lambda }}^{\mathrm{std}}$ and ${\mathrm{\Lambda }}^{\mathrm{all}}$ under the absolute or normalized error criterion, and show that the power of ${\mathrm{\Lambda }}^{\mathrm{std}}$ is the same as the one of ${\mathrm{\Lambda }}^{\mathrm{all}}$ for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Representation and factorization theorems for almost-Lp-spaces

We extend the notions of $p$-convexity and $p$-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost $p$-convex and almost $p$-concave space is order isomorphic to an almost-$L^p$-space. The class of almost-$L^p$-spaces contains, in particular, direct sums of (infinitely many) $L^p$-spaces with different norms, that are not in general $p$-convex – nor $p$-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for $p$-concave operators acting in $p$-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than $L^p$-spaces.     

Torsion completions are bounded

Given a commutative ring A and a finitely generated ideal I, we prove that $I^{\infty }$-torsion A-modules that are also I-adically complete (or merely derived I-complete) must have bounded $I^{\infty }$-torsion, i.e., they are killed by $I^n$ for some $n\ge 0$.     

Sobolev H1 geometry of the symplectomorphism group

For a closed symplectic manifold $(M,\omega )$ with compatible Riemannian metric g we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the $H^1$ metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the $H^1$ metric carries conjugate points via some simple examples.     

Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

Fock space on C∞ and Bose-Fock space

In this paper, we introduce the Fock space on ${\mathbb{C}}^{\infty }$ and obtain an isomorphism between the Fock space on ${\mathbb{C}}^{\infty }$ and Bose-Fock space. Based on this isomorphism, we obtain representations of some operators on the Bose-Fock space and answer a question in [2]. As a physical application, we study the Gibbs state.     

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-K-invariant $L^1$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-K-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-K-invariance condition and strong $L^{\infty }$ convergence.     

New families of flag-transitive linear spaces

In this paper, we construct new families of flag-transitive linear spaces with $q^{2n}$ points and $q^2$ points on each line that admit a one-dimensional affine automorphism group. We achieve this by building a natural connection with permutation polynomials of ${\mathbb{F}}_{q^2}$ of a particular form and following the scheme of Pauley and Bamberg in (2008) [14].     

On the symbol-pair distances of repeated-root constacyclic codes of length 2ps

Let $p$ be an odd prime, and $\lambda$ be a nonzero element of the finite field ${\mathbb{F}}_{p^m}$. The $\lambda$-constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are classified as the ideals of quotient ring $\frac{{\mathbb{F}}_{p^m}\left[x\right]}{〈x^{2p^s}?\lambda 〉}$ in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such $\lambda$-constacyclic codes of length $2p^s$ are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are established, which produce many new MDS symbol-pair codes with good parameters.     

Solution of the Navier–Stokes problem

A new a priori estimate for solutions to Navier–Stokes equations is derived. Uniqueness and existence of these solutions in ${\mathbb{R}}^3$ for all $t>0$ is proved in a class of solutions locally differentiable in time with values in $H^1\left({\mathbb{R}}^3\right)$, where $H^1\left({\mathbb{R}}^3\right)$ is the Sobolev space. By the solution a solution to an integral equation is understood. No smallness restrictions on the data are imposed.