Completeness of a dilation system ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ on the standard Lebesgue space $L^2(0,1)$ is considered for 2-periodic functions $\varphi$. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space $H^2\left({𝔻}_2^{\infty }\right)$ on the Hilbert multidisc ${𝔻}_2^{\infty }$. Several simple sufficient conditions are exhibited, which include however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). For instance, each of the following conditions implies cyclicity...

Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure ${\mu }_S:={\sum }_{a\in S}(1-|a\left|²\right)ⁿ{\delta }_a$ is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure ${\mu }_S$ bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual bounded sequence...

Let $L:=-\Delta +V$ be a Schrödinger operator on ${ℝ}^n$ with $n\ge 3$ and $V\ge 0$ satisfying ${\Delta }^{-1}V\in L^{\infty }\left({ℝ}^n\right)$. Assume that $\varphi :{ℝ}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,·)$ is an Orlicz function, $\varphi (·,t)\in {𝔸}_{\infty }\left({ℝ}^n\right)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on ${ℝ}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}\left({ℝ}^n\right)\ni f\mapsto wf\in H_{\varphi }\left({ℝ}^n\right)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}\left({ℝ}^n\right)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }\left({ℝ}^n\right)$ under some assumptions on $\varphi$. As applications, the author further obtains the...