Let $G$ be a locally compact group and $\rho$ the left Haar measure on $G$. Given a non-negative Radon measure $\mu$, we establish a necessary condition on the pairs $\left(q,p\right)$ for which $\mu$ is a multiplier from $L^q\left(G,\rho \right)$ to $L^p\left(G,\rho \right)$. Applied to ${ℝ}^n$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7]. When $G$...

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta +\frac{k}{x_n}\frac{\partial }{\partial x_n}$ on ${ℝ}^n$ is proved. In this note there is shown that in the cases $k\ne 0$, $k\ne 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.

Let $p^{\text{'}}$ and $q^{\text{'}}$ be points in ${ℝ}^n$. Write $p^{\text{'}}\sim q^{\text{'}}$ if $p^{\text{'}}-q^{\text{'}}$ is a multiple of $(1,...,1)$. Two different points $p$ and $q$ in ${ℝ}^n/\sim$ uniquely determine a tropical line $L(p,q)$ passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p^{\text{'}}$ and $q^{\text{'}}$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also...

We study spectral properties of transfer operators for diffeomorphisms $T:X\to X$ on a Riemannian manifold $X$. Suppose that $\Omega$ is an isolated hyperbolic subset for $T$, with a compact isolating neighborhood $V\subset X$. We first introduce Banach spaces of distributions supported on $V$, which are anisotropic versions of the usual space of $C^p$ functions $C^p\left(V\right)$ and of the generalized Sobolev spaces $W^{p,t}\left(V\right)$, respectively. We then show that the transfer operators associated to $T$ and a smooth weight $g$ extend boundedly to these...

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1\<p\<\infty$. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel T^n-T^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2\<q\<\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, the sequence ${\left(\left[T^n\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left(T^n\left(x\right)\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left(M_n\left(T\right)x\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$, where $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...