110 research outputs found

### Spaces H^1 and BMO on ax+b-groups

Let S be the semidirect product of R^d and R^+ endowed with the Riemannian
symmetric space metric and the right Haar measure: this is a Lie group of
exponential growth. In this paper we define an Hardy space H^1 and a BMO space
in this context. We prove that the functions in BMO satisfy the John-Nirenberg
inequality and that BMO may be identified with the dual space of H^1. We then
prove that singular integral operators which satisfy a suitable integral
Hormander condition are bounded from H^1 to L^1 and from L^{\infty} to BMO. We
also study the real interpolation between H^1, BMO and the L^p spaces

### Heat maximal function on a Lie group of exponential growth

Let G be the Lie group R^2\rtimes 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 show that the maximal function
associated with the heat kernel of the Laplacian \Delta is bounded from the
Hardy space H^1 to L^1. We also prove that the heat maximal function does not
provide a maximal characterization of the Hardy space H^1.Comment: 18 page

### 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

### Boundedness from H^1 to L^1 of Riesz transforms on a Lie group of exponential growth

Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$
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 Riesz transforms $S_{ij}=\Delta^{-1}X_iX_j$ and
$R_{ij}=X_iX_j\Delta^{-1}$ are not.Comment: This paper will be published in the "Annales de l'Institut Fourier

### Spectral multipliers for Laplacians with drift on Damek-Ricci spaces

We prove a multiplier theorem for certain Laplacians with drift on
Damek-Ricci spaces, which are a class of Lie groups of exponential growth. Our
theorem generalizes previous results obtained by W. Hebisch, G. Mauceri and S.
Meda on Lie groups of polynomial growth.Comment: 13 page

### Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the
left-invariant Riemannian symmetric space structure and the right Haar measure
$\rho$, which is a Lie group of exponential growth. Hebisch and Steger in
[Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$
admits a Calder\'on--Zygmund decomposition which involves a particular family
of sets, called Calder\'on--Zygmund sets. In this paper, we first show the
existence of a dyadic grid in the group $S$, which has {nice} properties
similar to the classical Euclidean dyadic cubes. Using the properties of the
dyadic grid we shall prove a Fefferman--Stein type inequality, involving the
dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function.
As a consequence, we obtain a complex interpolation theorem involving the Hardy
space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009),
277--295].Comment: Math. Z. (to appear

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