Related papers: Spectral ergodic Banach problem and flat polynomia…

Let $(G/\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\geq 2$. We adapt an argument of Parry to show that $L^2(G/\Gamma)$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite…

We introduce a notion being a $k$-fold Lebesgue function for measure preserving transformations, where any $2$-fold Lebesgue function is just ordinary Lebesgue. We discuss how this new metrical isomorphisms invariant of dynamical systems is…

A complex polynomial $P(z) = c_0 + c_1 z +...+ c_n z^n$ is called unimodular if $|c_j| = 1$, $j = 0,...,n$. Littlewood asked the question (1966) on how close a unimodular polynomial come to satisfying $|P(z)| \approx \sqrt{n+1}$ if $n \ge…

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish corresponding…

For a graph $G$ on $n$ vertices with normalized Laplacian eigenvalues $0 = \lambda_1(G) \leq \lambda_2(G) \leq \cdots \leq \lambda_n(G)$ and graph complement $G^c$, we prove that \begin{equation*} \max\{\lambda_2(G),\lambda_2(G^c)\}\geq…

We present an elementary problem on analytic polynomials with coefficients $\pm 1$ or in $\{0,\pm 1 \}$ which implies Riemann hypothesis. It is turns out that this problem is a particular case of the weak form of a flat polynomials problem…

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $\Omega = \big(\Omega^1,\Omega^2,\Omega^3\big)$ of closed $2$-forms. One wants to construct a coframing $\omega = \big(\omega^1,\omega^2,\omega^3\big)$ of $M$…

In this paper, we study the existence of solutions of the equation $(-\Delta)_1^s u=f$ in a bounded open set with Lipschitz boundary $\Omega\subset \Rn$, vanishing on $\Co \Omega$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of…

We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$. The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha}$-regular free boundaries by…

We consider the problem of finding on a given Euclidean domain $\Omega$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(\lambda(-A)) = 1$. This generalizes a…

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace $\t$, and $\{\alpha_ t\} $ a strongly continuous extension to $L^p(M,\t)$ of a semigroup of absolute contractions on $L^1 (M,\tau)$. By means…

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact K\"{a}hler manifold are the same as those of $\mathbb{C}P^n$…

We show that smoothness implies norm-controlled inversion: the smoothness of an element $a$ in a Banach algebra with a one-parameter automorphism group is preserved under inversion, and the norm of the inverse $a^{-1}$ is controlled by the…

The note is devoted to multiple mixing, spectrum, rank and self-joinings of measure-preserving transformations. We recall famous open problems, discuss related questions and some known results. A hypothetical example of an automorphism of…

On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\Delta^2 u -\alpha \Delta u +\beta u = u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that $\alpha^2 \geq 4…

We give a topological interpretation of the space of $L^2$-harmonic forms on Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We also give a Chern-Gauss-Bonnet formula for the $L^2$-Euler characteristic of some of…

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our…

We prove that the faithful and uniqueness of norm properties are stable in different product algebras such as direct-sum product algebra, convolution product algebra, and module product algebra. Further, we exhibit that these properties are…

For any natural $n$, and real $\alpha\geq 0$ we construct an ergodic automorphism $T$ such that its tensor powers $T^{\otimes n}$ have singular spectra if $n\leq 1+\alpha /2$, and Lebesgue if $n\, > 1+\alpha/2$.

Spectral morphisms between Banach algebras are useful for comparing their K-theory and their "noncommutative dimensions" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral…