๐”– Bobbio Scriptorium
โœฆ   LIBER   โœฆ

Asymmetry of Convolution Norms on Lie Groups

โœ Scribed by A.H. Dooley; Sanjiv Kumar Gupta; Fulvio Ricci

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
151 KB
Volume
174
Category
Article
ISSN
0022-1236

No coin nor oath required. For personal study only.

โœฆ Synopsis

We prove that on most connected non-commutative Lie groups there exists a convolution operator which is bounded on L p but unbounded on L q for every q not belonging to the interval with endpoints 2 and p. Furthermore, the kernel of such an operator can be supported on an arbitrary neighbourhood of the identity.

2000 Academic Press

1. Introduction

Convolution operators on locally compact abelian groups enjoy the following symmetry property: if such an operator is bounded on L p , for some p # (1, ), then it is also bounded on the dual space L p$ with the same norm. This fact is based on the remark that a convolution operator T is equal to its own transpose under the duality ( f, g) = | G f (x) g(&x) dx.

๐Ÿ“œ SIMILAR VOLUMES

Analyticity and Injectivity of Convoluti
Analyticity and Injectivity of Convolution Semigroups on Lie Groups
โœ Hiroshi Kunita ๐Ÿ“‚ Article ๐Ÿ“… 1999 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 165 KB

We prove that all continuous convolution semigroups of probability distributions on an arbitrary Lie group are injective. Let [+ t , t>0] be a continuous convolution semigroup of probability distributions on a Lie group G. For each t>0, we set T t f (x)= G f (xy) + t (dy) for a bounded continuous fu

Hardyโ€“Littlewoodโ€“Sobolev Theory on Lie G
Hardyโ€“Littlewoodโ€“Sobolev Theory on Lie Groups with Lorentz Norms
โœ E. Tychopoulos ๐Ÿ“‚ Article ๐Ÿ“… 1997 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 124 KB

In this article we give a Lorentz space version of the HardyแŽLittlewoodแŽSobolev theory for operator semigroups defined on Lie groups.