Let $G$ be the Lie group ${ℝ}^2⋉{ℝ}^{+}$ endowed with the Riemannian symmetric space structure. Let $X_0,X_1,X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second-order Riesz transforms $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not. ...

Given a rational function $\varphi \left(T\right)$ on ${\mathbb{P}}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure ${\mu }_{\varphi ,v}$ on the Berkovich space ${\mathbb{P}}_{\mathrm{Berk},\mathrm{v}}^1/{ℂ}_v$ such that if $\left\{z_n\right\}$ is a sequence of points in ${\mathbb{P}}^1\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\mathrm{Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$. The proof uses a polynomial lift $F(x,y)=(F_1(x,y),F_2(x,y))$ of $\varphi$ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is...

Let $ℒ$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let $S_{R}f:={\int }_0^{R}d{E}_{\lambda }f,R\ge 0,$ denote the associated “spherical partial sums,” where $ℒ={\int }_0^{\infty }\lambda d{E}_{\lambda }$ is the spectral resolution of $ℒ.$ We prove that $S_{R}f\left(x\right)$ converges a.e. to $f\left(x\right)$ as $R\to \infty$ under the assumption $log(2+ℒ)f\in L^2\left(G\right).$

Let $\varphi$ be a psh function on a bounded pseudoconvex open set $\Omega \subset {ℂ}^n$, and let $ℐ\left(m\varphi \right)$ be the associated multiplier ideal sheaves, $m\in {\mathbb{N}}^{☆}$. Motivated by global geometric issues, we establish an effective version of the coherence property of $ℐ\left(m\varphi \right)$ as $m\to +\infty$. Namely, given any $B⋐\Omega$, we estimate the asymptotic growth rate in $m$ of the number of generators of $ℐ{\left(m\varphi \right)}_{|B}$ over ${𝒪}_{\Omega }$, as well as the growth of the coefficients of sections in $\Gamma (B,ℐ(m\varphi \left)\right)$ with respect to finitely many generators globally defined on $\Omega$. Our approach relies on proving...

Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function $$N_f\left(x\right)=\#\left\{0\le n\<x\left|f\right(n)\mid n\right\}$$ under a mild restriction on the values of $f$. When $f=s_q$, the base $q$ sum of digits function, the integers counted by $N_f$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.