372 research outputs found
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
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
On the boundary convergence of solutions to the Hermite-Schr\"odinger equation
In the half-space , we consider the
Hermite-Schr\"odinger equation ,
with given boundary values on .
We prove a formula that links the solution of this problem to that of the
classical Schr\"odinger equation. It shows that mixed norm estimates for the
Hermite-Schr\"odinger equation can be obtained immediately from those known in
the classical case. In one space dimension, we deduce sharp pointwise
convergence results at the boundary, by means of this link.Comment: 12 page
On the maximal operator of a general Ornstein-Uhlenbeck semigroup
If is a real, symmetric and positive definite matrix, and
a real matrix whose eigenvalues have negative real parts, we
consider the Ornstein--Uhlenbeck semigroup on with covariance
and drift matrix . Our main result says that the associated maximal
operator is of weak type with respect to the invariant measure. The
proof has a geometric gist and hinges on the "forbidden zones method"
previously introduced by the third author.Comment: 21 pages. Introduction revised. Some changes in Sections 3 and
Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex
We prove genuinely sharp two-sided global estimates for heat kernels on all
compact rank-one symmetric spaces. This generalizes the authors' recent result
obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar
heat kernel bounds are shown in the context of classical Jacobi expansions, on
a ball and on a simplex. These results are more precise than the qualitatively
sharp Gaussian estimates proved recently by several authors.Comment: 16 page
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