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1.
Denote by γ the Gauss measure on ℝ
n
and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space
\mathfrakh1g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator (rI+L)iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from
\mathfrakh1g{{\mathfrak{h}}^1}{{\rm \gamma}} to L
1γ. This result is in sharp contrast both with the fact that (rI+L)iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is bounded from H
1γ to L
1γ, where H
1γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007), and with the fact that in the Euclidean case (rI-D)iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space
\mathfrakh1\mathbbRn{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to L
1ℝ
n
. We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T{\mathcal{T}}, bounded on L
2
μ, and with a kernel satisfying certain analytic assumptions, is bounded from H
1
μ to L
1
μ if and only if it is bounded from
\mathfrakh1m{{\mathfrak{h}}^1}{\mu} to L
1
μ. Here H
1
μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and
\mathfrakh1m{{\mathfrak{h}}^1}{\mu} is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009). The case of translation invariant operators on homogeneous trees is also considered. 相似文献
2.
Let (ℋ
t
)
t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ
d
with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ
∞ the invariant measure for (ℋ
t
)
t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L
2(γ
∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ
t
)
t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect
to the invariant measure γ
∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on
L
p
(γ
∞) if 1<p<∞.
The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project
“HARP”. 相似文献
3.
S. Mauceri 《代数通讯》2013,41(11):5363-5367
Let R be a prime algebra over an infinite field and suppose that R is generated by its group of units U(R). We prove that if U(R) satisfies a group identity then R must be an integral domain. 相似文献
4.
Summary Let G/K be a rank one or complex non compact symmetric space of dimension l. We prove that if f Lp, 1p2, the Riesz means of order z of f with respect to the eigenfunction expansion of the Laplacian converge to falmost everywhere for Re(z)>(l, p). The critical index (l, p) is the same as in the classical result of Stein in the Euclidean case.The first author was supported by funds of the Ministero della Pubblica Istruzione. The work was done while the second author was a member of the Mathematical Sciences Research Institute at Berkeley, with a fellowship from the Consiglio Nazionale delle Ricerehe. 相似文献
5.
García-cuerva José Mauceri Giancarlo Sjögren Peter Torrea José-luis 《Potential Analysis》1999,10(4):379-407
We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails. 相似文献
6.
José García-Cuerva Giancarlo Mauceri Peter Sjögren José Luis Torrea 《Journal d'Analyse Mathématique》1999,78(1):281-305
The setting of this paper is Euclidean space with the Gaussian measure. We letL be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The main result is a multiplier
theorem, saying that a function ofL which is of Laplace transform type defines an operator of weak type (1,1) for the Gaussian measure. The (distribution) kernel
of this operator is determined, in terms of an integral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies
in particular to the imaginary powers ofL. It is also verified that the weak type constant of these powers increases exponentially with the absolute value of the exponent.
The four authors have received support from the European Commission via the TMR network “Harmonic Analysis”. The first and
last authors were also partially supported by the Spanish DGICYT, under grant PB97-0030. 相似文献
7.
We study a class of kernels associated to functions of a distinguished Laplacian on the solvable group AN occurring in the Iwasawa decomposition G = ANK of a noncompact semisimple Lie group G. We determine the maximal ideal space of a commutative subalgebra of L1, which contains the algebra generated by the heat kernel, and we prove that the spectrum of the Laplacian is the same on all Lp spaces, 1 ≤ p < ∞. When G is complex, we derive a formula that enables us to compute the Lp norm of these kernels in terms of a weighted Lp norm of the corresponding kernels for the Euclidean Laplacian on the tangent space. We also prove that, when G is either rank one or complex, certain Hardy-Littlewood maximal operators, which are naturally associated with these kernels, are weak type (1, 1). 相似文献
8.
Let (M, ??, ??) be a space of homogeneous type and denote by ${F^{C}_{c} (M)}$ the space of finite linear combinations of continuous (1, ??)-atoms with compact support. In this note we give a simple function theoretic proof of the equivalence on ${F^{C}_{c} (M)}$ of the H 1-norm and the norm defined in terms of finite linear combinations of atoms. The result holds also for the class of nondoubling metric measure spaces considered in previous works of the authors and Carbonaro. 相似文献
9.
Michael Cowling Shaun Disney Giancarlo Mauceri Detlef Müller 《Inventiones Mathematicae》1990,101(1):237-260
Research supported by the Australian Research Council and the Italian Ministero della Pubblica Istruzione 相似文献
10.
Let (ℋ
t
)
t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ
d
with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ
∞ the invariant measure for (ℋ
t
)
t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L
2(γ
∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ*
f(x)=sup
t≥o
|ℋ
t
f(x)| is of weak type 1 with respect to the invariant measure γ
∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L
p
(γ
∞) if and only if 1<p≤∞.
相似文献