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A remark on the asymmetry of convolution operators

Saverio Giulini — 1989

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

A convolution operator, bounded on L q ⁢ ( ℝ n ) , is bounded on L p ⁢ ( ℝ n ) , with the same operator norm, if p and q are conjugate exponents. It is well known that this fact is false if we replace ℝ n with a general non-commutative locally compact group G . In this paper we give a simple construction of a convolution operator on a suitable compact group G , wich is bounded on L q ⁢ ( G ) for every q ∈ [ 2 , ∞ ) and is unbounded on L p ⁢ ( G ) if p ∈ [ 1 , 2 ) .

A remark on the asymmetry of convolution operators

Saverio Giulini — 1989

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

A convolution operator, bounded on L q ⁢ ( ℝ n ) , is bounded on L p ⁢ ( ℝ n ) , with the same operator norm, if p and q are conjugate exponents. It is well known that this fact is false if we replace ℝ n with a general non-commutative locally compact group G . In this paper we give a simple construction of a convolution operator on a suitable compact group G , wich is bounded on L q ⁢ ( G ) for every q ∈ [ 2 , ∞ ) and is unbounded on L p ⁢ ( G ) if p ∈ [ 1 , 2 ) .

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

Michael Cowling;  Saverio Giulini;  Stefano Meda — 2001

Annales de l’institut Fourier

Let X be a symmetric space of the noncompact type, with Laplace–Beltrami operator - ℒ , and let [ b , ∞ ) be the L 2 ( X ) -spectrum of ℒ . For τ in ℂ such that Re τ ≥ 0 , let 𝒫 τ be the operator on L 2 ( X ) defined formally as exp ( - τ ( ℒ - 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

Michael Cowling;  Saverio Giulini;  Andrzej Hulanicki;  Giancarlo Mauceri — 1994

Studia Mathematica

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.

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