1,118 research outputs found

### Riesz transforms on solvable extensions of stratified groups

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 $G$. Here we prove weak type $(1,1)$,
$L^p$-boundedness for $p \in (1,2]$ and $H^1 \to L^1$ boundedness of the Riesz
transforms $Y \Delta^{-1/2}$ and $Y \Delta^{-1} Z$, where $Y$ and $Z$ are any
horizontal left-invariant vector fields on $G$, as well as the corresponding
dual boundedness results. At the crux of the argument are large-time bounds for
spatial derivatives of the heat kernel, which are new when $\Delta$ is not
elliptic.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1504.0386

### $L^p$ spectral multipliers on the free group $N_{3,2}$

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

### Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

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. These multiplier results are sharp if
$d_1 \ge d_2$. We discuss also an interesting phenomenon for weighted
Plancherel estimates for $d_1 <d_2$. The described spectral multiplier theorem
is the analogue of the result for the sublaplacian on the Heisenberg group
obtained by D. M\"uller and E.M. Stein and by W. Hebisch

### Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups

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 $\dim [\mathfrak{g},\mathfrak{g}] \leq 2$.Comment: 33 page

### Spectral multipliers on $2$-step groups: topological versus homogeneous dimension

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 $s > Q/2$. It is known that, for several $2$-step groups and
sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp
and in many cases it is possible to push it down to $d/2$. Here we show that,
for all $2$-step groups and sub-Laplacians, the sharp threshold is strictly
less than $Q/2$, but not less than $d/2$.Comment: 17 page

### From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

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.Comment: 32 page

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