Approximate identities on some homogeneous Banach spaces

摘要

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L p functions where translations are isometries, namely Marcinkiewicz spaces and Stepanoff spaces , 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M p ) or even unbounded (). We construct a function f that belongs to these spaces and has the property that all approximate identities converge to f pointwise but they never converge in norm. Keywords Approximate identities - Locally L p functions - Bounded p-means - Marcinkiewicz spaces - Stepanoff spaces - Wiener amalgams - Herz spaces - Continuity of translation - Semi-normed function spaces - Convolution operators Mathematics Subject Classification (2000) Primary 41A65 - Secondary 42A85 - 46E30 Communicated by K. H. Gröchenig.