TY - JOUR
T1 - Optimal domains and integral representations of Lp(G)-valued convolution operators via measures
AU - Okada, S.
AU - Ricker, W. J.
PY - 2007
Y1 - 2007
N2 - Given 1 ≤ p < ∞, a compact abelian group G and a measure μ ∈ M (G), we investigate the optimal domain of the convolution operator Cμ(p) : f → f * μ (as an operator from Lp (G) to itself). This is the largest Köthe function space with order continuous norm into which Lp (G) is embedded and to which Cμ(p) has a continuous extension, still with values in Lp(G). Of course, the optimal domain depends on p and μ. Whereas Cμ(p) is compact precisely when μ ∈ M 0(G), this is not always so for the extension of C μ(p) to its optimal domain (which is always genuinely larger than Lp(G) whenever μ ∈ M0 (G)). Several characterizations of precisely when the extension is compact are presented.
AB - Given 1 ≤ p < ∞, a compact abelian group G and a measure μ ∈ M (G), we investigate the optimal domain of the convolution operator Cμ(p) : f → f * μ (as an operator from Lp (G) to itself). This is the largest Köthe function space with order continuous norm into which Lp (G) is embedded and to which Cμ(p) has a continuous extension, still with values in Lp(G). Of course, the optimal domain depends on p and μ. Whereas Cμ(p) is compact precisely when μ ∈ M 0(G), this is not always so for the extension of C μ(p) to its optimal domain (which is always genuinely larger than Lp(G) whenever μ ∈ M0 (G)). Several characterizations of precisely when the extension is compact are presented.
KW - Convolution operator
KW - Optimal domain
KW - Vector measure in L(G)
UR - http://www.scopus.com/inward/record.url?scp=33947160168&partnerID=8YFLogxK
U2 - 10.1002/mana.200410491
DO - 10.1002/mana.200410491
M3 - Article
SN - 0025-584X
VL - 280
SP - 423
EP - 436
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
IS - 4
ER -