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

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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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Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth

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Cowling M., Giulini S., Hulanicki A., Mauceri G.
Studia Mathematica
|
1994
|
tom 111
|
nr 2
103-121
EN
We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^1(AN),$ but also for $L^p(AN)$, where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.
3
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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.
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