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

Peter Sjögren[1]; Maria Vallarino[2]

  • [1] Göteborg University and Chalmers University of Technology Department of Mathematical Sciences 412 96 Göteborg (Sweden)
  • [2] Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni Via R. Cozzi 53 20125 Milano (Italy)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 4, page 1117-1151
  • ISSN: 0373-0956

Abstract

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Let G be the Lie group ℝ 2 ⋉ 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 Δ = - ( X 0 2 + X 1 2 + X 2 2 ) . In this paper we consider the first order Riesz transforms R i = X i Δ - 1 / 2 and S i = Δ - 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 i j = X i Δ - 1 X j are bounded from H 1 to L 1 , while the transforms S i j = Δ - 1 X i X j and R i j = X i X j Δ - 1 , for i , j = 0 , 1 , 2 , are not.

How to cite

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Sjögren, Peter, and Vallarino, Maria. "Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth." Annales de l’institut Fourier 58.4 (2008): 1117-1151. <http://eudml.org/doc/10344>.

@article{Sjögren2008,
abstract = {Let $G$ be the Lie group $\{\mathbb\{R\}\}^2\ltimes\{\mathbb\{R\}\}^+$ 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 transforms $S_\{ij\}=\Delta ^\{-1\}X_iX_j$ and $R_\{ij\}=X_iX_j\Delta ^\{-1\}$, for $i,j=0,1,2$, are not.},
affiliation = {Göteborg University and Chalmers University of Technology Department of Mathematical Sciences 412 96 Göteborg (Sweden); Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni Via R. Cozzi 53 20125 Milano (Italy)},
author = {Sjögren, Peter, Vallarino, Maria},
journal = {Annales de l’institut Fourier},
keywords = {Singular integrals; Riesz transforms; Hardy space; Lie groups; exponential growth; singular integrals},
language = {eng},
number = {4},
pages = {1117-1151},
publisher = {Association des Annales de l’institut Fourier},
title = {Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth},
url = {http://eudml.org/doc/10344},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Sjögren, Peter
AU - Vallarino, Maria
TI - Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 4
SP - 1117
EP - 1151
AB - Let $G$ be the Lie group ${\mathbb{R}}^2\ltimes{\mathbb{R}}^+$ 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 transforms $S_{ij}=\Delta ^{-1}X_iX_j$ and $R_{ij}=X_iX_j\Delta ^{-1}$, for $i,j=0,1,2$, are not.
LA - eng
KW - Singular integrals; Riesz transforms; Hardy space; Lie groups; exponential growth; singular integrals
UR - http://eudml.org/doc/10344
ER -

References

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