# alessio martini

## Contact Details

Namealessio martini |
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## Pub CategoriesMathematics - Functional Analysis (14) Mathematics - Analysis of PDEs (13) Mathematics - Operator Algebras (2) Mathematics - Spectral Theory (2) Mathematics - Metric Geometry (1) |

## Publications Authored By alessio martini

We prove a sharp multiplier theorem of Mihlin-H\"ormander type for the Grushin operator on the unit sphere in $\mathbb{R}^3$, and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics. Read More

We prove a general multiplier theorem for symmetric left-invariant sub-Laplacians with drift on non-compact Lie groups. This considerably improves and extends a result by Hebisch, Mauceri, and Meda. Applications include groups of polynomial growth and solvable extensions of stratified groups. Read More

A sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation. Read More

Let $\Box_b$ be the Kohn Laplacian acting on $(0,j)$-forms on the unit sphere
in $\mathbb{C}^n$. In a recent paper of Casarino, Cowling, Sikora and the
author, a spectral multiplier theorem of Mihlin--H\"ormander type for $\Box_b$
is proved in the case $0

Let $G$ be a $2$-step stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. Read More

Let $G = N \rtimes A$, where $N$ is a stratified group and $A = \mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $\Delta$ on $G$. We prove a theorem of Mihlin-H\"ormander type for spectral multipliers of $\Delta$. Read More

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$. Read More

From a theorem of Christ and Mauceri and Meda it follows that, for a homogeneous sublaplacian $L$ on a $2$-step stratified group $G$ with Lie algebra $\mathfrak{g}$, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for $1 < p < \infty$ if the spectral multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$, where $Q = \dim \mathfrak{g} + \dim[\mathfrak{g},\mathfrak{g}]$ is the homogeneous dimension of $G$. Here we show that the condition can be pushed down to $s > d/2$, where $d = \dim \mathfrak{g}$ is the topological dimension of $G$, provided that $d \leq 7$ or $\dim [\mathfrak{g},\mathfrak{g}] \leq 2$. Read More

Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1 < p < \infty$ if $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s > d/2$ is sufficient. Read More

Let $L$ be the homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent group $N_{3,2}$ on 3 generators. We prove a theorem of Mihlin-H\"ormander type for the functional calculus of $L$, where the order of differentiability $s > 6/2$ is required on the multiplier. Read More

In a recent work by A. Martini and A. Sikora (arXiv:1204. Read More

We prove a number of results on the geometry associated to the solutions of evolution equations given by first-order differential operators on manifolds. In particular, we consider distance functions associated to a first-order operator, and discuss the associated geometry, which is sometimes surprisingly different to riemannian geometry. Read More

We study the Grushin operators acting on $\R^{d_1}_{x'}\times \R^{d_2}_{x"}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 - (\sum_{\jone=1}^{d_1}|x'_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x"_\jtwo}^2. \] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. Read More

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,... Read More

The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L_1,... Read More

This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,... Read More

In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are obtained for a pair of positive-self adjoint operators on a Hilbert space, whose spectral projectors satisfy a ``balance condition'' involving certain operator norms. This result is then applied to obtain uncertainty inequalities on Riemannian manifolds, Riemannian symmetric spaces of non-compact type, homogeneous graphs and unimodular Lie groups with sublaplacians. Read More