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Sharp endpoint results for imaginary powers and Riesz transforms on certain noncompact manifolds

100%
Mauceri G., Meda S., Vallarino M.
Studia Mathematica
|
2014
|
tom 224
|
nr 2
153-168
EN
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 X¹(M), introduced in previous work of the authors, to L¹(M).
2
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H¹ and BMO for certain locally doubling metric measure spaces of finite measure

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Carbonaro A., Mauceri G., Meda S.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 1
13-41
EN
In a previous paper the authors developed an H¹-BMO theory for unbounded metric measure spaces (M,ρ,μ) 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 $(ℝ^{d},ρ_{φ}, μ_{φ})$, where $dμ_{φ} = e^{-φ} dx$ and $ρ_{φ}$ is the Riemannian metric corresponding to the length element $ds² = (1+|∇φ|)² (dx₁² + ⋯ + dx²_{d})$. This generalizes previous work of the last two authors for the Gauss space.
3
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Invariant operators on function spaces on homogeneous trees

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Cowling M., Meda S., Setti A. G.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 1
53-61
4
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A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree

100%
Cowling M., Meda S., Setti A.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 1
223-232
EN
We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.
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