Optimal domains and integral representations of Lp(G)-valued convolution operators via measures
✍ Scribed by S. Okada; W. J. Ricker
- Publisher
- John Wiley and Sons
- Year
- 2007
- Tongue
- English
- Weight
- 213 KB
- Volume
- 280
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
Given 1 ≤ p < ∞, a compact abelian group G and a measure μ ∈ M (G), we investigate the optimal domain of the convolution operator $ C^{(p)}_{\mu} $: f ↦ f ∗︁ μ (as an operator from L^p^(G) to itself). This is the largest Köthe function space with order continuous norm into which L^p^(G) is embedded and to which $ C^{(p)}_{\mu} $ has a continuous extension, still with values in L^p^(G). Of course, the optimal domain depends on p and μ. Whereas $ C^{(p)}_{\mu} $ is compact precisely when μ ∈ M~0~(G), this is not always so for the extension of $ C^{(p)}_{\mu} $ to its optimal domain (which is always genuinely larger than L^p^(G) whenever μ ∈ M~0~(G)). Several characterizations of precisely when the extension is compact are presented. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)