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

Estimates for functions of the Laplacian on manifolds with bounded geometry

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel.Comment: 19 page