## Abstract

Given 1 ≤ p <∞, a compact abelian group G and a function g ∈ L1 (G), we identify the maximal (i.e. optimal) domain of the convolution operator C

_{g}^{ (p)}: f ↦ f * g (as an operator from L^{p}(G) to itself). This is the largest Danach function space (with order continuous norm) into which L^{p}(G) is embedded and to which Cg (p) has a continuous extension, still with values in L^{p}(G). Of course, the optimal domain depends on p and g. Whereas C_{g}^{(p)}is compact, this is not always so for the extension of C_{g}^{(p)}to its optimal domain. Several characterizations of precisely when this is the case are presented.Original language | English |
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Pages (from-to) | 525-546 |

Number of pages | 22 |

Journal | Integral Equations and Operator Theory |

Volume | 48 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2004 |

## Keywords

- Convolution operator
- Optimal domain
- Vector measures in Lp(G)

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