For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [fgl08a, fgl08]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products are restricted to those given by multiplication with left uniformly continuous functions. This algebra, , is canonically isomorphic to a twisted L¹-algebra. For amenable G that is rigidly symmetric as a discrete group we show the following result: An element of is invertible in if and only if it is invertible as a bounded operator on L²(G). This report is about work in progress. Complete details and further results will be given in a paper still in preparation.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-6,
author = {Gero Fendler and Karlheinz Gr\"ochenig and Michael Leinert},
title = {Convolution-dominated integral operators},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {121-127},
zbl = {1220.47035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-6}
}
Gero Fendler; Karlheinz Gröchenig; Michael Leinert. Convolution-dominated integral operators. Banach Center Publications, Tome 89 (2010) pp. 121-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-6/