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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...

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 .

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

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 ( A N ) , but also for L p ( A N ) , where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.

Contact and conformal maps on Iwasawa N groups

Michael CowlingFilippo De MariAdam KorányiHans Martin Reimann — 2002

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The action of the conformal group O 1 , n + 1 on R n may be characterized in differential geometric terms, even locally: a theorem of Liouville states that a C 4 map between domains U and V in R n whose differential is a (variable) multiple of a (variable) isometry at each point of U is the restriction to U of a transformation x g x , for some g in O 1 , n + 1 . In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group G on the space G / P , where P is a parabolic subgroup. We solve...

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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