112 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 , where is a stratified group and
acts on via automorphic dilations. Homogeneous sub-Laplacians on and
can be lifted to left-invariant operators on and their sum is a
sub-Laplacian on . Here we prove weak type ,
-boundedness for and boundedness of the Riesz
transforms and , where and are any
horizontal left-invariant vector fields on , 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 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 be the Lie group given by the semidirect product of and
endowed with the Riemannian symmetric space structure. Let be a
distinguished basis of left-invariant vector fields of the Lie algebra of
and define the Laplacian . In this paper we
consider the first order Riesz transforms and
, for . We prove that the operators , but
not the , are bounded from the Hardy space to . We also show
that the second order Riesz transforms are bounded
from to , while the Riesz transforms and
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
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