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H¹ and BMO for certain locally doubling metric measure spaces of finite measure

Andrea CarbonaroGiancarlo MauceriStefano Meda — 2010

Colloquium Mathematicae

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...

Riesz potentials and amalgams

Michael CowlingStefano MedaRoberta Pasquale — 1999

Annales de l'institut Fourier

Let ( M , d ) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q ( M ) is a space which looks locally like L p ( M ) but globally like L q ( M ) . We consider the case where the measure μ ( B ( x , ρ ) of the ball B ( x , ρ ) with centre x and radius ρ behaves like a polynomial in ρ , and consider the mapping properties between amalgams of kernel operators where the kernel ker K ( x , y ) behaves like d ( x , y ) - a when d ( x , y ) 1 and like d ( x , y ) - b when d ( x , y ) 1 . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

Sharp estimates for the Ornstein-Uhlenbeck operator

Giancarlo MauceriStefano MedaPeter Sjögren — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure γ on d . We prove a sharp estimate of the operator norm of the imaginary powers of on L p ( γ ) , 1 < p < . Then we use this estimate to prove that if b is in [ 0 , ) and M is a bounded holomorphic function in the sector { z : m o d arg ( z - b ) < arcsin | 2 / p - 1 | } and satisfies a Hörmander-like condition of (nonintegral) order greater than one on the boundary, then the operator M ( ) is bounded on L p ( γ ) . This improves earlier results of the authors with J. García-Cuerva...

L p - L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III

Michael CowlingSaverio GiuliniStefano Meda — 2001

Annales de l’institut Fourier

Let X be a symmetric space of the noncompact type, with Laplace–Beltrami operator - , and let [ b , ) be the L 2 ( X ) -spectrum of . For τ in such that Re τ 0 , let 𝒫 τ be the operator on L 2 ( X ) defined formally as exp ( - τ ( - b ) 1 / 2 ) . In this paper, we obtain L p - L q operator norm estimates for 𝒫 τ for all τ , and show that these are optimal when τ is small and when | arg τ | is bounded below π / 2 .

A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree

Michael G. CowlingStefano MedaAlberto G. Setti — 2010

Colloquium Mathematicae

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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