Higher order Riesz transforms on noncompact symmetric spaces
In this note we prove various sharp boundedness results on suitable Hardy type spaces for Riesz transforms of arbitrary order on noncompact symmetric spaces of arbitrary rank.
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arXiv subjects
G. Mauceri
Publications and source records attributed to G. Mauceri.
Riesz Transforms and Spectral Multipliers of the Hodge-Laguerre Operator
On $\mathbb{R}^d_+$, endowed with the Laguerre probability measure $μ_α$, we define a Hodge-Laguerre operator $\mathbb{L}_α=δδ^*+δ^* δ$ acting on differential forms. Here $δ$ is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives $\partial_{x_i}$ are replaced by the "Laguerre derivatives" $\sqrt{x_i}\partial_{x_i}$, and $δ^*$ is the adjoint of $δ$ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure $μ_α$. We prove dimension-free bounds on $L^p$, $1<p<\infty$, for the Riesz transforms $δ\mathbb{L}_α^{-1/2}$ and $δ^* \mathbb{L}_ α^{-1/2}$. As applications we prove the strong Hodge-de Rahm-Kodaira decomposition for forms in $L^p$ and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in $L^p$. We also prove that for suitable functions $m$ the operator $m(\mathbb{L}^α)$ is bounded on $L^p$, $1<p<\infty$.
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Sharp endpoint results for imaginary powers and Riesz transforms on certain noncompact manifolds
In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the imaginary powers of the Laplacian and the Riesz transform are bounded from the Hardy space X^1(M), introduced in previous work of the authors, to L^1(M).
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Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds
In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Y^h(M) of the Hardy-type space X^h(M), introduced in a previous paper of the authors, as the class of all locally square integrable functions satisfying suitable BMO-like conditions, where the role of the constants is played by the space of global k-quasi-harmonic functions. Furthermore we prove that Y^h(M) is also the dual of the space X^k_fin(M) of finite linear combination of X^k-atoms. As a consequence, if Z is a Banach space and T is a Z-valued linear operator defined on X^k_fin(M), then T extends to a bounded operator from X^k(M) to Z if and only if it is uniformly bounded on X^k-atoms. To obtain these results we prove the global solvability of the generalized Poisson equation L^ku=f with f in L^2_loc(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.
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A maximal function characterisation of the Hardy space for the Gauss measure
In dimension one we give a maximal function characterisation of the Hardy space H^1(g) for the Gauss measure g, introduced by G. Mauceri and S. Meda. In arbitrary dimension, we give a description of the nonnegative functions in H^1(g) and use it to prove that L^p(g) is a contained in H^1(g) for 1<p\le\infty.
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Equivalence of norms on finite linear combinations of atoms
Let M be a space of homogeneous type and denote by F^\infty_{cont}(M) the space of finite linear combinations of continuous (1,\infty)-atoms. In this note we give a simple function theoretic proof of the equivalence on F^\infty_{cont}(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 A. Carbonaro and the authors.
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Hardy type spaces on certain noncompact manifolds and applications
In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces on M, the sequence Y^1(M/, Y^2(M), ... of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace--Beltrami operator L on M. Under the additional condition that the volume of the geodesic balls of radius r is controlled by C r^a e^{2\sqrt{b} r} for some real number a and for all large r, we prove also an endpoint result for first order Riesz transforms D L^{-1/2}. In particular, these results apply to Riemannian symmetric spaces of the noncompact type.
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Atomic decomposition of Hardy type spaces on certain noncompact manifolds
In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X^k(M), introduced in a previous paper of the authors, have an atomic characterization. As an application, we prove that the Riesz transforms of even order 2k are bounded from X^k(M) to L^1(M)and on L^p(M) for 1<p<\infty.
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Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator
In the setting of Euclidean space with the Gaussian measure g, we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein-Uhlenbeck semigroup. These operators are known to be bounded on L^p(g), for 1<p<\infty. We determine which of them are bounded from H^1(g) to L^1(g) and from L^\infty(g) to BMO(g). Here H^1(g) and BMO(g) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.
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Comparison of spaces of Hardy type for the Ornstein-Uhlenbeck operator
Denote by g the Gauss measure on R^n and by L the Ornstein-Uhlenbeck operator. In this paper we introduce a local Hardy space h^1(g) of Goldberg type and we compare it with the Hardy space H^1(g) introduced in a previous paper by Mauceri and Meda. We show that for each each positive r the imaginary powers of the operator rI+L are unbounded from h^1(g) to L^1(g). This result is in sharp contrast both with the fact that the imaginary powers are bounded from $H^1(g}$ to L^1(g), and with the fact that for the Euclidean laplacian Δand the Lebesgue measure λ) the imaginary powers of rI-Δare bounded from the Goldberg space h^1(λ) to L^1(λ). We consider also the case of Riemannian manifolds M with Riemannian measure m. We prove that, under certain geometric assumptions on M, an operator T, bounded on L^2(m), and with a kernel satisfying certain analytic assumptions, is bounded from H^1(m) to L^1(m) if and only if it is bounded from h^1(m) to L^1(m). Here H^1(m) denotes the Hardy space on locally doubling metric measure spaces introduced by the authors in arXiv:0808.0146, and h^1(m) is a Goldberg type Hardy space on M, equivalent to a space recently introduced by M. Taylor. The case of translation invariant operators on homogeneous trees is also considered.
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The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup
Let (H_t) 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_\infty the invariant measure for (H_t). Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L^2(γ_\infty). We prove that if the matrix R generates a one-parameter group of periodic rotations then the maximal operator associated to the semigroup is of weak type 1 with respect to the invariant measure. We also prove that the maximal operator associated to an arbitrary normal Ornstein-Uhlenbeck semigroup is bounded on L^p(γ_\infty) if and only if 1<p\le \infty.
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H^1 and BMO for certain locally doubling metric measure spaces of finite measure
In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,ρ,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and "isoperimetric" property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R^n,ρ_ϕ, m_ϕ), where dm_ϕ=e^{-ϕ} dx and ρ_ϕ$ is the Riemannian metric corresponding to the length element ds^2=(1+ |gradϕ|)^2(dx_1^2+...+dx_n^2)$. This generalizes previous work of the last two authors for the Gauss space.
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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.
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